The Legacy of Peter Wynn

Abstract

After the death of Peter Wynn in December 2017, manuscript documents he left came to our knowledge. They concern continued fractions, rational (Padé) approximation, Thiele interpolation, orthogonal polynomials, moment problems, series, and abstract algebra. The purpose of this paper is to analyze them and to make them available to the mathematical community. Some of them are in quite good shape, almost finished, and ready to be published by anyone willing to check and complete them. Others are rough notes, and need to be reworked. Anyway, we think that these works are valuable additions to the literature on these topics and that they cannot be left unknown since they contain ideas that were never exploited. They can lead to new research and results. Two unpublished papers are also mentioned here for the first time.

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mathematics
Article
The Legacy of Peter Wynn
Claude Brezinski 1, F. Alexander Norman 2and Michela Redivo-Zaglia 3,*


Citation: Brezinski, C.; Norman, F.A.;
Redivo-Zaglia, M. The Legacy of
Peter Wynn. Mathematics 2021,9, 1240.
https://doi.org/10.3390/
math9111240
Academic Editor: Francesco Aldo
Costabile
Received: 30 March 2021
Accepted: 27 April 2021
Published: 28 May 2021
Publisher’s Note: MDPI stays neutral
with regard to jurisdictional claims in
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iations.
Copyright: © 2021 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
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Attribution (CC BY) license (https://
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4.0/).
1Laboratoire Paul Painlevé, Université de Lille, CNRS, UMR 8524, F-59000 Lille, France;
claude.brezinski@univ-lille.fr
2Department of Mathematics, University of Texas at San Antonio, One UTSA Circle,
San Antonio, TX 78249, USA; sandy.norman@utsa.edu
3Department of Mathematics “Tullio Levi-Civita”, University of Padua, Via Trieste 63, 35121 Padua, Italy
*Correspondence: michela.redivozaglia@unipd.it
Abstract:
After the death of Peter Wynn in December 2017, manuscript documents he left came to our
knowledge. They concern continued fractions, rational (Padé) approximation, Thiele interpolation,
orthogonal polynomials, moment problems, series, and abstract algebra. The purpose of this paper
is to analyze them and to make them available to the mathematical community. Some of them are
in quite good shape, almost finished, and ready to be published by anyone willing to check and
complete them. Others are rough notes, and need to be reworked. Anyway, we think that these
works are valuable additions to the literature on these topics and that they cannot be left unknown
since they contain ideas that were never exploited. They can lead to new research and results. Two
unpublished papers are also mentioned here for the first time.
Keywords:
orthogonal polynomials; extrapolation methods; Padé approximation; continued frac-
tions; rational interpolation; complex analysis; software; abstract algebra
Contents
Introduction ................................................ 2
The Discovery ............................................... 4
Mathematical Background ....................................... 7
The Shanks Transformation and the ε-Algorithms ........................ 7
Padé Approximation ......................................... 9
Continued Fractions ......................................... 9
Rational Interpolation ........................................ 11
The Legacy ................................................. 11
Main Documents ........................................... 11
Complex Analysis and Continued Fractions ........................ 11
Interpolation .......................................... 17
The ε-Algorithm ........................................ 25
Project for a Book ....................................... 32
Algebra ............................................. 33
Software ............................................. 35
Unpublished Typewritten Documents ............................... 36
Other Documents ........................................... 36
Drafts on Analysis ....................................... 36
Drafts on Algebra ....................................... 37
Personal Documents ...................................... 38
References ................................................. 40
References of Peter Wynn ...................................... 40
Mathematics 2021,9, 1240. https://doi.org/10.3390/math9111240 https://www.mdpi.com/journal/mathematics

Mathematics 2021,9, 1240 2 of 45
Translations by Peter Wynn ..................................... 44
General Bibliography ......................................... 44
1. Introduction
Peter Wynn (1931–2017) was a mathematician, a numerical analyst, and a computer sci-
entist (see in Figure 1a photo of him taken in 1975). In his scientific life he produced 109 pub-
lications (see References [
1
–
109
]), and he translated two books from Russian
[110,111]
. He
is mostly known for his discovery of the
ε
-algorithm [
3
], a recursive method for the imple-
mentation of the Shanks transformation for scalar sequences [
112
], for its extensions to the
vector, matrix, and confluent cases [
12
,
13
,
23
,
24
], and for his numerous reports and papers
on Padé approximants and continued fractions. His works influenced a generation of pure
and numerical analysts, with an important impact on the creation of new methods for the
acceleration of scalar, vector, matrix, and tensor sequences, on the approximation of func-
tions, and on iterative procedures for the solution of fixed point problems. Volume 80, No.
1 of the journal Numerical Algorithms was dedicated to him with his full biography. More
recently, a complete analysis of all his works was provided in Reference [
113
], together
with those of other scientists who worked and are still working on these domains.
Figure 1. Peter Wynn in 1975. © C. Brezinski.
Thus, one can wonder why it was necessary to publish an additional paper on Wynn’s
work. During the last years of his life, Peter Wynn was living in Zacatecas, Mexico. Each
year, he had to come back to the United States for some administrative reasons. On
one occasion, he was visiting friends in San Antonio and left them boxes containing
mathematical documents he did not want, for some unknown reason, to keep with him in
Mexico. In January 2020, C.B. was contacted by F.A.N., a colleague of these friends, who
informed him of the existence of these documents. This is how Wynn’s legacy came to
light. Then, the authors of this paper decided to analyze these unpublished works.
As everyone can understand, it was a quite difficult task. Only a part of the documents
have been extracted from the boxes and studied. The handwritten lists made by Wynn for
indicating the contents of the boxes show that he put together several kinds of documents.
What he named “rough notes” are very difficult to read and understand. In these lists,
he often indicated what he called “notes” and, in this case, they are usually well written
and understandable. The best documents are designated as “paper”. Moreover, Wynn
sometimes wrote new notes on the back of another document! In addition, he also made

Mathematics 2021,9, 1240 3 of 45
Xerox copies of documents, and inserted them into the boxes. However, his lists help us to
try to identify the kind of document we were considering. But, sometimes, this was quite
impossible. Moreover, most of the documents we found have no date and, often, pages
are not numbered. When unnumbered sheets of paper are stacked on top of each other,
without any separation, it has sometimes been difficult to know where a document begins
and where it ends. It is also possible that Wynn himself mixed up some texts. For these
reasons, certain groupings of pages may be questionable. Thus, we apologize in advance
for all possible mistakes contained in this paper.
Let us motivate the potential reader by giving an idea of the main themes covered by
Wynn. The most important documents left by Wynn, which are almost complete and in
a good shape are the following. One of them concerns Bürmann series over a field; they
generalize Taylor series and are used in the reversion of series. Another one (187 pages)
is on stability and
F
-functions that play a role in the solution of the differential equation
y0(t) = Ay(t)
, variation diminishing functions, interpolating rational functions, exponen-
tial fitting forms. The Hamburger-Pick-Nevanlinna problem is treated in a document of
179 pages. There is a document on continued fraction transformations of the Euler-Mclaurin
series that has 202 pages; it contains applications to various series. The convergence of
associated continued fractions, and truncation error bounds for Thiele’s continued fractions
are the topics of another document. Then, we analyze various documents on interpolation.
The first one is about functional interpolation, in which a recursive algorithm, which seems
to be new, is given for constructing interpolating rational functions. Interpolation by the use
of rational functions is studied in another document. Wynn gave two recursive algorithms
for their computation. They also seem to be new. A document extends a report of Wynn
on the abstract theory of the
ε
-algorithm [
69
]. The
ε
-algorithm is applied to sequences of
elements of a ring. An interesting pedagogical document is on iterated complex number
spirals. We present some numerical experiments illustrating Wynn’s ideas. There is a
document that looks like a book project on extrapolation, Padé approximation, continued
fractions, and orthogonal polynomials. It can certainly serve as a basis for lectures on
continued fractions, Padé approximation, and the
ε
-algorithm. Other documents are on
algebra. One of them (266 pages) is on
S
-rings. It is quite theoretical, without any applica-
tion nor reference to the literature. A second one treats factorisations of a triangular matrix.
There are also unfinished manuscripts on various topics, which can be of interest. Two
unpublished papers of Wynn are also mentioned here for the first time. The documents
left by Wynn show the intellectual process leading to the elaboration of new results until
their publication. They are also a testimony on the human side of research by describing
the friendship and the collaboration between researchers, and their mutual influence.
The purpose of this paper is to make this legacy available to the international math-
ematical community. It contains a description of the unpublished manuscripts of Wynn.
They offer many new results and developments. Despite the fact that not all documents
have been sorted yet, we decided to propose them immediately since we think that the
research must go on. We hope to encourage some readers to resume the work of Wynn and
bring it to an end. We are sure that several of his ideas are worth pursuing.
All the documents extracted up to now from the boxes left by Wynn have been digi-
tized, and they can be downloaded from the following URL (Legacy Archive: Peter Wynn).
The main material is mentioned in this paper. The unusable or incomplete documents we
found are not listed here, but they are also inserted in the same website together with a
small description. Other information on Peter Wynn can also be found in the site Mathemat-
ics Research of the Department of Mathematics at the University of Texas at San Antonio
(UTSA) https://mathresearch.utsa.edu/Legacy/Peter-Wynn/ (accessed on 30 April 2021).
The history of the discovery of these documents is told in Section 2with the testimony
of F.A.N. and of the friends of Wynn who inherited them. For readers who are not familiar
with the topics touched upon by Wynn, a short mathematical introduction is provided in
Section 3. More details can easily be found in the literature, particularly in Reference [
113
].

