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A NEW TYPE OF HERMITE MATRIX POLYNOMIAL SERIES
EMILIO DEFEZ∗AND MICHAEL M. TUNG[
Abstract. Conventional Hermite polynomials emerge in a great diversity of
applications in mathematical physics, engineering, and related fields. However,
in physical systems with higher degrees of freedom it will be of practical interest
to extend the scalar Hermite functions to their matrix analogue. This work
introduces various new generating functions for Hermite matrix polynomials
and examines existence and convergence of their associated series expansion
by using Mehler’s formula for the general matrix case. Moreover, we derive
interesting new relations for even- and odd-power summation in the generating-
function expansion containing Hermite matrix polynomials. Some new results
for the scalar case are also presented.
1. Introduction
During the last decade, orthogonal matrix polynomials have become increas-
ingly important for numerical computation and its analysis. In particular, Hermite
matrix polynomials were introduced and studied for the first time in [8, 10] and
subsequently received considerable attention for its application in the solutions of
matrix differential equations [2–5,12,13]. In the scalar case, Hermite polynomials
emerge in the context of quantum mechanics and optics, mathematical and nuclear
physics among other areas of highly practical interest. Only recently, a new kind of
series expansion involving conventional Hermite polynomials was introduced in [7]
in order to describe new field states in quantum optics—yet without any rigorous
proof of existence and convergence in general.
In this article, we present an entirely new class of series expansion for Hermite
matrix polynomials, which also includes the scalar polynomial expansion of [7] as
a particular case. Besides, we will provide an accurate analysis of convergence.
To start with, in Section 2, we define the new Hermite matrix polynomial series.
Section 3 illustrates the reduction to the scalar case, given in [7]. Throughout
this work, we denote by Cr×rthe set of all the complex square matrices of size r.
Furthermore, by Θ and Iwe denote the zero and the identity matrix, respectively.
If f(z), g(z) are holomorphic functions on an open set Ω ⊂C, and if the eigenvalues
σ(A)⊂Ω, then f(A), g(A) represent the images of functions fand gacting on the
matrix Asuch that f(A)g(A) = g(A)f(A), see [6, p.558]. If Re(z)>0 for every
eigenvalue z∈σ(A), we say that matrix Ais positive stable, In this case, we use
√A=A1/2= exp (log (A)/2) for the image of the function z1/2= exp (log (z)/2)
2000 Mathematics Subject Classification. 33C45, 33C47, 15A16.
Key words and phrases. Hermite matrix polynomials, Hermite polynomials, generating
functions.
The authors thank the Spanish Ministerio de Econom´ıa y Competitividad and the European
Regional Development Fund (ERDF) for financial support under grant TIN2014-59294-P.
1
2 EMILIO DEFEZ∗AND MICHAEL M. TUNG[
acting on the matrix A. Note that log (z) as usual denotes the principal branch of
the complex logarithm.
A matrix polynomial of degree nis an expression of the form Pn(t) = Antn+
An−1tn−1+··· +A1t+A0, where tis a real variable and Aj∈Cr×rfor 0 ≤j≤n.
The usual 2-norm of a quadratic matrix Ais denoted by kAk2.
2. The new Hermite matrix polynomial series
For any positive stable matrix A∈Cr×r, the associated Hermite matrix polyno-
mials [8] are defined by the following three-term recurrence for any non-zero positive
integer m∈N:
H−1(x, A)=Θ, H0(x, A) = I, Hm(x, A) = x√2AHm−1(x, A)−2(m−1)Hm−2(x, A).
(2.1)
This definition fulfills for any n∈N0(positive integers including zero) the relations
H2n+1(0, A)=Θ, H2n(0, A)=(−1)n(2n)!
n!I. (2.2)
For the following derivations we need to use two well-known results. The first result
was demonstrated in [1] and establishes the following upper bound:
kH2n(x, A)k2≤gn(x),kH2n+1(x, A)k2≤ |x|
A
2−1/2
2
2gn(x)
n+ 1 ,
gn(x)=22n(2n+ 1)!
n!exp 5
2kAk2x2, n ∈N0.
(2.3)
The second result was demonstrated in [9] and is Mehler’s formula in the matrix
case:
X
n≥0
Hn(x, A)Hn(y, A)
2nn!tn= (1−t2)−1
2exp 2xyt −(x2+y2)t2
2(1 −t2)A, x, y ∈R,|t|<1.
(2.4)
Now we are in the position to prove the following theorem:
Theorem 2.1.Let A∈Cr×rbe a positive stable matrix and x∈R. Then for
|t|<1/16:
A(s;x, t, A) := X
n≥0
H2n+s(x, A)
n!tn=
Hsx
√1+4t, A
√1+4ts+1 exp 2tx2
1+4tA, s ∈N0.
(2.5)
Proof 2.1.First we will prove that the matrix series A(s;x, t, A) is convergent for
any fixed integer s∈N0. Taking into account (2.3), being s= 2lan even number,
one obtains
H2n+s(x, A)
n!tn
2
=
H2n+2l(x, A)
n!tn
2≤gn+l(x)|t|n
n!.
