# Relation between Borweins’ Cubic Theta Functions and Ramanujan’s Eisenstein Series

Authors: B. R. Srivatsa Kumar; Shruthi; D. Anu Radha
Journal: Journal of Applied Mathematics
 (2021)

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## Abstract

Two-dimensional theta functions were found by the Borwein brothers to work on Gauss and Legendre’s arithmetic-geometric mean iteration. In this paper, some new Eisenstein series identities are obtained by using (p, k)-parametrization in terms of Borweins’ theta functions.


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## Body

## 1. Introduction

The seriesLq, Mq, and Nq by Eisenstein are defined below, primarily mentioned by Ramanujan in his second notebook [1].
(1)Lq≔1−24∑m=1∞mqm1−qm,(2)Mq≔1+240∑m=1∞m3qm1−qm,(3)Nq≔1−504∑m=1∞m5qm1−qm.Borwein brothers [2, 3] established the consecutive three two-dimensional theta functions on cubic; as parallel to Jacobi’s theta function identity and the arithmetic-geometric mean iteration of Guass and Legendre, they are stated as follows for ω=exp2πi/3:
aq≔∑m,n=−∞∞qm2+mn+n2,bq≔∑m,n=−∞∞ωm−nqm2+mn+n2,cq≔∑m,n=−∞∞qm+1/32+m+1/3n+1/3+n+1/32.One can easily verify thata0=b0=1 and c0=0. By using Euler binomial theorem [2, 3], Borwein brothers found an expression for bq and cq in terms of infinite products, namely,
(5)bq=f13f3,(6)cq=3q1/3f33f1,(7)a3q=b3q+c3q,wherefk is defined as
fk≔qk;qk∞=∏m=1∞1−qmk,q<1.Proofs of (5)–(7) can be found in [2, 3]. Berndt et al. [4] demonstrated the following identities by using Ramanujan’s elliptic functions in the theory of signature 3, namely,
(9)Mq≔aqa3q+8c3q,(10)Nq≔a6q−20a3qc3q−8c3q,(11)Mq3≔19aq9a3q−8c3q,(12)Nq3≔a6q−43a3qc3q+827c3q.In the current work, usingp,k-parameters introduced by Alaca et al. [5–7], the new Eisenstein series identities are obtained which connects aq, bq, and cq; these are the examples of sum-to-product identities. Using the classical theory of elliptic functions and modular equations of degree 3, Chan [8] proved (9) and (12). Moreover, Liu [9, 10] provided alternate proofs of (9)–(12) by using the simple elliptic functions. The identities of Ramanujan’s Eisenstein series Pn of weight 2 found in [10, 11] have been proved by Bhuvan [12]. Xia and Yao [11] also obtained several Eisenstein series identities that includes Borweins’ theta functions containing bq, bq2, bq4, cq, cq2, and cq4 by using (p, k)-parametrization. Further, in 2020, Shruthi and Srivatsa Kumar [13] obtained new Eisenstein series involving Borweins’ cubic theta functions. This paper is classified as follows. Section 2 is devoted for preliminary results which are required to prove our main results in Section 3.


