# A Note on Certain Modular Equations about Infinite Products of Ramanujan

Authors: Hong-Cun Zhai
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

Ramanujan proposed additive formulae of theta functions that are related to modular equations about infinite products. Employing these formulaes, we derived some identities on infinite products. In
the same spirit, we also could present elementary and simple proofs of certain Ramanujan's
modular equations on infinite products.


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## Body

## 1. Introduction

Q
-theory is undoubtedly one of the most famous and useful mathematical theorems, such as Andrews-Askey type integral [1]
(1)∫cd‍(qt/c,qt/d,ft,rst;q)∞(at,bt,et,st;q)∞ϕ23[r,bt,cgrst,bc;q,sg]dqt=d(1-q)(q,c/d,qd/c,abcd,df,rsd;q)∞(ac,bc,ad,bd,sd,de;q)∞×∑k=0∞‍(f/e,ad,sd,bd;q)k(ce)k(q,df,rsd,abcd;q)k×ϕ23[r,bdqk,acdqkgrsdqk,abcdqk;q,sg],
Askey-Roy type integral [2]
(2)∫-ππ((qeiθ(cρ)Pn(eiθ,f)Pm(eiθ,g)i×(ρeiθd,qde-iθ,ρce-iθ,qeiθ(cρ);q)∞)i×((aeiθ,beiθ,ce-iθ,de-iθ;q)∞)-1∑)dθ=2π(af;q)n(bg;q)m(abcd,ρc/d,dq/(ρc),ρ,q/ρ;q)∞anbm(q,bc,bd,ac,ad;q)∞×∑k=0n‍(q-n,ac,ad;q)kqk(q,af,abcd;q)kϕ23[q-m,bc,bdbg,abcdqk;q,q],
Moment integrals [3]
(3)∫-∞∞‍Pn(w,c)Pm(w,d)(aw,bw;q)∞dα(s,t)(w)=(ac;q)n(bd;q)m(abst;q)∞anbm(as,at,bs,bt;q)∞×∑k=0n‍(q-n,as,at;q)kqk(q,ac,abst;q)kϕ23[q-m,bs,btbd,abstqk;q,q],
(wherePn(a,b)=(a-b)⋯(a-bqn-1)),q-Fractional Calculus Equations [4] andq-Calculus [5]. For more information, please refer to [1–5].The theta functions are very useful tool in researchingq-series, especially in dealing with the form of the equation similar to above formulas, whose left-hand side is summation and right-hand side is integral. The additive identities of theta are one of the important of Ramanujan's contributions. Using it, we gave elementary and simple proofs of certain Ramanujan's modular equations on infinite products. For more information, please refer to [1–7].In his notebook [8, pages 34–38], Ramanujan defines the following theta functions:
(4)f(a,b):=∑n=-∞∞‍an(n+1)/2bn(n-1)/2=(-a;ab)∞(-b;ab)∞(ab;ab)∞,|ab|<1,(5)φ(q):=f(q,q)=∑n=-∞∞‍qn2=(-q;-q)∞(q;-q)∞,(6)ψ(q):=f(q,q3)=∑n=1∞‍qn(n-1)/2=(q2;q2)∞(q;q2)∞,
where
(7)(a;q)∞:=∏n=0∞‍(1-aqn),|q|<1,
sometimes written as
(8)(a,b,c,…;q)∞=(a;q)∞(b;q)∞(c;q)∞⋯.
The infinite products are from the Jacobi triple product identity [8, page 35].In the course of deduction, we used the following simple fact [9, 10]:
(9)(ak;qk)∞=(a,aωk,…,aωkk-1;q)∞,(a,q)∞=(a,aq,aq2,…,aqk-1;qk)∞.
By definition of Ramanujan theta functions one can easily verify the following identities [8, page 45]:
(10)f(a,b)+f(-a,-b)=2f(a3b,ab3),f(a,b)-f(-a,-b)=2af(ba,aba4b4).
From (4), ifab=cd=p, we have
(11)f(a,b)f(c,d)=∑m,n=-∞∞‍p(m2+n2)/2-(m+n)/2amcn.
Thus settingm-n=2jandm+n=2k, we find that
(12)f(a,b)f(c,d)+f(-a,-b)f(-c,-d)=∑n,m=-∞m,n,even+∞‍p(m2+n2)/2-(m+n)/2amcn=2∑j,k=-∞∞‍pj2+k2-kaj+kck-j=2∑j,k=-∞∞‍pk(k-1)(ac)kpj(j-1)(bc)-j=2f(ac,bd)f(ad,bc).
Thus whenab=cd, we have
(13)f(a,b)f(c,d)+f(-a,-b)f(-c,-d)=2f(ac,bd)f(ad,bc).
Similarly we have that
(14)f(a,b)f(c,d)-f(-a,-b)f(-c,-d)=2af(bc,abcdb/c)f(bd,abcdb/d).
The special case of these identities can be written as the following form by using Jacobian theta function [6, 7]:
(15)2θ1(x+y∣2τ)θ1(x-y∣2τ)=θ3(y∣τ)θ4(x∣τ)-θ3(x∣τ)θ4(y∣τ).
The authors of [6, 7] give simple proofs and very important use of it.In the above two identities, puttingc=aandd=b, we easily obtain
(16)f2(a,b)+f2(-a,-b)=2f(a2,b2)φ(ab),(17)f2(a,b)-f2(-a,-b)=4af(ba,aba2b2)ψ(a2b2).


