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A new look on the generating function for the number of divisors

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Abstract

The q-binomial coefficients are specializations of the elementary symmetric functions. In this paper, we use this fact to give a new expression for the generating function of the number of divisors. As corollaries, we obtained new connections between partitions and divisors.
Journal of Number Theory 149 (2015) 57–69
Contents lists available at ScienceDirect
Journal of Number Theory
www.elsevier.com/locate/jnt
A new look on the generating function
for the number of divisors
Mircea Merca
Department of Mathematics, University of Craiova, A.I. Cuza 13, Craiova,
200585, Romania
a r t i c l e i n f o a b s t r a c t
Article history:
Received 31 August 2014
Received in revised form 1 October
2014
Accepted 2 October 2014
Available online 5 December 2014
Communicated by David Goss
MSC:
11A25
11P81
11P84
05A17
05A19
Keywords:
Divisors
Lambert series
Partitions
The q-binomial coefficients are specializations of the elemen-
tary symmetric functions. In this paper, we use this fact to
give a new expression for the generating function of the num-
ber of divisors. As corollaries, we obtained new connections
between partitions and divisors.
© 2014 Elsevier Inc. All rights reserved.
1. Introduction
Any series of the form
n=1
an
qn
1qn,|q|<1,
E-mail address: mircea.merca@profinfo.edu.ro.
http://dx.doi.org/10.1016/j.jnt.2014.10.009
0022-314X/© 2014 Elsevier Inc. All rights reserved.
58 M. Merca / Journal of Number Theory 149 (2015) 57–69
where the an(n =1, 2, ...)are real numbers is called a Lambert series. These series
are well known class of series in analytic function theory and number theory and are
mentioned in the classical texts by Abramowitz and Stegun [1, pp. 826–827], Bromwich
[7, pp. 102–103], Chrystal [8, pp. 345–346], Hardy and Wri g ht [13, pp. 257–258], Knopp
[18, pp. 448–452], MacMahon [24, pp. 26–32], Pólya and Szegő [28, pp. 125–129], and
Titchmarsh [30, pp. 160–161]. Lambert series have been elegantly used in a varie ty of
contexts of Ramanujan’s research works. The dimension provided by Ramanujan inspired
Andrews and Berndt [5] to prove a lot of identities given by Ramanujan.
Lambert series are natural generalizations of the following formula related to the
theory of numbers:
n=1
qn
1qn=
n=0
τ(n)qn,|q|<1.(1)
In multiplicative number theory, the divisor function τ(n)is defined as the number of
divisors of n, unity and nitself included, i.e.,
τ(n)=
d|n
1.
We use the convention that τ(n) =0for n 0.
Due to Clausen’s [9], we have the following identity:
n=1
qn
1qn=
n=1
1+qn
1qnqn2,|q|<1.(2)
In this paper, motivated by these results, we shall prove:
Theorem 1. For |q| <1,
n=1
qn
1qn=1
(q;q)
n=1
(1)n1nqn+1
2
(q;q)n
,(3)
where
(a;q)n=(1a)(1 aq)1aq2···1aqn1
is the q-shifted factorial, with (a; q)0=1.
Some consequences of this result can be easily derived.
M. Merca / Journal of Number Theory 149 (2015) 57–69 59
Corollary 1.1. The number of divisors of ncan be expressed as a finite discrete convolu-
tion:
τ(n)=
n
k=1
a(k)p(nk),
where
a(n)=
k=1
(1)k1kq(n, k),
p(n)is the number of partitions of n, and q(n, k)denotes the number of partitions of n
into exactly kdistinct parts (q(0, k) =1and q(n, k) =0for nnegative).
Note that a(n)is in fact a finite sum, since q(n, k) =0for k>n.
Example. The a(n)sequence begins as follows for n >0,
1,1,1,1,3,0,2,1,2,1,2,4,1,1,4,2,1,3,1, ...
