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Journal of Applied Mathematics
Volume 2012, Article ID 181264, 14 pages
http://dx.doi.org/10.1155/2012/181264
Research Article

Eisenstein Series Identities Involving the Borweins' Cubic Theta Functions

Department of Mathematics, Jiangsu University, Jiangsu, Zhenjiang 212013, China

Received 28 March 2012; Revised 15 May 2012; Accepted 23 May 2012

Academic Editor: Ferenc Hartung

Copyright © 2012 Ernest X. W. Xia and Olivia X. M. Yao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Based on the theories of Ramanujan's elliptic functions and the (p, k)-parametrization of theta functions due to Alaca et al. (2006, 2007, 2006) we derive certain Eisenstein series identities involving the Borweins' cubic theta functions with the help of the computer. Some of these identities were proved by Liu based on the fundamental theory of elliptic functions and some of them may be new. One side of each identity involves Eisenstein series, the other products of the Borweins' cubic theta functions. As applications, we evaluate some convolution sums. These evaluations are different from the formulas given by Alaca et al.

1. Introduction

Let and denote the sets of positive integers and complex numbers, respectively. Throughout the paper, we always assume that 𝑞 and |𝑞|<1.

In their paper, J. M. Borwein and P. B. Borwein [1] introduced the following three functions: 𝑎(𝑞)=𝑚,𝑛=𝑞𝑚2+𝑚𝑛+𝑛2,𝑏(𝑞)=𝑚,𝑛=𝜔𝑚𝑛𝑞𝑚2+𝑚𝑛+𝑛2,𝑐(𝑞)=𝑚,𝑛=𝑞(𝑚+1/3)2+(𝑚+1/3)(𝑛+1/3)+(𝑛+1/3)2,(1.1) where 𝜔=𝑒2𝜋𝑖/3. These functions are now called the Borweins' cubic theta functions. The Borwein brothers [1] derived representations for 𝑏(𝑞) and 𝑐(𝑞) in terms of infinite products, namely, 𝑏(𝑞)=(𝑞;𝑞)3𝑞3;𝑞3,(1.2)𝑐(𝑞)=3𝑞1/3𝑞3;𝑞33(𝑞;𝑞),(1.3) where (𝑎;𝑞)=𝑖=01𝑎𝑞𝑖.(1.4) The function 𝑎(𝑞) has the following representation derived by the Borwein brothers [1] and Berndt [2]: 𝑎(𝑞)=𝑞;𝑞22𝑞2;𝑞2𝑞3;𝑞62𝑞6;𝑞6𝑞+4𝑞4;𝑞4𝑞12;𝑞12𝑞2;𝑞4𝑞6;𝑞12.(1.5) Elementary proofs of (1.2), (1.3), and (1.5) can be found in [3]. The Borwein brothers [1] also proved the following well-known relation for 𝑎(𝑞), 𝑏(𝑞) and 𝑐(𝑞), namely: 𝑎3(𝑞)=𝑏3(𝑞)+𝑐3(𝑞).(1.6)

In his second notebook [4], Ramanujan gave the definitions of the Eisenstein series 𝐿(𝑞), 𝑀(𝑞), and 𝑁(𝑞), namely, 𝐿(𝑞)=124𝑛=1𝑛𝑞𝑛1𝑞𝑛,(1.7)𝑀(𝑞)=1+240𝑛=1𝑛3𝑞𝑛1𝑞𝑛,(1.8)𝑁(𝑞)=1504𝑛=1𝑛5𝑞𝑛1𝑞𝑛.(1.9)

Utilizing Ramanujan's elliptic functions in the theory of signature 3, Berndt et al. [5] proved the following representations for 𝑀(𝑞), 𝑀(𝑞3), 𝑁(𝑞), and 𝑁(𝑞3), namely: 𝑎𝑀(𝑞)=𝑎(𝑞)3(𝑞)+8𝑐3,𝑀𝑞(𝑞)(1.10)3=19𝑎(𝑞)9𝑎3(𝑞)8𝑐3,(𝑞)(1.11)𝑁(𝑞)=𝑎6(𝑞)20𝑎3(𝑞)𝑐3(𝑞)8𝑐3𝑁𝑞(𝑞),(1.12)3=𝑎64(𝑞)3𝑎3(𝑞)𝑐38(𝑞)+𝑐273(𝑞).(1.13) Chan [6] proved (1.10) and (1.11) by employing the classical theory of elliptic functions and modular equations of degree 3. Based on the fundamental theory of elliptic functions, Liu [7, 8] provided different proofs of (1.10), (1.11), (1.12), and (1.13). He also discovered some striking Eisenstein series identities.