Since X
n≥0
gn+l(x)|t|n
n!is convergent for |t|<1/16, the matrix series A(2l;x, t, A) is
convergent in any compact real interval. On other hand, if s= 2l+ 1 is an odd
A NEW TYPE OF HERMITE MATRIX POLYNOMIAL SERIES 3
number, then
H2n+s(x, A)
n!tn
2
=
H2(n+l)+1(x, A)
n!tn
2≤ |x|
(A/2)−1/2
2
2gn+l(x)|t|n
(n+l+ 1)n!.
Since X
n≥0
|x|
(A/2)−1/2
22gn+l(x)|t|n
(n+l+ 1)n!is convergent for |t|<1/16, the matrix
series A(2l+ 1; x, t, A) is convergent in any compact real interval. Thus, the series
A(s;x, t, A) is convergent for all fixed integers s∈N0, what was to be shown.
For proving formula (2.5), we will use the induction method. We put y= 0
in formula (2.4) and use relations (2.2). Thus, Mehler’s formula (2.4) reduces to
(see [1] for details):
X
n≥0
(−1)nH2n(x, A)
22nn!t2n= (1 −t2)−1
2exp −x2t2
2(1 −t2)A, x ∈R,|t|<1/2.(2.6)
Taking u=−t2/4, we can rewrite (2.6) in the form
X
n≥0
H2n(x, A)
n!un= (1 + 4u)−1
2exp 2ux2
1+4uA, x ∈R,|u|<1/16.(2.7)
Observe that (2.7) is exactly A(0; x, u, A). Thus, formula (2.5) is true for s= 0.
Next, we proceed to prove that formula (2.5) is also true for s= 1. After using
the three-term recurrence (2.1) for m= 2n+ 1, multiplying each term by tn/n!
with |t|<1/16, and finally applying the sum from n= 1 to infinity, one arrives at
X
n≥1
H2n+1(x, A)
n!tn=x√2AX
n≥1
H2n(x, A)
n!tn−4X
n≥1
nH2n−1(x, A)
n!tn.(2.8)
To start summation at n= 0, we rewrite (2.8) in the form
(1 + 4t)X
n≥0
H2n+1(x, A)
n!tn=H1(x, A)−x√2AH0(x, A) + x√2AX
n≥0
H2n(x, A)
n!tn.
(2.9)
Applying (2.1), we conclude that H1(x, A) = x√2A, H0(x, A) = I. Now, using
(2.7), we may simplify (2.9) to the form:
A(1; x, t, A) = x√2A
√1+4t3exp 2tx2
1+4tA=
H1x
√1+4t, A
√1+4t2exp 2tx2
1+4tA,
(2.10)
and formula (2.5) is also true for s= 1.
Up to this point we only have shown that the inductive statement A(s;x, u, A)
holds for s= 0,1. Next, we proceed with the inductive step, showing that if
A(l;x, u, A) holds for 0 ≤l≤s−1, then A(s;x, u, A) is also true.
Again, we use the recurrence relation (2.1) for m= 2n+sto obtain
H2n+s(x, A) = x√2AH2n+s−1(x, A)−2(s−1)H2n+s−2(x, A)−4nH2n+s−2(x, A).
(2.11)
4 EMILIO DEFEZ∗AND MICHAEL M. TUNG[
Multiplying (2.11) by tn/n! with |t|<1/16, and applying the sum from n= 1 to
infinity, we deduce
X
n≥1
H2n+s(x, A)
n!tn=√2AX
n≥1
H2n+s−1(x, A)
n!tn−2(s−1) X
n≥1
H2n+s−2(x, A)
n!tn
−4X
n≥1
nH2n+s−2(x, A)
n!tn.(2.12)
We can rewrite (2.12) in the form
A(s;x, t, A) = Hs(x, A)−x√2AHs−1(x, A)−2(s−1)Hs−2(x, A) (2.13)
+x√2AA(s−1; x, t, A)−2(s−1)A(s−2; x, t, A)−4tA(s;x, t, A).
The first three terms of equation (2.13) reduce to Hs(x, A)−x√2AHs−1(x, A)−
2(s−1)Hs−2(x, A) = Θ by using recurrence (2.1) for m=s. Next, we can simplify
expression (2.13) to
(1 + 4t)A(s;x, u, A) = x√2AA(s−1; x, u, A)−2(s−1) A(s−2; x, u, A).(2.14)
Using the induction hypothesis, we have
A(s−1; x, u, A) =
Hs−1x
√1+4t, A
√1+4tsexp 2tx2
1+4tA
A(s−2; x, u, A) =
Hs−2x
√1+4t, A
√1+4ts−1exp 2tx2
1+4tA
.(2.15)
Substituting (2.15) in (2.14), one finally arrives at
(1 + 4t)A(s;x, u, A) = x√2A
Hs−1x
√1+4t, A
√1+4tsexp 2tx2
1+4tA
−2(s−1)
Hs−2x
√1+4t, A
√1+4ts−1exp 2tx2
1+4tA
=
exp 2tx2
1+4tA
√1+4ts−1"x√2A
√1+4tHs−1x
√1+4t, A−2(s−1)Hs−2x
√1+4t, A#
=
Hsx
√1+4t, A
√1+4ts−1exp 2tx2
1+4tA,
which completes this proof.