For n =9, we have
p(8) + p(7) p(6) p(5) 3p(4) 2p(2) + p(1) + 2p(0) = 3
and τ(9) equals 3because the three divisors in questions are: 1, 3and 9.
Corollary 1.2. Let nbe a positive integer. Then
k=1
(1)k1kq(n, k)=
k=−∞
(1)kτnk(3k1)/2.
We see that the generalized pentagonal numbers appear in this relation accompanied
by the number of divisors function:
q(n, 1) 2q(n, 2) + 3q(n, 3) 4q(n, 4) + 5q(n, 5) −···
=τ(n)τ(n1) τ(n2) + τ(n5) + τ(n7) τ(n12) −···
Surprisingly, this relation was not observed for many years.
In the final section of the paper, we derive a nice identity that combines triangular
and pentagonal numbers.
60 M. Merca / Journal of Number Theory 149 (2015) 57–69
2. Proof of Theorem 1
Being given a set of variables {x1, x2, ..., xn}, recall [23] that the kth elementary
symmetric function ek(x1, x2, ..., xn)is given by
ek(x1,x
2, ..., xn)=
1i1<i2<···<ikn
xi1xi2···xik
for k=1, 2, ..., n. We set e0(x1, x2, ..., xn) =1by convention. For k<0or k>n, we
set ek(x1, x2, ..., xn) =0.
The elementary symmetric functions are characterized by the following identity of
formal power series in t:
E(t)=
n
k=0
ek(x1,...,x
n)tk=
n
k=1
(1 + xkt)
For k=1, 2, ..., n, we consider that 1 +xkt =0. On the one hand, we have
d
dt lnE(t)=
n
k=1
d
dt ln(1 + xkt)=
n
k=1
xk
1+xkt.(4)
On the other hand, we can write
d
dt lnE(t)=n
k=1
1
1+xkt n
k=1
kek(x1,...,x
n)tk1.(5)
Thus, by (4) and (5), we derive the identity
n
k=1
xk
1+xkt=n
k=1
1
1+xkt n
k=1
kek(x1,...,x
n)tk1,(6)
where x1, x2, ..., xnand tare independent vari ables such that 1 +xkt =0for k=
1, 2, ..., n.
The q-analogue of the classical binomial coefficient is called the q-binomial coefficient
and is defined by
n
k=(q;q)n
(q;q)k(q;q)nk,if k∈{0,1,...,n},
0,otherwise.
Replacing xkby qk1in (6), we obtain
n
k=1
qk1
1+qk1t=1
(t;q)n
n
k=1
kqk
2n
ktk1,(7)
M. Merca / Journal of Number Theory 149 (2015) 57–69 61
where we have invoked the fact that
ek1,q,...,q
n1=qk
2n
k.
The case t =qof (7) can be written as
n
k=1
qk
1qk=1
(q;q)n
n
k=1
(1)k1kqk+1
2n
k.(8)
For |q| <1, we have
lim
n→∞ n
k=1
(q;q)k
and the theorem is proved.
3. Proofs of Corollaries 1.1 and 1.2
Recall [2, Theorem 13-3, p. 162] that the partition function p(n)has the generating
function
n=0
p(n)qn=1
(q;q)
,|q|<1.
In this context, the formula (3) can be written as
n=1
τ(n)qn=
n=0
p(n)qn
n=1
(1)n1nqn+1
2
(q;q)n,|q|<1.
On the other hand, the generating function of q(n, k)[4, Theorem 11.4.1, p. 559] is
given by
n=0
q(n, k)qn=qk+1
2
(q;q)k
,|q|<1.
For |q| <1, we have
k=1
(1)k1kqk+1
2
(q;q)k
=
k=1
n=0
(1)k1kq(n, k)qn
=
n=0
k=1
(1)k1kq(n, k)qn
=
n=1
a(n)qn
because q(0, k) =0for any positive integer k.