In this paper, using the parameters 𝑝 and 𝑘 introduced by Alaca et al. [911] (see (2.1)), we deduce some Eisenstein series identities involving the Borweins' cubic theta functions with the help of the computer. These identities are examples of sum-to-product identities. Some of these identities were proved by Liu [7, 8] based on the fundamental theory of elliptic function and some of them may be new.

This paper is organized as follows. In Section 2, we gave the (𝑝,𝑘)-parametrization of 𝑚𝐿(𝑞𝑚)𝐿(𝑞), and 𝑀(𝑞𝑙) for 𝑚{2,3,4,6,12}, 𝑙{1,2,3,6}, which are due to Alaca et al. [911]. Section 3 is devoted to deriving some Eisenstein series identities involving the Borweins' cubic theta functions. As corollaries of our results, in Section 4, we derive new representations for the convolution sums 𝑖+6𝑗=𝑛𝜎(𝑖)𝜎(𝑗) and 𝑖+12𝑗=𝑛𝜎(𝑖)𝜎(𝑗) and contrast them with the known evaluations due to Alaca and Williams [11] and Alaca et al. [10].

2. Eisenstein Series 𝐿(𝑞), 𝑀(𝑞) and Parameters 𝑝, 𝑘

In this section, we gave the parametric representations for 𝑚𝐿(𝑞𝑚)𝐿(𝑞), 𝑀(𝑞𝑙), 𝑏(𝑞𝑖) and 𝑐(𝑞𝑖) for 𝑚{2,3,4,6,12}, 𝑙{1,2,3,6} and 𝑖{1,2,4} in terms of the parameters 𝑝 and 𝑘 first defined by Alaca and Williams [11], namely, 𝜑𝑝=𝑝(𝑞)=2(𝑞)𝜑2𝑞32𝜑2𝑞3,𝜑𝑘=𝑘(𝑞)=3𝑞3.𝜑(𝑞)(2.1)