Mathematics 2021,9, 1240 4 of 45
Some of the documents left by Wynn in San Antonio are analyzed and commented in
Section 4. Some others will be analyzed in a second paper if they are of interest.
All quotations from Wynn are in italics. Inside a quotation, our own comments (when
necessary) are in roman characters into square brackets. When Wynn mentions a reference
given at the end of his text, we replaced it by that of our own bibliography numbered in
arabic figures between square brackets
[·]
. The documents of Wynn are numbered in bold
italics arabic figures. For referring to them in the text, these numbers are placed between
curly brackets
{·}
to distinguish them from the bibliographical references. Concerning the
references of Wynn, we were able to update them, by inserting new DOI, MR (Mathematical
Reviews) and Zbl (Zentralblatt reviews) numbers, and by adding newly discovered papers.
Let us also mention that in 1960, the journal Mathematical Tables and Other Aids to Computation
(known in short as MTAC) changed its name and became Mathematics of Computation, and
that from its Volume 5, the journal Revue française de traitement de l’information, Chiffres
became simply Chiffres. Several papers and communications of Wynn, when he was at
the Stichting Mathematisch Centrum (now Centrum Wiskunde & Informatica) in Amsterdam,
from 1960 to 1964, can also be found at the CWI’s Institutional Repository at the address
https://ir.cwi.nl/ (accessed on 30 April 2021).
2. The Discovery
On 14 January 2020, C.B. received the following message from F.A.N.:
I have just now read your recent remembrance of Peter Wynn appearing in Numerical Algorithms.
I was charmed—thank you for that. But I am also very much interested because, serendipitously,
I have just today “inherited” from a retired colleague several rather heavy boxes of Wynn’s papers
which had been left with my colleague for safekeeping (or convenience) some years earlier.
I have not yet opened the boxes, but I will begin examining the contents soon. The reason
I am writing you is to ask if you would be willing to answer a couple of questions, which I
pose below.
As I begin looking at the material, I could find (1) drafts of papers that might be of interest
to mathematicians, (2) personal items that might be of interest to family or friends, (3) miscellany
that would be of little interest to anyone, or (4) items for which I am at a loss to know whether to
keep or discard [...]
Here are my questions to you:
In case I find material that looks as if it might have some mathematical import, would you
be willing to take look at it, or could you suggest someone who would be?
The colleague he was speaking about is Manuel Philip Berriozábal. He was born in
1931. He was awarded the Bachelor of Science degree in mathematics from Rockhurst
College in 1952, a Master of Science degree in mathematics from the University of Notre
Dame in 1956, and a Ph.D. in mathematics from the University of California at Los Angeles
(UCLA) in 1961. After serving for one year as a lecturer at UCLA, he joined the faculty at
Tulane University as an Assistant Professor. Four years later, he moved to the University of
New Orleans as an Associate Professor. He was promoted to Professor six years later. In
1975, Manuel Berriozábal married Maria Antonietta Rodriguez (see Figure 2). He joined
the faculty at the University of Texas at San Antonio (UTSA) in 1976, and in 1979, he
started the now nationally recognized Prefreshman Engineering Program (PREP) at UTSA.
San Antonio PREP received a Presidential Award for Excellence in Science, Mathematics
and Engineering Mentoring and a La Promesa Program Award from the National Latino
Children’s Institute. Several years ago, TexPREP (Texas Prefreshman Engineering Program)
received a special commendation from the Texas Senate. These accomplishments caught
the attention of the Washington, DC-based Quality Education for Minorities Network and
resulted in Berriozábal being named one of the six Giants in Science at a conference held
in February 1998. In May 1998, he was a recipient of the San Antonio “I Have a Dream”
Foundation Endeavors Award. PREP has also been replicated on eight college campuses
in eight states outside of Texas. It was during his professorship at the Louisiana State
University in New Orleans that he became a close friend of Peter Wynn.

Mathematics 2021,9, 1240 5 of 45
Figure 2. Maria Antonietta and Manuel Philip Berriozábal. ©M.A. and M.P. Berriozábal.
Of course, C.B. and M.R.-Z. were very much interested in the contents of these boxes
since they were in the final preparation of their book which contains, among others, an
analysis of all the publications of Peter Wynn [
113
]. F.A.N. began to look at them, to scan a
lot of their material, and to send it to us. The correspondence with him was completely
taken up by M.R.-Z., since the same day C.B. received F.A.N.’s first message, he had to
enter a hospital for a health problem.
On 15 January, F.A.N. wrote to C.B. and M.R.-Z.:
I have opened two of the four boxes, looking thoroughly through the contents of one of them.
FIRST BOX
The first box was a smaller one containing several hundreds of pages of handwritten notes
and a few other random items. In this and the next message, I have attached several of the smaller
items that might be indicative of material to come. Nothing is dated so I have no idea whether
this material simply anticipates papers that have been completed and published or represents
new work. In addition, not much is paginated, so I generally presume that the order in which
pages appear is the order in which they were completed and assembled. Some of this material
was left with Dr. Berriozábal here in San Antonio after Peter had visited Mexico and there are
some suggestions (e.g., written accounts of receipts) that suggest that some of the material was
produced while he was living in Mexico. Whether this was after his last publication in 1981, I do
not know.
To Do List:
The first document above appears to be a list of various topics he wished to
address, mainly through the construction of relevant notes.
Projects:
This document lists a number of projects that he had planned and may well have
completed. Whether these “projects” have found their way into the literature as published papers,
I do not know. I’ve copied a snapshot from the document and you will note that some of the items
have been denoted with a “D__” . Perhaps this is Peter’s shorthand for “Done” or fait accompli. I
didn’t see in his publications any that specifically included, for example, “stratified commutative
ring” or “Bürmann series”, so perhaps this represents some work that hasn’t yet been published,
but will appear somewhere in his collection of notes. I did find the beginnings of what looks to be
a monograph on Interpolation Theory–the first in his list of projects and one that doesn’t seem to
have been completed at that time.
Duplicate List:
This may reflect a list of some of the items that Peter included in his boxes
of documents.
The next messages will include 2 larger documents, one a bibliography and the other a
more extensive list of documents.
Other items in this first box include the aforementioned piece of a monograph on Interpola-
tion Theory, as well as a couple of hundred pages of paginated notes of another document. In
that case, I see pages 70–300+ so I can’t be sure of the initial title. There is another paginated

Mathematics 2021,9, 1240 6 of 45
treatment of a topic (I don’t recall exactly what at the moment), a book review, and some hundreds
of other pages of notes that do not appear to fit together well.
SECOND BOX
I’ll mention here that the second box, one that originally held a dozen reams of office paper,
now holds about 20 kg of handwritten notes. I have not yet examined these notes in detail, and
when I do, I want to be careful not to disturb the assembly of any documents.
Bibliography:
This is likely not anything that you would use. I imagine Peter kept this
bibliography simply for reference purposes.
List of documents:
This seems to be a list of documents and/or things he felt he needed
to do, such as copy this or Xerox that. I don’t know if it has any intrinsic value, but it might
reflect documents included among his papers.
In a message to F.A.N., 17 January 2020, Maria Antonietta Berriozábal wrote:
We liked his visits. Each time he came it was without much notice and we would just go out to
eat. For me I was so very busy for so many years and so was Manny [her husband Manuel]
that a surprise visit took creativity to get the three of us together, but we did it for Peter. He liked
nice places. And I loved hearing he and Manny talk old times. He would also talk about living in
Mexico which he loved.
He was such a mystery to me because he was pretty much alone and he liked it that way.
I do not recall meeting Peter when we were in New Orleans which was August 1975 to
June 1976 [. . .]
What I do recall is that after we married I was sorting and clearing papers for Manny from
our apartment which he had lived in for many years and I came across a letter from Peter written
around the time Manny and I met. In that letter I gathered Peter was a bachelor friend of Manny
who was not the marrying kind and he was commenting that Manny had met “the one”. He was
happy for Manny but it seemed to me Peter was happy being alone. Now that I think of it I don’t
think I ever got to know who Peter was. He had a wonderful smile. He liked to laugh and joke
and was very very blond.
C.B. and M.R.-Z. wrote to Maria Berriozábal and sent her the photo of Wynn in-
serted above (also see References [113,114]). She answered on 21 January 2020:
[
. . .
]The photos are valuable. I met Peter when he was a much older man but still had that
incredibly beautiful smile and lots and lots of hair!
Each time he came to the US for many years, he would call Manny and we would go to
dinner. If I recall correctly, it was mostly on Peter’s nickel and as I stated to Sandy [F.A.N.] we
went to nice restaurants, although Manny may have won some times and we would go to our
famous Luby’s Cafeteria—always the frugal Manny.
I do not recall Peter in New Orleans when I lived there with Manny the first nine months
of our marriage. I left San Antonio to join him as he was waiting and hoping to join UTSA. The
New Orleans scholars community from the four universities and colleges there met socially and
regularly at people’s homes but I do not recall Peter in any of that circle. It could have been that
he was no longer in New Orleans. This was August 1975 to June 1976.
This morning when I read your note the words that rang in my head is how youth is wasted
on the young. I wish I had paid more attention and had retained the conversations with Peter,
but at that time my life was incredibly full and so was Manny’s. There were times when I had
to make major changes in my schedule to join Manny and Peter and I always did and looked
forward to those dinners. My questions to Peter were always about Mexico and his life there.
That is what I wish I could recall more. I just know that he was very happy there. I do not recall
what his conversations with Manny were all about and now Manny is forgetting a lot of things.
On one of these trips which had to occur every six months, I believe, because that is how he
kept his US citizenship active he said he had some boxes with him and asked if we could store
them for him for safekeeping. I recall his taking them out of his car. When I looked at them again
seriously in recent years I saw how big the boxes were and soon learned how heavy they were.
Yet, I seem to recall Peter carrying them by himself. Maybe a Dolly helped. I just know he trusted
us with them.
One of my last year’s resolutions was to clean the garage and I did. It took some weeks but
with some help I did it. Peter’s boxes were the only things that were still in there without any
resolution. We had been telling each other for a couple of years that Peter had probably died since