62 M. Merca / Journal of Number Theory 149 (2015) 57–69
Takin g into account the well-known Cauchy products of two power series
n=0
anqn
n=0
bnqn=
n=0n
k=0
akbnkqn,
Corollary 1.1 is proved.
For |q| <1, it is clear that the relation (3) can be written as
n=1
a(n)qn=(q;q)
n=1
τ(n)qn
=
n=−∞
(1)nqn(3n1)/2
n=1
τ(n)qn
=
n=1
k=−∞
(1)kτnk(3k1)/2qn,
where we have invoked the Euler pentagonal number theorem [3, Corollary 1.7, p. 11],
(q;q)=
n=−∞
(1)nqn(3n1)/2
and the Cauchy product of two power series. The proof of Corollary 1.2 is finished.
4. Other identities involving τ(n)
The aim of this section is to present new identities involving the number of divisors
function τ(n). For this purpose we use some well known partition identities.
An r-regular partition of n(r>1) is a partition of nwith no part divisible by r. The
number of r-regular partitions of nis denoted by br(n)and follows the convention that
br(0) =1. In classical representation theory, r-regular partitions of nlabel irreducible
r-modular representations of the symmetric group Snwhen ris prime [16]. More recently,
such partitions have been studied for their arithmetic properties in connection with the
theory of modular forms and Galois representations [12,19,27].
Corollary 4.1. For n >0, r>1,
k=−∞
(1)kτnr·k(3k1)/2=
n
k=1
a(k)br(nk).
M. Merca / Journal of Number Theory 149 (2015) 57–69 63
Proof. Elementary techniques in the theory of partitions [3] give the following generating
function for the number of r-regular partitions
n=0
br(n)qn=(qr;qr)
(q;q)
,|q|<1.(9)
Thus, for |q| <1we can write
qr;qr
n=1
τ(n)qn=
n=0
br(n)qn
n=1
a(n)qn
and then, using the Euler pentagonal number theorem, we get
n=−∞
(1)nqr·n(3n1)/2
n=1
τ(n)qn=
n=0
br(n)qn
n=1
a(n)qn.
Applying the Cauchy product of two power series, we arrive at our conclusion. 2
The case r=2of this corollary is given by
k=−∞
(1)kτnk(3k1)=
n
k=1
a(k)q(nk),
were q(n) denotes the number of partitions of nwith distinct parts. Recall [3, Corol-
lary 1.2, p. 5] that the number of partitions of nwith distinct parts is equal to the
number of partitions of nwith odd parts.
Example. For n =9, we have
q(8) +q(7) q(6) q(5) 3q(4) 2q(2) + q(1) + 2q(0) = τ(9) τ(7) τ(5).
Corollary 4.2. For n >0,
k=0
τnk(k+1)/2=
i+j+k=n
a(i)q(j)q(k).(10)
Proof. By the Jacobi triple product identity [6, Theorem 11],
q2;q2qx;q2q/x;q2=
n=−∞
xnqn2,|q|<1,x=0,
with xreplaced by q, we get
q2;q2q2;q22
=
n=0
qn2+n,|q|<1.
64 M. Merca / Journal of Number Theory 149 (2015) 57–69
Replacing q2by q, we have
(q;q)(q;q)2
=
n=0
qn(n+1)/2,|q|<1.
Thus, we deduce that
n=1
a(n)qn
n=0
q(n)qn2
=
n=0
qn(n+1)/2
n=1
τ(n)qn,|q|<1.
Equating coefficients of qnon each side give the result. 2
Let q(n) denote the number of partitions of ninto odd parts, each part being of two
kinds. For instance, for n =4, we consider kand kto be different versions of kand
so we have: 3 +1, 3 +1
, 3+1, 3+1
, 1 +1 +1 +1, 1 +1 +1 +1
, 1 +1 +1
+1
,
1 +1
+1
+1
and 1+1
+1
+1
. Thus q(4) =9. According to Sloane [29, A022567],
the generating function for q(n)is given by
n=0
q(n)qn=(q;q)2
,|q|<1.