Remark 2.1.Working with the series A(s;x, t, A) and its new results established
so far, we may continue to derive additional, previously unknown relations and
properties of the Hermite matrix series, altogether absent from the literature on
special functions.
A NEW TYPE OF HERMITE MATRIX POLYNOMIAL SERIES 5
For example, considering the combinations A(s;x, t, A) + A(s;x, −t, A) and also
A(s;x, t, A)− A(s;x, −t, A), for s∈N0, x ∈R,|t|<1/16 it immediately follows
that
X
n≥0
H4n+s(x, A)
(2n)! t2n=
Hsx
√1+4t, A
2√1+4ts+1 exp 2tx2
1+4tA
+
Hsx
√1−4t, A
2√1−4ts+1 exp −2tx2
1−4tA,
X
n≥0
H4n+s+2(x, A)
(2n+ 1)! t2n+1 =
Hsx
√1+4t, A
2√1+4ts+1 exp 2tx2
1+4tA
−
Hsx
√1−4t, A
2√1−4ts+1 exp −2tx2
1−4tA.
(2.16)
Considering now the combinations A(s+ 1; x, t, A) + A(s+ 1; x, −t, A) and also
A(s+1; x, t, A)−A(s+ 1; x, −t, A), for s∈N0, x ∈R, and |t|<1/16, it immediately
follows that
X
n≥0
H4n+s+1(x, A)
(2n)! t2n=
Hs+1 x
√1+4t, A
2√1+4ts+2 exp 2tx2
1+4tA
+
Hs+1 x
√1−4t, A
2√1−4ts+2 exp −2tx2
1−4tA,
X
n≥0
H4n+s+3(x, A)
(2n+ 1)! t2n+1 =
Hs+1 x
√1+4t, A
2√1+4ts+2 exp 2tx2
1+4tA
−
Hs+1 x
√1−4t, A
2√1−4ts+2 exp −2tx2
1−4tA.
(2.17)
3. The scalar Hermite polynomial series revisited
Clearly, all newly proposed relations for the Hermite matrix polynomials also
subsume the conventional scalar case. This puts us into the position to easily
recover the scalar formula (3.1) derived by the authors of [7] within the context of
quantum mechanics via what they call the quantum mechanical operator-Hermite
polynomial method.
6 EMILIO DEFEZ∗AND MICHAEL M. TUNG[
From a purely mathematical standpoint, we can deduce the following result from
Theorem 2.1:
Corollary 3.1.Let {Hn(x)}n≥0be the usual sequence of scalar Hermite polynomi-
als. Then for s∈N0, x ∈R,|t|<1/4:
X
n≥0
H2n+s(x)
n!tn=
Hsx
√1+4t
√1+4ts+1 exp 4tx2
1+4t,(3.1)
X
n≥0
H4n+s(x)
(2n)! t2n=
Hsx
√1+4t
2√1+4ts+1 exp 4tx2
1+4t+
Hsx
√1−4t
2√1−4ts+1 exp −4tx2
1−4t,
X
n≥0
H4n+s+2(x)
(2n+ 1)! t2n+1 =
Hsx
√1+4t
2√1+4ts+1 exp 4tx2
1+4t−
Hsx
√1−4t
2√1−4ts+1 exp −4tx2
1−4t,
X
n≥0
H4n+s+1(x)
(2n)! t2n=
Hs+1 x
√1+4t
2√1+4ts+2 exp 4tx2
1+4t+
Hs+1 x
√1−4t
2√1−4ts+2 exp −4tx2
1−4t,
X
n≥0
H4n+s+3(x)
(2n+ 1)! t2n+1 =
Hs+1 x
√1+4t
2√1+4ts+2 exp 4tx2
1+4t−
Hs+1 x
√1−4t
2√1−4ts+2 exp −4tx2
1−4t.
(3.2)
Proof 3.1.It is well known that when A= 2 with matrix dimension r= 1, the
Hermite matrix polynomials Hn(x, A) reduce to their scalar counterparts, the usual
Hermite polynomials Hn(x). After substituting these values into formulas (2.5),
(2.16) and (2.17), we recover formulas (3.1)–(3.2).
What remains is to prove the change in the interval of convergence, having
|t|<1/4 instead of |t|<1/16. For the scalar case, we use the following bound
derived by Cramer [11],
|Hn(x)| ≤ K2n/2√n!ex2/2, K = 1.086435,(3.3)
to finally obtain the interval of convergence |t|<1/4.
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∗Instituto de Matem´
atica Multidisciplinar, Universitat Polit`
ecnica de Val`
encia,
Camino de Vera, s/n, 46022 Valencia, Spain
E-mail address:edefez@imm.upv.es
[Instituto de Matem´
atica Multidisciplinar, Universitat Polit`
ecnica de Val`
encia,
Camino de Vera, s/n, 46022 Valencia, Spain
E-mail address:mtung@imm.upv.es