Mathematics 2021,9, 1240 7 of 45
we did not hear from him anymore. He quit coming. Manny and I would comment from time to
time that we still had those boxes and that we needed to do something about them.
A year ago when the boxes were the last unsettled things in the garage I told our gardener
who was helping me with the project the story of the boxes. He said what if it is money. Since
it had been so long that Peter had left the boxes with us I gave myself permission to open them.
Hilario, my helper, even wondered if they were full of money! I only opened two boxes and all I
saw were reams of papers with math problems. Pages and pages and pages. When Sandy [F.A.N.]
helped us with Manny’s papers after he retired I decided to ask him to go over the papers and
determine what they were. Manny looked at them but had no idea what the material was. When
Sandy [F.A.N.] took the boxes out I saw how at least one of them had folders of some kind so the
papers were not only sheets of math problems. I hope someone who has Peter’s interest and his
career uppermost in their mind will continue to review it and possibly record it for posterity.
To close, thank you again for honoring Peter’s work.
3. Mathematical Background
Some documents written by Wynn are pure algebra or complex analysis. They require,
at least for some of them, quite a good knowledge of these fields. However, it is impossible
to give herein a full account of the definitions and notions necessary for their complete
understanding. We will restrict ourselves to the most specialized topics addressed by
Wynn in the domain of numerical analysis, namely the Shanks transformation and the
ε
-algorithms, Padé approximation, continued fractions, and rational interpolation. In
this section, we only present the definitions and the main algebraic properties that are
sufficient for understanding most of the documents analyzed. The fundamental question
of convergence is not addressed. We refer to References [113,115–119] for more details.
3.1. The Shanks Transformation and the ε-Algorithms
When a sequence
(Sn)
of numbers is slowly converging to its limit
S
, and when
one has no access to the process building it, it can be transformed into a new sequence
(or a set of new sequences), which, under some assumptions, converge(s) faster to the
same limit. Many such sequence transformations exist and have been studied; see Refer-
ences
[115,117–119]
. Among them, one of the most well known, studied, and used is due
to Shanks [
112
]. It consists in transforming
(Sn)
into a set of sequences denoted
{ek(Sn)}
,
indexed by kand n, and defined by
ek(Sn) =

SnSn+1· · · Sn+k
∆Sn∆Sn+1· · · ∆Sn+k
.
.
..
.
..
.
.
∆Sn+k−1∆Sn+k· · · ∆Sn+2k−1


1 1 · · · 1
∆Sn∆Sn+1· · · ∆Sn+k
.
.
..
.
..
.
.
∆Sn+k−1∆Sn+k· · · ∆Sn+2k−1

,k,n=0, 1, . . . . (1)
These numbers can be recursively computed by the scalar
ε
-algorithm of Wynn [
3
], whose
rule is
ε(n)
k+1=ε(n+1)
k−1+ (ε(n+1)
k−ε(n)
k)−1,k,n=0, 1, . . . , (2)
with
∀n
,
ε(n)
−1=
0 and
ε(n)
0=Sn
, and it holds
ε(n)
2k=ek(Sn)
,
k
,
n=
0, 1,
. . .
The
ε(n)
2k+1
are only
intermediate results with no interest for our purpose. In fact, quantities with a different
parity can be eliminated from
(2)
, thus leading to the cross-rule of Wynn [
48
] which only
links those of the same parity
(ε(n)
k+2−ε(n+1)
k)−1+ (ε(n+2)
k−2−ε(n+1)
k)−1= (ε(n+2)
k−ε(n+1)
k)−1+ (ε(n)
k−ε(n+1)
k)−1,
with the initial conditions ε(n)
−2=∞,ε(n)
−1=0, and ε(n)
0=Snfor n=0, 1, . . .

Mathematics 2021,9, 1240 8 of 45
It can be proved that
∀n
,
ε(n)
2k=S
if and only if the sequence
(Sn)
satisfies a linear
difference equation of order k
a0(Sn−S) + a1(Sn+1−S) + · · · +ak(Sn+k−S) = 0, n=0, 1, . . . ,
where the coefficients
ai
are such that
a0ak6=
0 and
a0+· · · +ak6=
0. In other words, if
and only if Snhas the form
Sn=S+
p
∑
i=1
Ai(n)rn
i+
q
∑
i=p+1
[Bi(n)cos(bin) + Ci(n)sin(bin)]eωin+
m
∑
i=0
ciδin,
where
Ai
,
Bi
and
Ci
are polynomials in
n
such that, if
di
is the degree of
Ai
plus 1 for
i=
1,
. . .
,
p
, and the maximum of the degrees of
Bi
and
Ci
plus 1 for
i=p+
1,
. . .
,
q
,
one has
m+1+
p
∑
i=1
di+2
q
∑
i=p+1
di=k,
with the conventions that the second sum vanishes if there are no complex zeros, and
m=−
1 if there is no term in
δin
(Kronecker’s symbol). The set of such sequences is named
the kernel of the transformation. Since many sequences produced by iterative methods
have this form or are close to it, it explains the success of this algorithm.
In Reference [
24
], Wynn extended his algorithm to the case where the
Sn
are vectors
or square matrices. In the matrix case, the significance of the power
−
1 in
(2)
is clear. For
vectors, the inverse
y−1
of a vector
y
is defined as
y−1=y/(y
,
y)
, thus leading to the vector
ε-algorithm.
Similarly, when a function
f(t)
is slowly converging to its limit
S
when
t
tends to
infinity, it can be transformed into a set of functions converging faster to
S
under certain
assumptions. For that purpose, Wynn extended his algorithm to that case by proposing the
first confluent form of the ε-algorithm whose rule is, for all t,
εk+1(t) = εk−1(t) + (ε0
k(t))−1,k=0, 1, . . . ,
with
ε−1(t) =
0 and
ε0(t) = f(t)
. Again the
ε2k+1(t)
are intermediate computation. It can
be proved that, for all
t
,
ε2k(t) = S
if and only if
f
satisfies the differential equation of order
k
a0(f(t)−S) + a1f0(t) + · · · +akf(k)(t) = 0,
with a0ak6=0, that is, in other words,
f(t) = S+
p
∑
i=1
Ai(t)erit+
q
∑
i=p+1
[Bi(t)cos(bit) + Ci(t)sin(bit)]eωit,
where
Ai
,
Bi
and
Ci
are polynomials in
t
such that, if
di
is the degree of
Ai
plus 1 for
i=
1,
. . .
,
p
, and the maximum of the degrees of
Bi
and
Ci
plus 1 for
i=p+
1,
. . .
,
q
, one has
p
∑
i=1
di+2
q
∑
i=p+1
di=k.
Moreover, the
ε2k(t)
can be expressed by a ratio of determinants quite similar to
(1)
, but in
which the derivatives of
f
are replacing the powers of the difference operator
∆
(see, for
example, Reference [113] (p. 24) or Reference [115] (p. 257)).

Mathematics 2021,9, 1240 9 of 45
3.2. Padé Approximation
Let fbe a formal power series
f(t) =
∞
∑
i=0
citi,
in which the coefficients
ci
and the variable
t
can be complex. A Padé approximant of
f
is a
rational function with a numerator of degree
p
at most and a denominator of degree
q
at most
such that its power series expansion agrees with
f
as far as possible, that is up to the degree
p+qinclusively. Such an approximant is denoted [p/q]fand, by construction, it holds
[p/q]f(t)−f(t) = O(tp+q+1),(t→0).
Let us set [p/q]f(t) = Np(t)/Dq(t)with
Np(t) = a0+a1t+· · · +aptpand Dq(t) = b0+b1t+· · · +bqtq.
Then, linearizing the conditions of the definition as
f(t)Dq(t)−Np(t) = O(tp+q+1)
leads
to the relations
a0=c0b0
a1=c1b0+c0b1
.
.
.
ap=cpb0+cp−1b1+· · · +cp−qbq
0=cp+1b0+cpb1+· · · +cp−q+1bq
.
.
.
0=cp+qb0+cp+q−1b1+· · · +cpbq
with the convention that
ci=
0 for
i<
0 which allows to treat simultaneously the cases
p≤qand p≥q.
Setting
b0=
1 allows to solve the system formed by the preceding last
q
equations for
the coefficients
b1
,
. . .
,
bq
. Knowing the
bi
’s, the first
p+
1 equations directly provide the
ai
’s.
It holds
[p/q]f(t) = 
tqfp−q(t)tq−1fp−q+1(t)· · · fp(t)
cp−q+1cp−q+2· · · cp+1
.
.
..
.
..
.
.
cpcp+1· · · cp+q
.
tqtq−1· · · 1
cp−q+1cp−q+2· · · cp+1
.
.
..
.
..
.
.
cpcp+1· · · cp+q

,
where fmdenotes the partial sum of fup to the term of degree minclusively.
It is easy to see from
(1)
that, applying the
ε
-algorithm to
Sn=∑n
i=0citi
leads to
ε(n)
2k= [n+k/k]f(t)
. Let
g
be the reciprocal series of
f
defined by
f(t)g(t) =
1 (it exists if
and only if
c06=
0). If the
ε
-algorithm is applied to the sequence of the partial sums of
g
,
then ε(n)
2k= [n+k/k]g(t) = 1/[k/n+k]f(t).
3.3. Continued Fractions
Acontinued fraction is an expression of the form
C=b0+a1
b1+a2
b2+a3
b3+a4
...
.