In this context, the identity (10) can be written as
k=0
τnk(k+1)/2=
n
k=1
a(k)q(nk),n>0.
Corollary 4.3. For n >0,
k=0
(1)k(2k+1)τnk(k+1)/2=
i+j+k=n
a(i)e(j)e(k),
where
e(n)=qe(n)qo(n),
and qe(n), respectively qo(n)denotes the number of partitions of ninto an even, respec-
tively an odd number of distinct parts.
Proof. To prove the corollary, we consider another special case of the Jacobi triple prod-
uct identity [4, Corollary 10.4.2, p. 500],
(q;q)3
=
n=0
(1)n(2n+1)qn(n+1)/2,|q|<1.
M. Merca / Journal of Number Theory 149 (2015) 57–69 65
By Andrews and Eriksson [6, Theorem 4, p. 25], we have
e(n)=(1)k,if n=k(3k±1)/2,
0,otherwise.
In other words, e(n)is the coefficient of qnin the Euler function (q; q). For |q| <1, we
deduce that
n=1
a(n)qn
n=0
e(n)qn2
=
n=0
(1)n(2n+1)qn(n+1)/2
n=1
τ(n)qn
and the proof follows easily. 2
Corollary 4.4. For n >0,
k=−∞
τnk2=
i+j+k+l=n
a(i)q(j)ps(k)ps(l)
where ps(n)denotes then number of self-conjugate partitions of n.
Proof. By the Jacobi triple product identity, with xreplaced by 1, we have
n=−∞
qn2=(q;q)(q;q)q;q22
,|q|<1.(11)
Recall [6, Eq. (3.4), p. 18] that the number of self-conjugate partitions of nis equal to
the number of partitions of ninto odd distinct parts. So the generating function of ps(n)
is given by
n=0
ps(n)qn=q;q2,|q|<1.
For |q| <1, we get
n=−∞
qn2
n=1
τ(n)qn=
n=1
a(n)qn
n=1
q(n)qn
n=1
ps(n)qn2
.
Equating coefficients of qnon each side give the result. 2
An overpartition of the nonnegative integer nis a partition of nwhere the first occur-
rence of parts of each size may be overlined. Let ¯p(n) denote the number of overpartitions
of n. For example, the overpartitions of the integer 3 are:
3,¯
3,2+1,¯
2+1,2+¯
1,¯
2+¯
1,1+1+1 and ¯
1+1+1.
66 M. Merca / Journal of Number Theory 149 (2015) 57–69
We see that ¯p(3) =8. Properties of ¯p(n)have been the subject of many recent studies [10,
14,17,20–22,25]. Recently, Hirschhorn and Sellers [15] studied the arithmetic properties
of overpartitions using only odd parts. The number of overpartitions into odd parts is
denoted by ¯q(n)and the generating function for ¯q(n)is given by
n=0
¯q(n)qn=(q;q)q;q2,|q|<1.
Takin g into account the identity (11), we obtain
k=−∞
τnk2=
i+j+k=n
a(iq(j)ps(k).
We remark that ¯q(n)is equal to the number of partitions of 2nin which all odd parts
occur with multiplicity 2and the even parts occur with multiplicity 1[29, A080054].
Corollary 4.5. For n >0,
k=−∞
(1)kτnk2=
i+j+k+l=n
a(i)q(j)f(k)f(l)
where
f(n)=pe(n)po(n),
and pe(n), respectively po(n)denotes the number of partitions of ninto even, respectively
odd number of parts.
Proof. We take into account the case x =1of the Jacobi triple product identity,
n=−∞
(1)nqn2=(q;q)(q;q)q;q22
,|q|<1,
and the fact that the generating function of f(n)is given by [11, p. 38, Eq. (22.14)]
n=0
f(n)qn=(q;q)1
=q;q2,|q|<1.2
Corollary 4.6. For n >0,
k=−∞
(6k+1)τnk(3k+1)/2=
i+j+k+l+m=n
a(i)e(j)e(k)f(l)f(m)
M. Merca / Journal of Number Theory 149 (2015) 57–69 67
where
e(n)=qe(n)qo(n)and f(n)=pe(n)po(n).