## 2. Preliminaries

Alaca and Williams [14] ascertained the parametric representations for mLqm−Lq, Mqn, bqi, and cqi for m∈2,3,4,6,12, n∈1,2,3,6, and i∈1,2,4 in terms of parameters p and k, namely,
p=pq=φ2q−φ2q32φ2q3,k=kq=φ3q3φq, where
φq=∑n=−∞∞qn2=−q:q2∞2q2;q2∞.Sinceφ0=1, one can verify p0=0 and k0=1. The sequential parametric representations in terms of p and k given by Alaca et al. [6, 7] and Alaca and Williams [14] for Mqn for n∈1,2,3,6,12 and mLqm−Lq for m∈2,3,4,6,12 are listed below.
Mq=k41+124p+964p2+2788p3+3910p4+2788p5+964p6+124p7+p8,Mq2=k41+4p+64p2+178p3+235p4+178p5+64p6+4p7+p8,Mq3=k41+4p+4p2+28p3+70p4+28p5+4p6+4p7+p8,Mq6=k41+4p+4p2−2p3−5p4−2p5+4p6+4p7+p8,Mq12=k41+4p+4p2−2p3−5p4−2p5+1/4p6+1/4p7+1/16p8,2Lq2−Lq=k21+4p+24p2+14p3+p4,3Lq3−Lq=2k21+8p+18p2+8p3+p4,4Lq4−Lq=3k21+6p+12p2+8p3,6Lq6−Lq=k25+22p+36p2+22p3+5p4,12Lq12−Lq=k211+34p+36p2+16p3+2p4.The following parametric representations in terms of the parametersp and k are given by Alaca et al. [5].
bq=k1−p4/31+2p1/32+p1/321/3,b−q=k1−p1/31+2p4/32+p1/321/3,bq2=k1−p2/31+2p2/32+p2/322/3,bq4=k2+p4/31−p1/31+2p1/324/3,cq=3k1+p4/3p1/321/3,c−q=−21/33kp1+p1/3,cq2=3kp2/31+p2/322/3,cq4=3kp4/31+p1/324/3.Now, we explain the concept ofp,k-parametrization as explained in [11]. Using the identities listed above, we derive the representations Ωp,k for Ωωq,ω−q,ωq2,ωq4 in terms of p and k. The said representation in terms of p,k is a polynomial in p and k, and we want to show that
(17)Ωp,k=∑i=1naiTi,where every ai is a rational number and Ti is the product involving Mqn or mLqm−Lq. Substituting the representations for Ti in terms of p and k in (17), then both sides of (17) are functions of p and k. Further on equating the coefficients of pikj on both sides of (17), a set of linear equations in ai are deduced. If these equations have a solution, then by using Maple, we solve the equations and evaluate the values of ai, which gives the Eisenstein series identities involving Borweins’ cubic theta functions.