## 2. Main Results

The sums and products of infinite are used in many domains of mathematics, such as Partition Functions [11–14], Fractal Geometry [9], Fractional Calculus [10], Fractal Time Series [4], and so on. Then the equations of it are concentrated by several mathematicians and engineers [15–18]. At the same time, it can be used in dynamic equations, differential equations [19], and partial differential equations [20].This paper has two main purposes. The first is to derive the identities as follows: for|q|<1,
(18)(q;q)∞2(q3;q3)∞2+3(-q;q)∞2(-q3;q3)∞2=4ω(q;q)∞2(q3;q3)∞2f(qω,q3ω2)f(qω2,q3ω)f2(ω,qω2),(19)(q;q)∞2(q3;q3)∞2-3(-q;q)∞2(-q3;q3)∞2=2(q;q)∞2(q3;q3)∞2f(ω2,qω)f2(ω,qω2)φ(q),(20)(q;q)∞2(q5;q5)∞2-5(-q;q)∞2(-q5;q5)∞2=4ζ(q;q)∞2(q5;q5)∞2×f(qζ2,qζ3)f(ζ,q2ζ4)f(ζ3,q2ζ2)f(qζ,qζ4)f2(ζ,qζ4)f2(ζ2,qζ3),
in whichω=exp(2iπ/3)andζ=exp(2iπ/5). In the same way, we are able to give the simple and elementary proofs of the following identities of Ramanujan [8, 11, 12]:
(21)(-q;q)∞2(-q3;q3)∞2+(q;q)∞2(q3;q3)∞2=2(-q3;q6)∞2(q3;q6)∞2(-q2,-q4;q6)∞,(22)(-q;q)∞2(-q3;q3)∞2-(q;q)∞2(q3;q3)∞2=4q(-q6;q6)∞2(q3;q6)∞2(-q,-q5;q6)∞,(23)(-q;q)∞2(-q5;q5)∞2-(q;q)∞2(q5;q5)∞2=4q(-q,-q2,-q3,-q4,-q6,-q7,-q8,-q9;q10)∞(q5;q10)∞4.