Proof. Considering the identity [4, p. 545, Exercise 14]
n=−∞
(6n+1)qn(3n+1)/2=(q;q)3
q;q22
,
the proof follows easily. 2
5. Concluding remarks
A new factorization has been introduced in this paper for the generating function for
number of divisors,
n=1
τ(n)qn=1
(q;q)
n=1
(1)n+1 nqn+1
2
(q;q)n
.
As a corollary of this result, we obtained formulas that combine the number of divisors
function τ(n)and the number of partitions of ninto exactly kdistinct parts.
We remark that
q(n, k)=pknk(k+1)/2,(12)
where pk(n) denotes the number of partitions of nwith no part greater than k. This
follows from
n=1
k=1
(1)k1kq(n, k)qn=
k=1
(1)k1kqk+1
2
(q;q)k
=
k=1
(1)k1kqk+1
2
n=0
pk(n)qn
=
k=1
n=0
(1)k1kpk(n)qn+k(k+1)/2
=
n=1
k=1
(1)k1kpknk(k+1)/2qn,
where we have invoked the generating function of pk(n)[2, Theorem 13-1, p. 161],
n=0
pk(n)qn=1
(q;q)k
,|q|<1
and the fact that pk(n) =0for nnegative.
68 M. Merca / Journal of Number Theory 149 (2015) 57–69
Moreover, the identity (12) has a simple combinatorial proof. We start from a partition
of ninto kdistinct parts and then we subtract a staircase of size k(i.e. subtract kto the
largest part, k1to the second largest one, etc., and 1to the smallest parts). The result
is a partition of n k(k+1)/2into at most kparts. Conjugating, we get a partition of
n k(k+1)/2with no part greater than k.
Thus we can write the following identity [26]
p1(n1) 2p2(n3) + 3p3(n6) 4p4(n10) + 5p5(n15) −···
=τ(n)τ(n1) τ(n2) + τ(n5) + τ(n7) τ(n12) −···
in which triangular and pentagonal numbers appear together.
For 0 <q<1, we have
n
k=1
qk
1qk=
n
k=1
qk1+qk+q2k+q3k+···
>
n
k=1
τ(k)qk.
The following inequality is immediate from (8).
Corollary 5.1. Let nbe a positive integer. For 0 <q<1,
n
k=1
τ(k)qk<1
(q;q)n
n
k=1
(1)k+1kqk+1
2n
k.
Acknowledgments
The author appreciates the anonymous referees for their comments on the original
version of this paper. Special thanks go to Dr. Oana Merca for the careful reading of the
manuscript and helpful remarks.
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Chapter
Let \(\alpha \) and \(\beta \) be two nonnegative integers such that \(\beta < \alpha \). For an arbitrary sequence \(\{a_n\}_{n\geqslant 1}\) of complex numbers, we investigate linear combinations of the form \(\sum _{k\geqslant 1} S(\alpha k-\beta ,n) a_k\), where S(k, n) is the total number of k’s in all the partitions of n into parts not congruent to 2 modulo 4. The general nature of the numbers \(a_n\) allows us to provide new connections between partitions and functions from multiplicative number theory.
... An excellent treatment of some surprising connections between partitions and divisors can be seen in [5]. Other connections between partitions and divisors have been recently published in [3,4]. ...
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In this paper, the author gives two linear homogeneous recurrence relations to compute the values of Euler’s partition function p(n) when n is odd. These recurrences do not involve the generalized pentagonal numbers and provide new connections between the partition function p(n) and the positive divisor function σx(n) for x = 0 and x = 1. Some relations involving the divisor functions .σx(n) and the parity of the partition function p(n) are presented in this context.