Mathematics 2021,9, 1240 10 of 45
For evident typographical reasons, it is written as
C=b0+a1
b1+a2
b2+a3
b3+· · ·
The numbers
ak
and
bk
are called the
k
th partial numerator and partial denominator, respec-
tively, ak/bkis the kth partial quotient, and
Cn=b0+a1
b1+· · · +an−1
bn−1+an
bn
is called the
n
th convergent of the continued fraction
C
(even if the sequence
(Cn)
does not
converge). A continued fraction is said to converge if the sequence
(Cn)
converges when
n
tends to infinity.
After reducing to the same denominator, Cncan be written as Cn=An/Bn. It can be
computed by the recurrence relationships
Ak=bkAk−1+akAk−2
Bk=bkBk−1+akBk−2,k=1, 2, . . .
with
A0=b0,A−1=1
B0=1, B−1=0.
Let
(Cn)
be the sequence of convergents of the continued fraction
C
and let
(Cpn)
be a
subsequence. The continued fraction
C0=b0
0+a0
1
b0
1
+a0
2
b0
2
+a0
3
b0
3
+· · ·
whose convergents satisfy C0
n=Cpnis given by
a0
n=Cpn−1−Cpn
Cpn−1−Cpn−2
,b0
n=Cpn−Cpn−2
Cpn−1−Cpn−2
,
with
b0
0=Cp0
,
b0
1=
1 and
a0
1=Cp1−Cp0
. This operation is called a contraction of the
continued fraction. Usually, pn=2n.
The analytic theory of continued fractions is concerned with continued fractions whose
partial numerators and/or denominators are functions of the complex variable
z
. Let us
consider the continued fraction
C=b0+a1z
1+a2z
1+a3z
1+· · ·
From the recurrence relations, we see that
A2k−1
,
A2k
and
B2k
are polynomials of degree
k
in
z
and that
B2k−1
is a polynomial of degree
k−
1. The expansions of
Ck
and
Ck−1
in ascending powers of
z
agree up to the term of degree
k−
1 inclusively. It is possi-
ble to choose
b0
,
a1
,
a2
,
. . .
so that the expansion of
Ck
agrees with that of a given series
f(z) = c0+c1z+c2z2+· · ·
up to the term of degree
k
. This continued fraction is called
the continued fraction corresponding to the series
f
. By a contraction of this continued
fraction, with
pk=
2
k
as explained above, we obtain a continued fraction whose conver-
gent
C0
k
agrees with that of
f
up to the term of degree 2
k
. This is the continued fraction
associated to the series
f
. Thus, by the uniqueness property of Padé approximants, we have
C2k= [k/k]f(z)and C2k+1= [k+1/k]f(z).

Mathematics 2021,9, 1240 11 of 45
3.4. Rational Interpolation
Consider the continued fraction
C(n)(x) = α(n)
0+x−xn
α(n)
1
+x−xn+1
α(n)
2
+· · · ,
with
α(n)
k=$(n)
k−$(n)
k−2
for
k=
1, 2,
. . .
, and
α(n)
0=$(n)
0=f(xn)
, and where the scalars
$(n)
k
are recursively computed by
$(n)
k+1=$(n+1)
k−1+xn+k+1−xn
$(n+1)
k−$(n)
k
(3)
with
$(n)
−1=
0 and
$(n)
0=f(xn)
. The quantities
$(n)
2k
can be expressed as a ratio of two
determinants.
The
k
th convergent
C(n)
k
of this continued fraction satisfies the interpolation conditions
C(n)
k(xi) = f(xi)
for
i=n
,
. . .
,
n+k
. The quantities
$(n)
k+1
are the reciprocal differences of
f
.
They formed the $-algorithm used by Wynn for rational extrapolation at infinity [2].
Let us now consider the confluent reciprocal differences of a function fdefined by
$k+1(t) = $k−1(t) + k+1
$0
k(t)
with
$−1(t) =
0 and
$0(t) = f(t)
. This formula will be used as an extrapolation method
for functions by Wynn, and named the confluent form of the $-algorithm [12].
Thiele’s expansion of a function fconsists in the continued fraction
f(t+h) = f(t) + h
α1(t)+h
α2(t)+· · · ,
with αk(t) = $k(t)−$k−2(t)for k=1, 2, . . . Replacing tby 0 and hby x, we get
f(x) = f(0) + x
α1(0)+x
α2(0)+· · ·
The successive convergents
Ck(x) = Ak(x)/Bk(x)
of this continued fraction are such that
f(x)−Ck(x) = O(xk+1). Since Padé approximants are uniquely defined, it holds
C2k(x) = [k/k]f(x),C2k+1(x) = [k+1/k]f(x).
4. The Legacy
Let us now describe the various documents contained in the boxes left by Wynn at his
friends’ house in San Antonio. All, except two, are handwritten.
4.1. Main Documents
We tried to sort the documents by themes. However, our classification is only an
attempt since there are many connections between the topics, the documents are not dated,
and, maybe, some of them contain pages coming out from various sources since they are
not numbered and, maybe, inserted by Wynn in disorder into the boxes.
4.1.1. Complex Analysis and Continued Fractions
1 Bürmann series over a field
These titled notes (52 pages) are devoted to Bürmann series. They are mentioned in
documents
{26–28}
, and were probably written at the same time as {
14
}, since both
titles are listed together.

Mathematics 2021,9, 1240 12 of 45
The Bürmann series of a function
f
is a generalized form of a Taylor series in which,
instead of a series in powers of
z−z0
, we have a series in powers of the analytic
function
φ(z)−φ(z0)
. It is used in the reversion of series which consists, starting
from
z=f(w)
, in expressing
w
as
w=g(z)
. The problem was considered by
Joseph Louis Lagrange (1736–1813) in 1770 [
120
] and generalized by Hans Heinrich
Bürmann (?–1817). A report on Bürmann’s theorem by Joseph-Louis Lagrange and
Adrien-Marie Legendre (1752–1833) appeared in Rapport sur deux mémoires d’analyse
du professeur Bürmann, Mémoires de l’Institut National des Sciences et Arts: Sciences
Mathématiques et Physiques, vol. 2, pages 13–17 (1799). An exhaustive treatment
of this topics is given in Reference [
121
] (pp. 55 ff.). The Lagrange-Bürmann series,
as it is often called, also allows for two functions
f
and
g
, both holomorphic in the
neighborhood of a point, to be expanded in a power series of the other one in two
overlapping regions. Series reversion related to Hankel determinants, continued
fractions, and combinatorics as explained in Section 6.10.4 of Reference [
113
] about
combinatorics; also see Reference [122]. Wynn wrote:
This paper is directed towards the transformation of series expansions [...]
The general results derived are illustrated by application to a problem concerning the trans-
formation of asymptotic relationships.
Notation. Let n >0be fixed finite integer.
(1) With
w
a fixed point in
C
, the finite part of the complex plane,
Ω(w)
an open set of
points in
C\{w}
with limit point at
w
,
¯
Ω(w)≡ {w}SΩ(w)
, and
p
,
q
mappings of
¯
Ω(w)
into C,
p(z)'(w,Ω)q(z)
means that p(z)−q(z) = o{(z−w)n}as z tends through Ω(w)to w.
(2) Let
M
be a nonvoid set of points in
C
. For each
w∈M
, let
Ω(w)
,
¯
Ω(w)
be as in (1) and
p(w,·),q(w,·):¯
Ω(w)→C.
p(w,z)'[M,Ω]q(w,z)
means that for each
w∈M
,
p(w
,
z)'(w,Ω)q(z)
.[We have]
¯
Ω=Sw∈M¯
Ω(w)
and
Ω0=Sw∈MΩ0(w)where {Ω0(w):z−w for all z ∈¯
Ω(w)}.
The main problem considered has a simple form as follows: let
aj
,
fj∈C(j:n)
[This
notation means that
j
runs from 1 to
n
. The notation
(j:
0,
n)
means that
j
goes from
0 to n]with a16=0and, with w ∈Cfixed, let a,f:¯
Ω(w)→Cbe such that
a(z)'(w,Ω)a(w) +
n
∑
j=1
aj(z−w)j
f(z)'(w,Ω)f(w) +
n
∑
j=1
fj(z−w)j.
Determine gi∈Cfor which
f(z)'(w,Ω)f(w) +
n
∑
i=1
gi{a(z)−a(w)}j.
In the general form of the problem
aj
,
fj:M→C(j:n)
are mappings with
a1(w)6=
0for
all w ∈M; for each w ∈M, a(w,·):¯
Ω(w)→Cis such that
a(w,z)'[M,Ω]a(w,w) +
n
∑
j=1
aj(w)(z−w)j
the mapping f :¯
Ω→Csatisfies the relationship
f(z)'[M,Ω]f(w) +
n
∑
j=1
fj(w)(z−w)j