## 3. Main Results

Theorem 1.
One hasb4q4b2q2=16−12∑m=1∞3mq12m1−q12m+mq4m1−q4m−4mq2m1−q2m,b4q2b2q4=−1−6∑m=1∞3mq6m1−q6m−16mq4m1−q4m+mq2m1−q2m,b3−qbq2b2q=1+13∑m=1∞195mq12m1−q12m−619mq6m1−q6m+159mq4m1−q4m+175mq3m1−q3m+51mq2m1−q2m+39mqm1−qm,c4q2cqc3−q=225128+132∑m=1∞−135mq12m1−q12m+81mq6m1−q6m−189mq4m1−q4m−171mq3m1−q3m+639mq2m1−q2m+351mqm1−qm,c4qcq2c3−q=1−14∑m=1∞−660mq12m1−q12m+108mq6m1−q6m+2228mq4m1−q4m−1545mq3m1−q3m−60mq2m1−q2m+25mqm1−qm,c4q2cq4c3−q=255256−164∑m=1∞1620mq12m1−q12m−702mq6m1−q6m+1332mq4m1−q4m+81mq3m1−q3m−1206mq2m1−q2m+225mqm1−qm,c3qcq222/3c2−q=94∑m=1∞5220mq12m1−q12m−156mq6m1−q6m+12mq4m1−q4m+11mq3m1−q3m+2mq2m1−q2m+mqm1−qm,b4q2b2−qb2qb2q4=1+12∑m=1∞36mq12m1−q12m−54mq6m1−q6m+4mq4m1−q4m+9mq3m1−q3m+2mq2m1−q2m+mqm1−qm,b4q2bqb−q=1−3∑m=1∞18mq12m1−q12m−9mq6m1−q6m−2mq4m1−q4m+mq2m1−q2m,b3qbq2b−qbq4=1−12∑m=1∞9mq3m1−q3m−8mq2m1−q2m+mqm1−qm,b3q4bq2bqb−q=1+3∑m=1∞−9mq12m1−q12m−mq4m1−q4m+2mq2m1−q2m,b4q2bqb2−qbq4=1−3∑m=1∞16mq4m1−q4m+9mq3m1−q3m+20mq2m1−q2m+3mqm1−qm,b4q2bqb−qb2q4=1−6∑m=1∞−16mq4m1−q4m+9mq3m1−q3m+4mq2m1−q2m+mqm1−qm,b4q2bq4b2qb−q=1+3∑m=1∞−9mq12m1−q12m+mq4m1−q4m−mq2m1−q2m+mqm1−qm,b4q2bq4bqb2−q=1−3∑m=1∞9mq12m1−q12m+3mq4m1−q4m−5mq2m1−q2m+mqm1−qm.Proof.
If suppose to be the case(19)a512Lq12−Lq+a46Lq6−Lq+a34Lq4−Lq+a23Lq3−Lq+a12Lq2−Lq=b4q4b2q2.On equating the various powers ofp4k2,p3k2,p2k2,pk2,k2 on both sides of the above, it is observed that
11a5+5a4+3a3+2a2+a1=16,34a5+22a4+18a3+16a2+14a1=32,36a5+36a4+36a3+36a2+24a1=24,16a5+22a4+24a3+16a2+14a1=8,2a5+5a4+2a2+a1=1.On solving the above five equations, one can see that(21)a5=32,a4=0,a3=12,a2=0,a1=−2.Substituting (21) in (19) and using (1), the desired result is obtained. In the same way, applying this technique, the following identities are also derived:
b4q2b2q4=−34Lq−6Lq6+14Lq−4Lq4−14Lq−2Lq2,b3−qbq2b2q=6524Lq−12Lq12−61972Lq−6Lq6+5324Lq−4Lq4+1724Lq−2Lq2,c4q2cqc3−q=−45256Lq−12Lq12+27256Lq−6Lq6−63256Lq−4Lq4−57256Lq−3Lq3−213256Lq−2Lq2,c4qcq2c3−q=5596Lq−12Lq12−316Lq−6Lq6−55796Lq−4Lq4+51596Lq−3Lq3+516Lq−2Lq2,c4q2cq4c3−q=−45512Lq−12Lq12+39512Lq−6Lq6−111512Lq−4Lq4−9512Lq−3Lq3+201512Lq−2Lq2,c3qcq222/3c2−q=1332Lq−12Lq12−3932Lq−6Lq6+932Lq−4Lq4+1132Lq−3Lq3+332Lq−2Lq2,b4q2b2−qb2qb2q4=32Lq−12Lq12−92Lq−6Lq6+12Lq−4Lq4+32Lq−3Lq3+12Lq−2Lq2,b4q2bqb−q=−316Lq−12Lq12+316Lq−6Lq6+116Lq−4Lq4−116Lq−2Lq2,b3qbq2b−qbq4=−32Lq−3Lq3+2Lq−2Lq2,b3q4bq2bqb−q=−332Lq−12Lq12−132Lq−4Lq4+18Lq−2Lq2,b4q2bqb2−qbq4=−12Lq−4Lq4−38Lq−3Lq3+54Lq−2Lq2,b4q2bqb−qb2q4=−Lq−4Lq4+34Lq−3Lq3+12Lq−2Lq2,b4q2bq4b2qb−q=−332Lq−12Lq12−116Lq−2Lq2+132Lq−4Lq4,b4q2bq4bqb2−q=−332Lq−12Lq12+516Lq−2Lq2−332Lq−4Lq4.Theorem 2.