... The results proved in this article continue the spirit of [10,11,12,14,15,16,17,18,20] by connecting the seemingly disparate branches of additive and multiplicative number theory. For further reading at the intersection of the additive and multiplicative branches of number theory, we recommend [1], [5,Ch. ...
... The results proved in this article continue the spirit of [10,11,12,14,15,16,17,18,20] by connecting the seemingly disparate branches of additive and multiplicative number theory. For further reading at the intersection of the additive and multiplicative branches of number theory, we recommend [1], [5, Ch. ...
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Let $\alpha$ and $\beta$ be two nonnegative integers such that $\beta < \alpha$. For an arbitrary sequence $\{a_n\}_{n\geqslant 1}$ of complex numbers, we consider the generalized Lambert series in order to investigate linear combinations of the form $\sum_{k\geqslant 1} S(\alpha k-\beta,n) a_k$, where $S(k,n)$ is the total number of non-overlined parts equal to $k$ in all the overpartitions of $n$. The general nature of the numbers $a_n$ allows us to provide connections between overpartitions and functions from multiplicative number theory.
... Note that this article continues the spirit of [2,[10][11][12][14][15][16]18,19] by connecting several famous special multiplicative functions with divisor sums over partitions which are additive in nature. ...
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We consider the number of k’s in all the partitions of n, and provide new connections between partitions and functions from multiplicative number theory. In this context, we obtain a new generalization of Euler’s pentagonal number theorem and provide combinatorial interpretations for some special cases of this general result.
... A related result was given by Merca [8]. In 2015, he proved ...
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Let $$ T(q)=\sum_{k=1}^\infty d(k) q^k, \quad |q|<1, $$ where $d(k)$ denotes the number of positive divisors of the natural number $k$. We present monotonicity properties of functions defined in terms of $T$. More specifically, we proved that $$ H(q) := T(q)- \frac{\log(1-q)}{\log(q)} $$ is strictly increasing in $ (0,1) $ while $$ F(q) := \frac{1-q}{q} \,H(q) $$ is strictly decreasing in $ (0,1) $. These results are then applied to obtain various inequalities, one of which states that the double-inequality $$ \alpha \,\frac{q}{1-q}+\frac{\log(1-q)}{\log(q)} < T(q)< \beta \,\frac{q}{1-q}+\frac{\log(1-q)}{\log(q)}, \quad 0<q<1, $$ holds with the best possible constant factors $\alpha=\gamma$ and $\beta=1$. Here, $\gamma$ denotes Euler's constant. This refines a result of Salem, who proved the inequalities with $\alpha=1/2$ and $\beta=1$.
... We see thatp(3) = 8. The properties of the overpartitions functionp(n) have been the subject of many recent studies [3,8,13,15,[17][18][19][21][22][23][24][25]. The partition functions p(n),p(n) and S e−o (n) are connected to r 4 (n) by the following result. ...
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We consider the function \(r_s(n)\) which gives the number of ways to write n as the sum of s squares. Since the generating functions for \(r_4(n)\) and \(r_8(n)\) are Lambert series, we use Merca’s factorization theorem for Lambert series to establish relationships between these functions and partitions into distinct parts. We also obtain convolutions involving overpartition functions as well as pentagonal recurrence formulas for \(r_4(n)\) and \(r_8(n)\). These results lead to new connections between divisors and partitions.
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We obtain a finite analogue of a recent generalization of an identity in Ramanujan's Notebooks. Differentiating it with respect to one of the parameters leads to a result whose limiting case gives a finite analogue of Andrews' famous identity for $\textup{spt}(n)$. The latter motivates us to extend the theory of the restricted partition function $p(n, N)$, namely, the number of partitions of $n$ with largest parts less than or equal to $N$, by obtaining the finite analogues of rank and crank for vector partitions as well as of the rank and crank moments. As an application of the identity for our finite analogue of the spt-function, namely $\textup{spt}(n, N)$, we prove an inequality between the finite second rank and crank moments. The other results obtained include finite analogues of a recent identity of Garvan, an identity relating $d(n, N)$ and lpt$(n, N)$, namely the finite analogues of the divisor and largest parts functions respectively, and a finite analogue of the Beck-Chern theorem. We also conjecture an inequality between the finite analogues of $k^{\textup{th}}$ rank and crank moments.