Mathematics 2021,9, 1240 13 of 45
and gi:M→Cfor which
f(z)'[M,Ω]f(w) +
n
∑
i=1
gi(w){a(w,z)−a(w,w)}i
are to be determined.
Then, Wynn described three variants of the problem corresponding to various properties
of the function a. The particular case of polynomials is treated. The text ends by:
The function
c(z) = ln(
1
+z)
is inverse to
a(z) = ez−
1.
c(eiθ) = ln{
2
(
1
+cos θ)}+
iθ/
2maps the segment
−π≤θ≤π
onto a curve
C
, symmetric about the real axis,
enclosing its nonpositive part, containing the real point
ln(z)
, the imaginary points
±iπ/
3,
and having as asymptotes the lines
z=±iπ/
2,
D
being the open domain bounded by
C
, a
maps Dbijectively onto the unit open disc.
2 On Stability Functions
The handwritten paper On stability functions (mentioned in {
26
} as “Paper, latest
version”) contains 187 pages with a bibliography at the end. Its first section is an
introduction and a presentation of the notations:
This paper is concerned with functions of the form
(1) f (z) = 1+az
1−az/2 +z2s(z)
where 0<a<∞, and
(2) s(z) = Z∞
0
dψ(t)
1+z2t
where
ψ(t)
is a nondecreasing function of bounded variation for 0
≤t≤∞
such that all
moments
(3) cν=Z∞
0tνdψ(t)
for ν=0, 1, . . . exist.
A function
f
of this form will be called an
F
-function. If the context permits, the notation
f∈F
or, where convenience dictates,
f(z)∈F
, will be used. The function
s
in the
representation (1) plays a significant role in the theory of the function
f
. A function of
the form (2) with
ψ
as described will be called an
S
-function; again the notations
s∈S
or
s(z)∈S will be used.
The mapping properties of F- and S-functions, in particular, will be investigated.
Then, Wynn stated that an
F
-function
f
is real for finite negative values of
z
with
f(z)>0
for all sufficiently small negative values of
z
, is asymptotically represented
as
z
tends to zero over an open set in the sector
π/
2, 3
π/
2 with the limit point 0,
and that the function
f
can be asymptotically represented by a series of the form
f(z) = ∑∞
ν=0cνzν
(formula scratched) which generates an associated continued frac-
tion whose even convergents map the closed infinite left half-plane
L
into the closed
unit disc
D
if and only if
f
is an
F
-function. The proof is given with a reference to his
paper (Reference [
98
]) for details. An example of an
F
-function is
ez
, and Wynn added:
The study of
F
-functions was motivated by the following consideration: With
A
a bounded
linear operator, the solution of the differential equation
(9) dy(t)
dt =Ay(t)
with y(0)prescribed, satisfies the relationship
(10) y(nh +h) = exp(Ah)y(nh)
for
n=
0, 1,
. . .
If 0
<h<∞
and the eigenvalues of
A
lie in
L
, those of
exp(Ah)
lie in
D
.
ky(nh)k
remains bounded and, indeed, decreases to zero as
n
increases indefinitely. An

Mathematics 2021,9, 1240 14 of 45
approximation
y∗(t)
to the solution of Equation (9) may be obtained by use of a Taylor series
method based upon use of an approximate identity
(11) m
∑
i=0
aidiy∗(t+h)
dti=
n
∑
i=0
bidiy∗(t)
dti.
Setting
m=n=r
and taking the
ai
and
bi
to be the denominator and numerator coefficients
of powers of
z
in
C2r(hz)
, where
C2r(z)
, with
r≥
1fixed, is a convergent of the continued
fraction associated with the exponential series, and setting
t=nh
, the special form of the
approximate identity (11) applied to Equation (10) yields the relationship
(12) y∗(nh +h) = C2r(Ah)y∗(nh)
for
n=
0, 1,
. . .
As a consequence of the mapping properties of
C2r(t)
described above, the
eigenvalues of
C2r(Ah)
lie in
D
, and the remarks concerning the behaviour of
ky(nh)k
apply
with equal force to
ky∗(nh)k
: the exact and approximate solutions of Equation (9) behave in
the same way. The practical details of the way in which relation (12) is implemented are not
of immediate concern; any method for the approximate solution of Equation (9) based upon
use of recursion (12) is stable.
Let us remember that the convergent
C2r
of the associated continued fraction to a
series is its
[r/r]
Padé approximant. Since the computation of Padé approximants to
the exponential function are highly numerically unstable (see References [
123
,
124
])
one can doubt the practical usefulness of the procedure mentioned.
Other examples of
F
-functions are given. Wynn claimed that they open up the possi-
bility of constructing stable schemes for the approximate solution of certain nonlinear
differential equations. A characterisation of
F
-functions which is independent of the
continued fraction theory is given. Based on the proof of this result, Wynn asserted that
it is possible to demonstrate the existence of functions with mapping properties less
specific than those of
F
-functions. Indeed, a number of
F
-functions can be derived from
a given
F
-function. This remark reminds the way some totally monotone sequences
can be derived from a totally monotone one as explained in Reference [
125
]. Then,
Wynn proved that
F
-functions are closed with respect to multiplication. Meromorphic
F-functions are then considered.
The next part of the work deals with variation diminishing functions. Let
(xi)
be a
sequence of real numbers. The transformation
(xi)7−→ (yj)
given by
yj=∑n
i=0an−ixi
,
j=
0, 1,
. . .
(it seems that
n
should be replaced by
j
), is said to be variation diminishing
if the number of changes of sign of the
yj
is less than or equal to the number of
changes of sign of the
xi
. Wynn wrote: Transformations of this type underly the theory
of many smoothing operations, and also occur in the numerical solution of certain partial
differential equations by iterative methods, and he referred to Reference [
126
]. Several
results are proved.
The next section of this manuscript is devoted to interpolation. Wynn wrote:
It is possible to construct a rational
F
-function whose derivative values agree with those of
a generating
F
-function up to prescribed orders not only at the origin but at a prescribed
sequence of points in
L
; furthermore this rational function may be derived by the use of
purely algebraic methods of rational function interpolation.
A theorem is proved and recurrences for the coefficients occurring in the interpolating
rational
F
-function are given. A long discussion, where orthogonal polynomials
play a role, follows. The algebraic problem of determining a rational function with
denominator and numerator of degrees equal to
m
and
m−
1 respectively, which
satisfies the interpolation conditions may be solved by a recursive process which
is described and justified in his Appendix 3 Extremal solutions of the Pick-Nevanlinna
problem. It could be of interest to code and test this algorithm, and to compare it with
the other existing ones.
Exponential fitting forms the topic of the following section. Since differential systems
of the form (9) constitute a very restricted class, Wynn now considered the system

Mathematics 2021,9, 1240 15 of 45
dy(x)/dx =f(y(x))
. Linearizing it yields
dy(x)/dx =f(
0
) + Jy(x)
where
J
is the
Jacobian matrix of f. Assuming its nonsingularity, we have
y(x+h) = eJh [y(x) + J−1f(0)] −J−1f(0).
Assuming that the eigenvalues
λj
of
J
are distinct, and
. . .
what follows is not clearly
stated and it seems that one page of the manuscript is missing. Anyway, Wynn
constructed a rational function such that its derivatives of prescribed orders agree
with those of eJh at the points λjh, and he wrote:
It is eminently desirable that a method for the construction of general rational functions
mapping
L
into
D
and having prescribed orders of contact with the exponential function
at the origin and other specified points in
L
should be made available. This is precisely the
service offered by Section [not identified] of this paper. In Theorem 4 [not identified],
the
F
-function producing the interpolation data is taken to be the exponential function; the
results of that theorem then show that the required function is obtained simply by means of
rational function interpolation.
A section on meromorphic
F
-functions follows in which the properties of the corre-
sponding continued fractions they generate are studied. The even and odd convergents
are examined.
The first Appendix has the title The asymptotic expansion of positive real functions. Ap-
pendix two is on The construction of functions belonging to certain classes. It deals with a
general theory of the derivation of functions of the form
F(λ) = Z∞
−∞(λ−t)−1dσ(t)
from others of the same form. In a previous Appendix (without a number) Wynn
already treated the same problem for
S
-functions. The third Appendix, already men-
tioned above, considers the Pick-Nevanlinna problem in a wider setting. Let us
remember that this problem consists in finding a holomorphic function
ϕ
that inter-
polates the data
λ1
,
. . .
,
λn∈D
subject to the constraint
|ϕ(λ)|<
1 for all
λ∈D
; see
Reference [127]. Wynn wrote:
It is clear from the above conspectus of results from the Pick-Nevanlinna theory, that the
solution of the problem of determining a rational function which satisfies the mapping and
interpolation conditions described above differs from that of constructing a rational function
which satisfies interpolation conditions alone in at least two respects: in the solution of the
first problem a combination of function-theoretic and algebraic methods is involved, while in
that of the second, algebraic methods are exclusively deployed; furthermore, each stage of the
solution of the first problem results, not in the construction of a single interpolating function
as is the case for the second problem, but in that of a family of functions with the required
properties.
Developments and theoretical results follow. A bibliography of 26 items ends the
paper. One can wonder why Wynn never published it.
3 The Hamburger-Pick-Nevanlinna problem
This manuscript contains 179 pages but with many portions scratched (it probably
contains a mixing of a draft of a paper, notes and rough notes all cited in {26,28}).
Let
p
,
q
,
r
be finite positive integers, let
aν(ν:
0, 2
p−
2
)
be finite real numbers, let
xk(k:q)
be distinct real argument values,
T(
1,
k) (k:q)
be the corresponding
finite positive integers, and
bk,ν(k:q|ν:
0, 2
T(
1,
k)−
1
)
sets of finite real valued
coefficients, let
λk(k:r)
be distinct argument values in
L
,
T(
2,
k) (k:r)
be the
corresponding finite positive integers, and
ck,τ(k:r|τ:
0,
T(
2,
k)−
1
)
sets of finite
complex valued coefficients. The problem is to determine a function
G
which satisfies
the asymptotic relationship