## 3. Modular Equations of Infinite Productions

In this section, we first give the two sets refinement about the identities (18) and (20).Theorem 1.
For|q|<1,
(24)(q;q)∞(q3;q3)∞-i3(-q;q)∞(-q3;q3)∞=2ω(q;q)∞(q3;q3)∞f(qω,q3ω2)f(ω,qω2),(q;q)∞(q3;q3)∞+i3(-q;q)∞(-q3;q3)∞=2(q;q)∞(q3;q3)∞f(qω2,q3ω)f(ω,qω2).Proof.
Note that1+ω+ω2=0andω-ω2=i3. By (9), we get that
(25)i3(-q;q)∞(-q3;q3)∞÷(q;q)∞(q3;q3)∞=(ω-ω2)(qω;q)∞(qω2;q)∞(-qω;q)∞(-qω2;q)∞=ω(1-ω)(1+ω)(qω;q)∞(qω2;q)∞(1+ω)(-qω;q)∞(-qω2;q)∞=-(ω;q)∞(qω2;q)∞(-ω;q)∞(-qω2;q)∞=-f(-ω,-qω2)f(ω,qω2).
From (10), we have
(26)f(ω,qω)-f(-ω,-qω)=2f(qω2,q3ω),(27)f(ω,qω)+f(-ω,-qω)=2ωf(qω,q3ω2).
Dividing byf(ω,qω), respectively, and then applying (25), we derive
(28)i3(-q;q)∞(-q3;q3)∞÷(q;q)∞(q3;q3)∞-1=-1-f(-ω,-qω2)f(ω,qω2)=-2ωf(qω,q3ω2)f(ω,qω2),(29)i3(-q;q)∞(-q3;q3)∞÷(q;q)∞(q3;q3)∞+1=1-f(-ω,-qω2)f(ω,qω2)=2f(qω2,q3ω)f(ω,qω2).
Multiplying by(q;q)∞/(q3;q3)∞, respectively, we complete the proofs of (24).Proof of (19).
Letα=exp(iπ/6); then it is easy to know that ω=iαandα+1/α=-3.
One has(30)3(-q;q)∞(-q3;q3)∞÷(q;q)∞(q3;q3)∞=-(α+1α)(qω;q)∞(qω2;q)∞(-qω;q)∞(-qω2;q)∞=-1α(1+α2)(qω;q)∞(qω2;q)∞(-qω;q)∞(-qω2;q)∞=-1α(1-iα)(1+iα)(qω;q)∞(qω2;q)∞(-qω;q)∞(-qω2;q)∞=-1α(1-ω)(1+ω)(qω;q)∞(qω2;q)∞(-qω;q)∞(-qω2;q)∞=-1α(1+ω)2(ω;q)∞(qω2;q)∞(-ω;q)∞(-qω2;q)∞=-ωαf(-ω,-qω2)f(ω,qω2)=-if(-ω,-qω2)f(ω,qω2).
Then we obtain that(31)1+3(-q;q)∞(-q3;q3)∞÷(q;q)∞(q3;q3)∞=1-if(-ω,-qω2)f(ω,qω2),1-3(-q;q)∞(-q3;q3)∞÷(q;q)∞(q3;q3)∞=1+i(-ω,-qω2)f(ω,qω2).
In (16), leta=ωandb=qω2then we have that
(32)f2(ω,qω2)+f2(-ω,-qω2)=2f(ω2,q2ω)φ(q).
Dividing byf2(ω,qω), respectively, we arrive at
(33)1+f2(-ω,-qω2)f2(ω,qω2)=2f(ω2,q2ω)φ(q)f2(ω,qω2).
Multiplying (31), combining with (33), and then multiplyed by(q;q)∞2/(q3;q3)∞2, we are able to obtain (19).Theorem 2.
For|q|<1,
(34)(q;q)∞(q5;q5)∞-5(-q;q)∞(-q5;q5)∞=2ζ(q;q)∞(q5;q5)∞f(qζ2,qζ3)f(ζ,q2ζ4)f(ζ,qζ4)f(ζ2,qζ3),(35)(q;q)∞(q5;q5)∞+5(-q;q)∞(-q5;q5)∞=2(q;q)∞(q5;q5)∞f(ζ3,q2ζ2)f(qζ,qζ4)f(ζ,qζ4)f(ζ2,qζ3).Proof.
First we recall thatζ5=1, ζ+ζ4-ζ2-ζ3=5 and 1+ζ+ζ2+ζ3+ζ4=0. Using (9), we have
(36)5(-q;q)∞(-q5;q5)∞÷(q;q)∞(q5;q5)∞=(ζ+ζ4-ζ2-ζ3)×(qζ;q)∞(qζ2;q)∞(qζ3;q)∞(qζ4;q)∞(-qζ;q)∞(-qζ2;q)∞(-qζ3;q)∞(-qζ4;q)∞=(ζ(1-ζ)(1-ζ2)(1+ζ)(1+ζ2)c×(qζ;q)∞(qζ2;q)∞(qζ3;q)∞(qζ4;q)∞)×((1+ζ)(1+ζ2)(-qζ;q)∞×(-qζ2;q)∞(-qζ3;q)∞(-qζ4;q)∞)-1=-f(-ζ,-qζ4)f(-ζ2,-qζ3)f(ζ,qζ4)f(ζ2,qζ3).
Then we know easily that
(37)1+5(-q;q)∞(-q5;q5)∞÷(q;q)∞(q5;q5)∞=1-f(-ζ,-qζ4)f(-ζ2,-qζ3)f(ζ,qζ4)f(ζ2,qζ3),1-5(-q;q)∞(-q5;q5)∞÷(q;q)∞(q5;q5)∞=1+f(-ζ,-qζ4)f(-ζ2,-qζ3)f(ζ,qζ4)f(ζ2,qζ3).
In (13) and (14), settinga=ζ, b=qζ4, c=ζ2, andd=qζ3, we get that
(38)f(ζ,qζ4)f(ζ2,qζ3)+f(-ζ,-qζ4)f(-ζ2,-qζ3)=2f(ζ3,q2ζ2)f(qζ4,qζ),f(ζ,qζ4)f(ζ2,qζ3)-f(-ζ,-qζ4)f(-ζ2,-qζ3)=2ζf(qζ2,qζ3)f(ζ,q2ζ4).
Dividing the above two equations byf(ζ,qζ4)f(ζ2,qζ3), respectively, and then combining with (37), we obtain that
(39)1+5(-q;q)∞(-q5;q5)∞÷(q;q)∞(q5;q5)∞=2f(ζ3,q2ζ2)f(qζ,qζ4)f(ζ,qζ4)f(ζ2,qζ3),(40)1-5(-q;q)∞(-q5;q5)∞÷(q;q)∞(q5;q5)∞=2ζf(qζ2,qζ3)f(ζ,q2ζ4)f(ζ,qζ4)f(ζ2,qζ3).
Multiplied by(q;q)∞/(q5;q5)∞, the identities (39) and (40) become (34) and (35).
Multiplying the two refinements in Theorems1 and 2, respectively, we obtain the identities (18) and (20). Using the same method, we can obtain refinement identities of (21) and (22) which are similar to Theorems 1 and 2; then we can deduce (21), (22), and (23) easily. The details of proofs are omitted.The following conclusion can be obtained easily.Corollary 3.
For|q|<1,
(41)(q;q)∞4(q3;q3)∞4-9(-q;q)∞4(-q3;q3)∞4=8ω(q;q)∞4(q3;q3)∞4×f(ω2,qω)f(qω,q3ω2)f(qω2,q3ω)f4(ω,qω2)φ(q).



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