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Using previous work by Merca, we show the partition function involving parts of k different magnitudes, shifted by the triangular numbers, equals the self convolution of the unrestricted partition function.
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We all recognize 0, 1, 1, 2, 3, 5, 8, 13,... but what about 1, 2, 4, 6, 3, 9, 12, 8, 10, 5, 15,...? If you come across a number sequence and want to know if it has been studied before, there is only one place to look, the On-Line Encyclopedia of Integer Sequences (or OEIS). Now in its 49th year, the OEIS contains over 220,000 sequences and 20,000 new entries are added each year. This article will briefly describe the OEIS and its history. It will also discuss some sequences generated by recurrences that are less familiar than Fibonacci's, due to Greg Back and Mihai Caragiu, Reed Kelly, Jonathan Ayres, Dion Gijswijt, and Jan Ritsema van Eck.
Chapter
Let a0, a1, a2,..., an,... be complex numbers not all zero. Let the power series $$ f(z) = {a_0} + {a_1}z + {a_2}{z^2} + \cdots + {a_n}{z^n} + \cdots $$ have radius of convergence R, R>0. If R = ∞, f(z) is called an entire function. Let 0 ≦ r < R. Then the sequence $$ \left| {{a_0}} \right|,\quad \left| {{a_1}} \right|r,\quad \left| {a{}_2} \right|{r^2},\quad \cdots, \;\left| {{a_n}} \right|{r^2},\; \cdots $$ tends to 0, and hence it contains a largest term, the maximum term, whose value is denoted by μ(r). Thus $$ \left| {{a_n}} \right|{r^n}\underline \leqslant \mu (r) $$ for n = 0, 1, 2, 3,..., r ≧ 0 [I, Ch. 3, § 3].
Chapter
Each of these two isolated pages has a connection with Ramanujan’s famous paper in which he gives the first proofs of the congruences p(5n+4)≡0 (mod 5) and p(7n+5)≡0 (mod 7). One of Ramanujan’s proofs hinges upon the beautiful identity $$\sum_{n=0}^{\infty}p(5n+4)q^n = 5 \frac{(q^5;q^5)_{\i}^5}{(q;q)_{\infty}^6}, \qquad |q|<1, $$ which is given on page 189. We provide a more detailed rendition of the proof given by Ramanujan, as well as a similarly beautiful identity yielding the congruence p(7n+5)≡0 (mod 7). On both pages, Ramanujan examines the more general partition function pr(n) defined by $$ \frac{1}{(q;q)_{\infty}^r}=\sum_{n=0}^{\infty}p_r(n)q^n, \qquad |q|<1. $$ In particular, he states new congruences for pr(n).
Chapter
The Rogers-Ramanujan identities are perhaps the most important identities in the theory of partitions. They were first proved by L.J. Rogers in 1894 and rediscovered by Ramanujan prior to his departure for England. Since that time, they have inspired a huge amount of research, including many analogues and generalizations. Published with the lost notebook is a manuscript providing 40 identities satisfied by these functions. In contrast to the Rogers-Ramanujan identities, the identities in this manuscript are identities between the two Rogers-Ramanujan functions at different powers of the argument. In other words, they are modular equations satisfied by the functions. The theory of modular forms can be invoked to provide proofs, but such proofs provide us with little insight, in particular, with no insight on how Ramanujan might have discovered them. Thus, for nearly a century, mathematicians have attempted to find “elementary” proofs of the identities. In this chapter, “elementary” proofs are given for each identity, with the proofs of the most difficult identities found only recently by Hamza Yesilyurt.