Mathematics 2021,9, 1240 16 of 45
λ2p−1(G(λ)−
2p−2
∑
ν=0
aνλ−ν−1)=O(1)
as
λ
tends to infinity in a sector of the form
π<γ≤arg(λ)≤δ<
2
π
, and satisfies
also the asymptotic relationship for k:q
(λ−xk)−2T(1,k)(G(λ)−
2T(1,k)−1
∑
ν=0
bk,ν(λ−xk)ν)=O(1)
as
λ
tends to
xk
over an open set contained in
L
with limit point at
xk
, for
k:r
, and
moreover satisfies the interpolation conditions
λτG(λk)/τ!=ck,τ
for
τ:
0,
T(
2,
k)−
1,
and finally maps
L
into
D
(or
U
?). According to Wynn The proof of the principal theorem
of this section is largely based upon results, due to M. Riesz, in the theory of linear functionals.
Since this part is a draft in quite bad shape, we will not pursue its analysis.
The document contains another interesting section
Matrix criteria
In this section Akhiezer’s treatment of the simple Pick-Nevanlinna problem is extended to
the diminished Hamburger-Pick-Nevanlinna problem; conditions that are necessary and
sufficient for the solubility of the latter problem are established and, assuming this problem
to be nondegenerate, inclusion discs for its solutions are located.
As a preliminary, a Hermitian matrix is constructed from the data which leads to an
extension of a theorem on the existence of inclusion discs for the values of all solutions
to the Hamburger problem and the simple Pick-Nevanlinna problem.
Wynn concluded:
The results of the above theorem are exclusively concerned with the diminished version of
the Hamburger-Pick-Nevanlinna problem. It is possible to extend the method of proof to the
treatment of the Hamburger-Pick-Nevanlinna problem itself, to examine the structure of
the rational function solutions to this problem and its diminished form, and to describe the
relative positions of the inclusion discs deriving from a sequence of subordinate problems.
These matters are, however, more conveniently dealt with methods described in the following
section, in which explicit expressions for the general solutions to the interpolation problems
concerned are described.
Unfortunately, this following section of the manuscript is not under a form which allows
to give a clear account of it.
4 Continued fraction transformations of the Euler-Mclaurin series
The document is 202 pages long. It dates from December 1976. The first part of the
document is in a quite good shape (probably a draft of the paper mentioned in {
26
},
with an abstract and a bibliography) with some corrections done with a pencil. Not all
pages have been written with the same pen. Some of them are missing or not in the
right order since all of them are not numbered (they could be some additional notes on
integral transform and analytic continuation). Moreover, references to some formulas
are missing, and some authors quoted in the text are not listed into the bibliography
given at the end. An in-depth study of this document is needed to fully understand it
if possible to exploit it.
It begins by a long abstract from which we extract the main points.
Results concerning the convergence of forward diagonal sequences of quotients in the Padé
table are given. In particular, it is shown that, if (
∗
)
fν=R∞
0tνdσ(t) (ν=
0, 1,
. . .)
,
σ
being a bounded nondecreasing real valued function such that all moments (
∗
) exist,
and (
∗∗
)
fν=O{(χν)!ξν}(
0
<χ≤
2, 0
<ξ<∞)
then all forward diagonal se-
quences of Padé quotients derived from the series
∑∞
ν=0fνzν
converge uniformly over any
bounded region in the
z
-plane not containing any point of the nonnegative real axis to (
∗∗∗
)
f(z) = R∞
0(1−zt)−1dσ(t)
,
f(z)
being the le Roy or
(B
,
χ)
sum of the given series for all
finite z in the sector χπ/2 ≤arg(z)≤(4−χ)π/2.

Mathematics 2021,9, 1240 17 of 45
This result extends results given by various authors, and it can also be extended to
the case where the lower limit in (
∗
) and (
∗∗∗
) is replaced by
−∞
. These results are
applied to the delayed Euler-Mclaurin series
∞
∑
ν=0
bj+νD2j+ν+1Ψ(µ)k2ν,
(
j≥
0, and
D
being
d/dµ
) regarded as a series expansion in ascending powers of
k2
. Convergence results for the Padé approximants of this series are derived, and
also for the same series in which single zeros are inserted between its successive
terms. Applications to Stirling’s asymptotic expansion of the logarithm of the gamma
function, and to the asymptotic expansion of the generalised Riemann zeta-function
are presented.
5 Convergence and truncation error bounds for associated continued fractions
This is a short document (13 pages), probably only the notes mentioned in {26}.
In this work, he proposes to give a convergence theorem for the functions
$2r{ψ(µ)}
,
where, presumably,
$2r
is the 2
r
th convergent of Thiele’s continued fraction. He begins
to prove other results (not reproduced herein since they contain too many erasures)
upon which the proof of the theorem is based. It is:
Theorem 3. Let
[α
,
β]
be a fixed interval of the finite real axis, and
ξ(s)
be a bounded
nondecreasing real valued function for
α≤s≤β
, and not a step function with a finite
number of salti. Let µ∈(−∞,∞)×[α,β]be fixed and
ψ(µ) = Zβ
α(µ−s)−2dξ(s)
${ψ(µ)}=Zβ
α(µ−s)−1dξ(s).
Then limr=∞$2r{ψ(µ)}=${ψ(µ)}, and
|${ψ(µ)} − $2r{ψ(µ)}| ≤ (β−α)|ψ(µ)|p(µ;α,β)2r−1,r=1, 2, . . .
where p(µ;α,β) = |µ−α|1/2 − |µ−β|1/2 /|µ−α|1/2 +|µ−β|1/2.
According to what precedes, it seems that the sign that looks like
×
in
µ∈(−∞,∞)×[α,β]
has to be replaced by
/∈[α
,
β]
. We do not know if this result had been later rediscovered
by other researchers. Thus, it was interesting to reproduce it here.
4.1.2. Interpolation
6 Functional Interpolation
This very well handwritten manuscript has 52 pages and seems to be the paper
mentioned several times in all the list of documents, projects and activities, with a
bibliography with a last reference dating from 1984. The first section is Interpolation
and extrapolation with a subsection 1.1–Procrustean technique. This word was already
used by Wynn in the title of the published paper of Reference [
2
] where he introduced
a particular form of the
$
-algorithm for extrapolation at infinity by a rational function
in
n
(see Reference [
113
]). It describes situations where different properties are fitted
to an arbitrary standard. In the Greek myth, Procrustes was a son of Poseidon. He
compelled travelers to lie on a bed, he cut off their legs that were longer than bed, and
stretched the feet of those who were too small. The manuscript begins by:
Most general theories arise from investigations of particular problems, and in this respect the
theory to be described is not exceptional. By way of motivation, the problem of deriving an
extrapolation method from an interpolatory formula and its converse are considered.
It is first supposed that an interpolatory function of complex variables
(1) F(m)
r(d|y;λ) = F(m)
r(dm, . . . , dm+r|ym, . . . , ym+r;λ)

Mathematics 2021,9, 1240 18 of 45
is available for which (a)
F(m)
r(d|y
;
yν) = dν(ν∈Im,m+r
; throughout the paper
Ii,j
is
the sequence
i
,
i+
1,
. . .
,
i+j
;
Ii
is the sequence
i
,
i+
1,
. . .
;
I
is
I0
) and (b) for certain
distributions of the dνand yν
G(m)
r(d|y) = lim F(m)
r(d|y;λ) (λ→∞)
exists. The function
F(m)
r
serves as the basis of an extrapolation method: given the sequence
Sν(ν∈I)
the number
G(m)
r(d|y)
obtained by setting
dν=Sν(ν∈Im,m+r)
in
G(m)
r(d|y)
is an estimate of
lim Sν(ν→∞)
. (The numbers
yν
used are suggested by the process
producing the sequence
{Sν}
: the choice
yν=ν(ν∈I)
is natural; the choice
yν=
2
ν
arises, for example, in Romberg’s method of integration [2 references] in which the number
of integration subranges is doubled at each stage, F(m)
rbeing a polynomial in λ−1.)
Then, Wynn illustrates the method by taking
F(m)
2i
as the quotient of two polyno-
mials of degree
i
in
λ
. Subject to certain existence conditions, the coefficient of
λi
in the denominator is 1, and that in the numerator is
$(m)
2i(d|y)
which can be ex-
pressed as the quotient of two determinants involving
dτ
,
yτ
,
τ∈Im,m+2i
. In this
case
G(m)
2i=$(m)
2i(d|y)
. Replacing
dτ
by
Sτ
, the determinantal formula for
$(m)
2i(d|y)
gives the extrapolated limit. These numbers can be recursively computed by Thiele’s
reciprocal difference algorithm (see
(3)
in Section 3.4), and they can be displayed in a
two dimensional array for which Wynn does not use the usual notation but a new one,
and he writes:
[The process] involves numbers
εi,j
which may be set at the intersections of the full rows
and columns and of the half rows and columns of a chipped triangular array in which the
row index
i
ranges over
¯
I−1/2
(
¯
Ik
is the sequence
k
,
k+
1
/
2,
k+
1,
. . .
;
¯
I
is
¯
I0
) and the
column index
j
over the range
Ii
, the number
ε−1/2,−1/2
being missing. The numbers
εi,j
are constructed from the initial vales
ε0,j=Sj(j∈I)
,
ε0
0,j=
0
(j∈I)
(the dash is used
to indicate a displacement operation acting upon numbers with two suffixes, whose effect
is illustrated by the relationships
ε0
0,j=ε−1/2,j+1/2
and
ε0
i,j=εi−1/2,j+1/2
) by use of
the relationship
(3) (∆jεi,j)(∆iε0
i,j) = wi,j
for
i∈¯
I
,
j∈Ii
where
∆j
is the difference operator
∆jεi,j=εi,j+1−εi,j
and
∆i
is similarly
defined. With
wi,j= (yi+j+1−yj−i)−1(i∈¯
I,j∈Ii)
[it holds]
$(m)
2i(S|y) = εi,i+m(i,m∈I).
And he concludes with his personal sense of humor:
We have called an extrapolation method of the above type a Procrustean technique [
2
],
although in fitting function values to a sequence, i.e. the bed to the victim, we are a little
kinder than Procrustes is reputed to have been.
He continues:
If in formula (1)
λ
is fixed and one of the
yν
is very large,
eν
[not defined] is approxi-
mately equal to
lim F(m)
r(λ) (λ→∞)
: if
λ
is fixed and the
yτ6=yν
are fixed and finite,
lim F(m)
r(λ) = eν(λ→∞)
. This observation may be made in terms of the behaviour of
G(m)
r(d|y)
as
yν
tends to infinity, the other
yτ
remaining finite; expressed in terms of the
Sτit is that, under suitable conditions, lim G(m)
r(S|y) = Sν(yν→∞).
There should be an error in what Wynn wrote. Since he states that
λ
is fixed,
λ→∞
should be replaced by
yν→∞
(remark of one of the reviewers). Wynn concludes
that this property is, for example, satisfied by the ratio of determinants expressing
G(m)
r(S|y)which, thus, tends to Sνas yνtends to infinity.