One hasc3qc2−q21/3cq2=−17+162∑m=1∞m3qm1−qm−8m3q2m1−q2m−21m3q3m1−q3m+28m3q6m1−q6m,c3q4c2−q21/3cq2=1296∑m=1∞m3q2m1−q2m−6m3q3m1−q3m+5m3q6m1−q6m,c3−qc2qcq2=324∑m=1∞−m3qm1−qm−10m3q2m1−q2m−21m3q3m1−q3m+30m3q6m1−q6m,c3−qc2q2cq4=324∑m=1∞−m3qm1−qm+12m3q2m1−q2m+17m3q3m1−q3m−28m3q6m1−q6m,b2q2b2q4=524−112∑m=1∞95m3qm1−qm+270m3q2m1−q2m+912m3q3m1−q3m−582m3q6m1−q6m+95m3q12m1−q12m,b3qb2q2bq4=253+13∑m=1∞193m3qm1−qm−486m3q2m1−q2m−1221m3q3m1−q3m+7734m3q6m1−q6m220m3q12m1−q12m,b3qb2q4bq2=68572+136∑m=1∞2741m3qm1−qm−7650m3q2m1−q2m−17976m3q3m1−q3m+108150m3q6m1−q6m−3065m3q12m1−q12m,b5qb2q2=10+3∑m=1∞25m3qm1−qm−32m3q2m1−q2m−243m3q3m1−q3m+960m3q6m1−q6m−30m3q12m1−q12m,b5q2bq4=−136−13∑m=1∞95m3qm1−qm+297m3q2m1−q2m+912m3q3m1−q3m−2769m3q6m1−q6m+95m3q12m1−q12m,b5q2b−q=1324−112∑m=1∞91m3qm1−qm+450m3q2m1−q2m+1824m3q3m1−q3m−678m3q6m1−q6m+55m3q12m1−q12m,b5q4bq2=5940+112∑m=1∞55m3qm1−qm−162m3q2m1−q2m−528m3q3m1−q3m+4890m3q6m1−q6m−55m3q12m1−q12m,b4qbq4b2q2=553+13∑m=1∞418m3qm1−qm−774m3q2m1−q2m−3408m3q3m1−q3m+17454m3q6m1−q6m−490m3q12m1−q12m,b4q4bqb2q2=4324+112∑m=1∞59m3qm1−qm+18m3q2m1−q2m+60m3q3m1−q3m+5118m3q6m1−q6m−95m3q12m1−q12m.Proof.
If suppose to be the case(24)a1Mq+a2Mq2+a3Mq3+a4Mq6=c3qc2−q21/3cq2.On equating the various powers ofp4k2,p3k2,p2k2,pk2,k2 on both sides of the above, it is observed that
a1+a2+a3+a4=0,124a1+4a2+4a3+4a4=81,964a1+64a2+4a3+4a4=324,2788a1+178a2+28a3−2a4=48.On solving the above, it is easy to see that(26)a1=2740,a2=−275,a3=−56740,a4=18910.Substituting (26) in (24) and using (2), the desired result is obtained. In the same way, applying this technique, the following identities are deduced:
c3q4c2−q21/3cq2=275Mq2−1625Mq3+27Mq6,c3−qc2qcq2=−2720Mq+272Mq2+56720Mq3−812Mq6,c3−qc2q2cq4=−2720Mq+815Mq2+45920Mq3−1895Mq6,b2q2b2q4=−19576Mq+332Mq2+1960Mq3−97480Mq6+19576Mq12,b3qb2q2bq4=193720Mq−2740Mq2−407240Mq3+1289120Mq6−1136Mq12,b3qb2q4bq2=27418640Mq−8596Mq2−749360Mq3+3605288Mq6−6131728Mq12,b5qb2q2=516Mq−25Mq2−24380Mq3+272Mq6−38Mq12,b5q2bq4=−19144Mq+3380Mq2+1915Mq3−923240Mq6+9144Mq12,b5q2b−q=−912880Mq+532Mq2+1930Mq3−113480Mq6+11576Mq12,b5q4bq2=11576Mq−9160Mq2−1160Mq3+16396Mq6−11576Mq12,b4qbq4b2q2=209360Mq−4340Mq2−7115Mq3+2909120Mq6−4972Mq12,b4q4bqb2q2=592880Mq+1160Mq2+148Mq3+853480Mq6−19576Mq12.


## 4. Conclusion and Application

Employing the technique developed by Xia and Yao [11], one can derive several relations between Eisenstein Series and Borwiens’ cubic theta functions using (p, k)-parametrization. As an application, one can evaluate convolution sums involving divisor sum.



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*Source: 6614572-2021-05-10.xml*