Mathematics 2021,9, 1240 19 of 45
The second subsection is named 1.2–Interpolatory functions. He claims that The steps taken
in the above derivation of an extrapolation method from an interpolatory formula may be reversed.
Then, he shows how to reverse the recursive rule given in the preceding subsection.
[...] set
dν=d(yν)
,
eν(z) = e(yν
,
z)
and
fν(z) = f(yν
,
z) (ν∈I)
. Suppose that a
function of complex variables
G(m)
r(S|y) = G(m)
r(Sm, . . . , Sm+r|ym, . . . , ym+r)
for which
lim G(m)
r(S|y) = Sν
as
yν→∞(ν∈Im,m+r)
, obtained either from an
interpolatory or from an extrapolation method or in some other way is available. Set
H(m)
r(e,f|z) = G(m)
r(em(z), . . . , em+r(z)|fm(z)−1, . . . , fm+r(z)−1).
[...]Thus in view of the property attributed to
G(m)
r(S|y)
just described,
lim H(m)
r(e
,
f|z) =
dν
as
z→yν
over
Z
(
ν∈Im,m+r)
:[
Z
is an open set of points in
C
]
H(m)
r(e
,
f|z)
is
an interpolatory function. If the
G(m)
r(S|y) (r
,
m∈I)
may be computed by means of a
recursive process, appropriate modification yields a process for computing numerical values
of the H(m)
r(e,f|z) (r,m∈I).
Then, Wynn develops the particular case of the extrapolation estimate
$2i(S|y)
, and
he obtains a set of rational functions
r(m)
i,j(z)
. He proves that
lim r(m)
i,j(z) = dν
as
z→yν(ν∈Im,m+i+j). Imposing the further condition
eν(z)−dν=O{ fν(z)}(z→yν;ν∈I)
it follows that
(7) r(m)
i,j(z)−dν=O{ fν(z)}(z→yν;ν∈Im,m+i+j;m,i,j∈I).
The interpolatory function derived from the extrapolation estimate
$(m)
2i(d|y)
is
r(m)
i,i(z). He proves that lim r(m)
i,j(z) = dνas z→yν(ν∈Im,m+i+j). Then
Setting now
ε0
0,j=
0,
ε0,j=ej(z) (j∈I)
and
wi,j={fj−i(z)−1−fi+j+1(z)−1}(i∈
¯
I,j∈I1), r(m)
i,i(z) = εi,i+m(z) (i,m∈I).
In this result, the conditions of the simple case in which
(8) e(y,z) = d(y),f(y,z) = z−y
may be imposed upon
e
and
f
. Now
eν(z) = dν
is a constant, independent of
z(ν∈I)
and
fν(z)
is the difference
z−yν(ν∈I)
. In this case,
r(m)
i,i
is the quotient of two
i
th degree
polynomials in
z
, the rational function from which the extrapolation limit
$(m)
2i(S|y)
was
derived. The above process now reduces to an algorithm for rational function interpolation
due to Brezinski [128]of which a generalisation has been proposed by Cordellier [129].
The discussion of interpolatory formulæ and extrapolation methods is terminated by the
remark that under appropriate conditions cyclic derivation of extrapolation methods from
interpolatory formulæ and conversely may be repeated indefinitely.
Section 2 of this document is entitled Approximants of general order. In the simple case
(8),
r(m)
i,j
is a ratio of two polynomials of degree
i
. In this Section, he proposes to study
the more general system of approximants
r(m)
i,j
. The first subsection is Nonuniform
approximation. Under the condition (8), the relationship (7) reduces to
r(m)
i,j(z)−dν=
O(z−yν)
: approximation is uniform, the form of the function
z−yν
being the same
for all relevant
yν
. Wynn notices that a suitable choice of the functions
fν
in the non
simplified case, non uniform approximation is possible.
The next subsection is Remainder term formulæ. Wynn explains that

Mathematics 2021,9, 1240 20 of 45
In certain circumstances an interpolation property of the form (7) holding at points induces
on the function possessing it a corresponding property of approximation over a set containing
the points. By imposing severe restrictions upon the functions
e
and
f
it is possible in a
few lines to exhibit the
r(m)
i,j
as approximations to a function defined over
Z
, and to provide
associated remainder terms.
He imposes the conditions
e(y,z) = d(y),f(y,z) = φ(z)−φ(y)
with
φ0(y)6=
0 for all
y
in
Z
. By a straightforward (as he writes) adaptation of an
argument of Nörlund [130] (Ch. 15, §3), Wynn obtains the expression of the error.
The following subsection is on Algorithms for approximation evaluation in which Wynn
gives formulæ for the recursive computation of the approximant values
r(m)
i,j(z)
for a
fixed
z∈Z
. After quite long developments involving Lagrange forms and divided dif-
ferences, Wynn shows that his relationship (3) can be applied with
wi,j=fm+i+j+1(z)−1
to yield
r(m)
i,j(z) = εi,j(i
,
j∈I)
starting from two different sets of initializations. How-
ever, in (3), the
ε0
i,j
are intermediate computations which can be eliminated, and he
arrives at the rule
(23) ∆j{wi,j(∆jεi,j)−1}=∆i{w0
i,j(∆iεi−1,j+1)−1},
where
εi,−1=
0
(i∈I)
,
ε−1,j=∞(j∈I)
. The initializations are
ε0,j=L(m)
j(z)
in
the row by row order
i∈I
,
j∈I−1
or
εi,0 =M(m)
i(z) (i∈I)
in the column order
j∈I−1,i∈Iwith
L(m)
j(z) =
j
∑
ν=0
em+ν(z)
j
∏
τ=0
τ6=ν
fm+τ(z)
fm+τ(z)−fm+ν(z)(m,j∈I)
and
M(m)
i(z) = 




i
∑
ν=0
em+ν(z)−1i
∏
τ=0
τ6=ν
fm+τ(z)
fm+τ(z)−fm+ν(z)




−1
(m,j∈I).
Recursive relations for the
L(m)
j
’s and the
M(m)
i(z)
’s are also given. Using divided
differences, they are also expressed in Newton form. Determinantal formulæ are
related to the recursive rules given.
It is showed that particular cases for the
eν
and the
fν
give back the usual Lagrange
interpolation formula, the Neville-Aitken scheme, and Newton series. The work of
Stoer on interpolation by rational functions [
131
] and the variant of the
ε
-algorithm
due to Claessens [132] are also recovered as special cases.
The next subsection is named Termination. When
e(y
,
z)
is a polynomial or rational
function of f(y,z), termination of the algorithms previously given is proved.
The following subsection treats Confluence that is when some argument values coincide.
Wynn examines what happens to the previous formulæ and recursions. In that case,
for n∈I2,
(31) f (y,z)−
n−1
∑
ν=1
cν(y,z)x(y,z)ν=O{x(y,z)n}
(32) e(y,z)−d(y)−
n−1
∑
ν=1
bν(y,z)x(y,z)ν=O{x(y,z)n},

Mathematics 2021,9, 1240 21 of 45
and formulæ for the computation of the coefficients
cν
and
dν
are given. They im-
plement a truncated composition of polynomials. The
dν
’s are the coefficients of the
Newton series representations of the corresponding Lagrange forms, which are con-
fluent forms of the functional divided differences. They are related to the
bν
, but all
details are too complex to be given herein.
Then, comes a subsection on Zero finding algorithms. Under suitable conditions, the
above algorithm for the truncated composition of polynomials can be used for the
inversion of formal power series. Moreover
In so doing it serves as the basis of a number of algorithms for determining the zeros
of a function and motivates the use of the approximants
r(m)
i,j(z)
for the same purpose.
Setting
e(y
,
z) = z−y
and
f(y
,
z) = φ(z)−φ(y)
,
φ
being the function under treatment,
relationships of the form (31,32) hold with
x(y
,
z) = z−y
,
cν(y
,
z) = φ(ν)(y)/ν!(ν∈I1)
are Taylor series coefficients, d(y) = 0, b1(y,z) = 1and bν(y,z) = 0(ν∈I2).
[...] Taking the points y0,y1and y2to be confluent, the Lagrange forms [...]are
L(0)
0=z−y0
L(0)
1=L(0)
0−(φ0
0)−1{φ(z)−φ0}
L(0)
2=L(0)
1−(φ0
0)−3φ00
0{φ(z)−φ0}2/2
where
φ0=φ(y0)
,
. . .
,
φ00
0=φ00 (y0)
. Newton’s process
z=y0−φ0/φ0
0
is obtained
from
L(0)
1(z)
by equating the latter to zero after setting
φ(z) =
0. The third order process
z=y0− {φ0/φ0
0}−{1
2φ2
0φ00
0/φ03
0}
is obtained from
L(0)
2
in the same way. Applying
relationship (3) to the initial values
ε0,j=L(0)
j(z) (j∈I0,2
with
w0,0 =w1,1 =w1/2,1/2 =
{φ(z)−φ0}−1
(since
y1=y2=y0
) and using
ε1,1 =r(0)
1,1 (z)
as just described, the further
third order process z =y0−φ0φ0
0/{φ02−1
2φ0φ00
0}is obtained.
The artifice described above is capable of further application.
For example, if
y16=y2
,
L(1)
1(z)
leads to the method regula falsi
z= (y2φ1−
y1φ2)/(φ1−φ2)
. If
y0=y1
, then Wynn obtains two combinations of Newton’s
method and regula falsi
z=y−(φ0/φ0
0)− {φ2
0(φ2−φ0−(y2−y0)φ0
0}/{(φ2−φ0)2φ0
0}
z=y0+φ0/{[φ0/(y2−y0)] −[φ0
0φ2/(φ2−φ0)]}.
In the same way, an
n
th order single point iteration process can be obtained. The
subsection ends by
The more general theory yields multipoint processes (for a further application of the
ε
-
algorithm to the problem of finding the zeros of a function, see [133,134]).
The last subsection of this document is entitled Extensions of the Lagrange-Bürmann
expansion. Wynn claims that The above treatment of the confluent case offers an interpretation
of the theory of this paper. He first gives the Lagrange-Bürmann expansion of
d(z)
in
powers of
φ(z)−φ(yµ)
where
yµ
belongs to a close contour in the complex plane. Then,
he obtains an algorithm for determining the coefficients in the Lagrange-Bürmann
expansion and in an asymptotic version of it from the Taylor series coefficients of
d(z)−d(y)
and
φ(z)−φ(y)
at the point
y=yµ
. The case of confluence is also treated.
In addition, in this case, we do not know why Wynn never published this work since
it was ready to be submitted.
7 Interpolation by the use of rational functions
This handwritten complete paper of 90 pages with a bibliography of 27 items, the last
one dated 1979, is present in the projects and in the Lists of documents, and it was
never published. It seems to be related to the previous manuscript {6}.

Mathematics 2021,9, 1240 22 of 45
In the Section 1 of this document, titled The Thiele-Nörlund interpolation theory, Wynn
reminds how to construct rational interpolating functions
C(m)
2s
in which the numerator
and the denominator have degree
s
, and
C(m)
2s+1
with a numerator of degree
s+
1 and
a denominator of degree
s
and such that
C(m)
r(xi) = fi
for
i=m
,
. . .
,
m+r
, where
the
xi
and the
fi
are assumed to be complex numbers. After having constructed the
reciprocal differences
$(m)
r
by Thiele process (this is the
$
-algorithm
(3)
, where now
r
and mare arbitrary indexes)
$(m)
r+1=$(m+1)
r−1+ (xm+r+1−xm)($(m+1)
r−$(m)
r)−1
for r,m=0, 1, . . . with $(m)
−1=0(m=1, 2, . . .)and $(m)
0=fm(m=0, 1, . . .), it holds
C(m)
r(λ) = N(m)
r(λ)
D(m)
r(λ)=λ−xm
$(m)
1−$(m)
−1
+λ−xm+1
$(m)
2−$(m)
0
+· · · +λ−xm+r−1
$(m)
r−$(m)
r−2
.
The successive numerators and denominators are recursively computed by
N(m)
r(λ)=($(m)
r−$(m)
r−2)N(m)
r−1(λ) + (λ−xm+r−1)N(m)
r−2(λ)
D(m)
r(λ)=($(m)
r−$(m)
r−2)D(m)
r−1(λ) + (λ−xm+r−1)D(m)
r−2(λ)
with
N(m)
−1(λ) = 1, N(m)
0(λ) = fm,D(m)
−1(λ) = 0, D(m)
0(λ) = 1.
Wynn asserts that conditions to ensure that all rational functions
C(m)
r
can be con-
structed by the above scheme and that they have the required interpolation properties
can be formulated in terms of determinants, and that determinantal formulæ can also
be given for the numbers and the functions involved, and he claims that Such formulae
are made more concise by the use of a special notation. It takes 6 pages to establish these
notations. After quoting a remark in German by Nörlund [
130
] (Ch. 15, p. 420) that
Wynn finds perhaps a little exuberant, he writes that:
It is the principal purpose of this paper to point out that, using another very simple relationship
(namely, in particular, that, if
xi
is replaced by
(λ−xi)−1
,
$(m)
2r
becomes
C(m)
2r(λ))
many
results suggested by the behaviour of reciprocal differences may be obtained for convergents.
These procedures are described in Section 2 of this document, titled The
σ
- and
µ
-
algorithms. After explaining how to obtain them, Wynn writes
Theorem 1. Let
λ
be a fixed finite complex number unequal to
xi(i=
0, 1,
. . .)
. Set
zi=
(λ−xi)−1
for
i=
0, 1,
. . .
[...] Numbers
σ(m)
r(λ) (r
,
m=
0, 1,
. . .)
can be constructed
from the initial values
σ(m)
−1(λ) =
0
(m=
1, 2,
. . .)
,
σ(m)
0(λ) = fm(m=
0, 1,
. . .)
by use
of the relationship
(16) σ(m)
r+1(λ) = σ(m+1)
r−1(λ) + {zm+r+1(λ)−zm(λ)}{σ(m+1)
r(λ)−σ(m)
r(λ)}−1
with r,m=0, 1, . . . and, in particular,
σ(m)
2s(λ) = C(m)
2s(λ)
for s,m=0, 1, . . .
Then, the document becomes unclear. It seems that replacing in the above recurrence,
zi
by
zifi
(this is the unclear point), and renaming
µ(m)
r
’s the
σ(m)
r
’s, Wynn obtains
µ(m)
2s+1(λ) = C(m)
2s+1(λ)−1. A proof of this result is given.
Section 3 of this document is entitled Interpolation and extrapolation. It is interesting to
quote its introduction which contains general comments by Wynn on these topics.

Mathematics 2021,9, 1240 23 of 45
The simple observation that, if the argument values
xi
are replaced by functions
(λ−xi)−1
the reciprocal differences
$(m)
2s
become interpolating functions
C(m)
2s(λ)
has already produced
the
σ
-algorithm of relationship (16), the simplest and most economical method, subject to
the stated conditions, for evaluating these functions; the observation also leads directly, as
will be shown below, to new interpolatory theory. Once made, the observation is trivial, and
its implications are not difficult to work out; perhaps its most interesting feature is that it
has not been made before. The simple relationship between reciprocal differences and rational
interpolating functions was not so much discovered as forced upon the author’s attention
while working out the consequences of principles underlying the process of interpolation
and the transformation of divergent series. These two subjects have recently became in-
creasingly important in computational mathematics; new theory in what once might have
been considered dead subjects is constantly being developed (mention may be made of recent
generalisations of polynomial interpolation described in [
135
–
138
]); it is highly probable
that the principles concerned will find further applications, and for this reason they are
now outlined.
It is difficult to summarize what follows without quoting large parts of the document.
Moreover, some notations and their inferences are not clear enough. Basically, Wynn
comes back to the link between interpolation and extrapolation already discussed in
the document {
6
}, that he named Functional Interpolation. In particular, he considered
the following extended
ε
-algorithm (which contains the
$
-algorithm and some other
extensions [128])
(25) ε(m)
r+1=ε(m+1)
r−1+γ(m)
r(ε(m+1)
r−ε(m)
r)−1,
with
γ(m)
r=ψ(m+r+
1
)−ψ(m)
. He considers the particular case
γ(m)
r=
1, which
corresponds to the
ε
-algorithm. He reminds that, when applied to the partial sums of
a formal power series, this algorithm furnishes the Padé approximants belonging to
the lower half, diagonal included, of the Padé table, and that he derived (no reference)
various determinantal formulæ from a more general form of approximating fractions
given by Jacobi [
139
]. He also mentions that, in its special form, the problem was also
studied by Frobenius [
140
] and Padé [
141
], and he adds the following remark in which
he explains how he obtained his ε-algorithm
While idly investigating the formulæ resulting from the choice
γ(m)
r=
1in the relationship
(25) [...] the author noted that expressions obtained for the numbers
ε(m)
2r
were equivalent to
extrapolatory determinantal expressions, simplified versions of those due to Jacobi and used
by Frobenius, previously published by Schmidt [
142
]and republished by Shanks [
112
]. In
this way the ε-algorithm was discovered.
Then, Wynn notices that (25) has been used by Claessens with
γ(m)
r= (λ−xm+r+1)−1
,
and applied to the partial sums of the Newton interpolation series, for obtaining ratio-
nal interpolating functions [
132
], and he explains the theoretical basis of this algorithm
Claessens was led to the discovery of the extended
ε
-algorithm by the consideration of interpo-
latory continued fractions not of Thiele form, but of a form introduced by Kronecker [
143
]in
connection with a process initiated by Rosenhain [
144
]and Borchardt [
145
]for constructing
the resolvent of two polynomials from systems of their numerical values.
If all the
xi
tend to a common value
x
, the interpolation fractions tend to the Padé
approximants and the extended
ε
-algorithm (25) tends to the usual one. When
γ(m)
r=m+r+1, (25) gives back the $-algorithm studied by Wynn in Reference [2].
The following Section is on Lozenge algorithms, which are algorithms relating quantities
located at the four corners of a lozenge in a table where the lower index indicates
a columns and the upper index a descending diagonal. The
ε
-algorithm and its
generalizations [
146
], the
$
-algorithm, and the algorithms numbered by Wynn (23)

Mathematics 2021,9, 1240 24 of 45
(see the manuscript {
6
}), (16) and (25) (see above in this manuscript) enter into this class.
They share some algebraic properties when, instead of applying them to a sequence
(Sn)
, they are applied to
(aSn+b)
, when
γ(m)
r
in (25) is multiplied by a constant, and
when a fractional linear transformation is applied to the elements with an even lower
index (property named homographic invariance). In these algorithms, the quantities with
an odd lower index are only intermediate results with no interest for their purpose.
They can be eliminated and a new rule relating five quantities disposed at the center
and at the extremities of a cross are obtained. The first algorithm to have been treated
in that way is the