Alaca and Williams [11] derived the representations of 𝑀(𝑞), 𝑀(𝑞𝑖), and 𝑖𝐿(𝑞𝑖)𝐿(𝑞)(𝑖=2,3,6) in terms of 𝑝 and 𝑘. Equations (3.69), (3.70), (3.71), (3.72), (3.84), (3.87), and (3.89) in [11] are 𝑀(𝑞)=1+124𝑝+964𝑝2+2788𝑝3+3910𝑝4+2788𝑝5+964𝑝6+124𝑝7+𝑝8𝑘4,𝑀𝑞2=1+4𝑝+64𝑝2+178𝑝3+235𝑝4+178𝑝5+64𝑝6+4𝑝7+𝑝8𝑘4,𝑀𝑞3=1+4𝑝+4𝑝2+28𝑝3+70𝑝4+28𝑝5+4𝑝6+4𝑝7+𝑝8𝑘4,𝑀𝑞6=1+4𝑝+4𝑝22𝑝35𝑝42𝑝5+4𝑝6+4𝑝7+𝑝8𝑘4,𝑞2𝐿2𝐿(𝑞)=1+14𝑝+24𝑝2+14𝑝3+𝑝4𝑘2,𝑞3𝐿3𝐿(𝑞)=2+16𝑝+36𝑝2+16𝑝3+2𝑝4𝑘2,𝑞6𝐿6𝐿(𝑞)=5+22𝑝+36𝑝2+22𝑝3+5𝑝4𝑘2,(2.2) respectively. Alaca et al. [12, 13] also deduced the representations of 12𝐿(𝑞12)𝐿(𝑞), 𝑀(𝑞12) and 4𝐿(𝑞4)𝐿(𝑞) in terms of 𝑝 and 𝑘. Equations (3.12), and (3.19) in [12] and (3.13) in [13] are 𝑞12𝐿12𝐿(𝑞)=11+34𝑝+36𝑝2+16𝑝3+2𝑝4𝑘2,𝑀𝑞12=1+4𝑝+4𝑝22𝑝35𝑝42𝑝5+14𝑝6+14𝑝7+1𝑝168𝑘4,𝑞4𝐿4𝐿(𝑞)=24𝑝3+36𝑝2𝑘+18𝑝+32,(2.3) respectively. Alaca et al. [9] also derived the following parametric representations for 𝑏(𝑞) and 𝑐(𝑞) in terms of 𝑝 and 𝑘. From Theorems 1, 2, and 4 in [9], we have 𝑏(𝑞)=(1𝑝)((1𝑝)(1+2𝑝)(2+𝑝))1/3𝑘21/3,𝑐(𝑞)=3(1+𝑝)(𝑝(1+𝑝))1/3𝑘21/3,𝑏𝑞2=((1𝑝)(1+2𝑝)(2+𝑝))2/3𝑘22/3,𝑐𝑞2=3(𝑝(1+𝑝))2/3𝑘22/3,𝑏𝑞4=(2+𝑝)((1𝑝)(1+2𝑝)(2+𝑝))1/3𝑘24/3,𝑐𝑞4=3𝑝(𝑝(1+𝑝))1/3𝑘24/3.(2.4) We now describe our approach. Let 𝑅(𝑒(𝑞),𝑒(𝑞2),𝑒(𝑞4)) be a function, where 𝑒=𝑏 or 𝑒=𝑐. Utilizing the representations for 𝑏(𝑞), 𝑏(𝑞2), 𝑏(𝑞4), 𝑐(𝑞), 𝑐(𝑞2) and 𝑐(𝑞4) in terms of 𝑝 and 𝑘, we derive the representations 𝑅(𝑝,𝑘) for 𝑅(𝑒(𝑞),𝑒(𝑞2),𝑒(𝑞4)) in terms of 𝑝 and 𝑘. We select suitable 𝑅(𝑒(𝑞),𝑒(𝑞2),𝑒(𝑞4)) such that 𝑅(𝑝,𝑘) is a polynomial in 𝑝 and 𝑘. We want to show that 𝑅(𝑝,𝑘)=𝑠𝑖=1𝐶𝑖𝑇𝑖,(2.5) where each 𝐶𝑖 is a rational number and 𝑇𝑖 is a product involving 𝑚𝐿(𝑞𝑚)𝐿(𝑞) and 𝑀(𝑞𝑚). Substituting the representation for 𝑇𝑖 in terms of 𝑝 and 𝑘 into (2.5), then both sides of (2.5) are functions in 𝑝 and 𝑘. Equating the coefficients of 𝑝𝑖𝑘𝑗 on both sides of (2.5), we obtain some linear equations in 𝐶𝑖. If these equations have a solution, then we can use computer to solve the equations and determine the values of the 𝐶𝑖. We then obtain some Eisenstein series identities involving the Borweins' cubic theta functions.

3. Some Eisenstein Series Identities

In this section, we derive some Eisenstein series identities. In fact, utilizing our method, we can obtain many identities, here we just list some of them. Our main theorem can be stated as follows.

Theorem 3.1. One has 16𝑛=1𝑛𝑞𝑛1𝑞𝑛4𝑛𝑞2𝑛1𝑞2𝑛9𝑛𝑞3𝑛1𝑞3𝑛+16𝑛𝑞4𝑛1𝑞4𝑛=(𝑞;𝑞)6𝑞2;𝑞23𝑞12;𝑞12𝑞4;𝑞43𝑞3;𝑞32𝑞6;𝑞6=𝑏2𝑞(𝑞)𝑏2𝑏𝑞4,(3.1)1+3𝑛=1𝑛𝑞𝑛1𝑞𝑛𝑛𝑞2𝑛1𝑞2𝑛+𝑛𝑞4𝑛1𝑞4𝑛9𝑛𝑞12𝑛1𝑞12𝑛=𝑞4;𝑞46𝑞2;𝑞23𝑞3;𝑞3𝑞12;𝑞122𝑞6;𝑞6(𝑞;𝑞)3=𝑏2𝑞4𝑏𝑞2,𝑏(𝑞)(3.2)13𝑛=0𝑛𝑞𝑛1𝑞𝑛2𝑛𝑞2𝑛1𝑞2𝑛9𝑛𝑞3𝑛1𝑞3𝑛+18𝑛𝑞6𝑛1𝑞6𝑛=(𝑞;𝑞)3𝑞2;𝑞23𝑞3;𝑞3𝑞6;𝑞6𝑞=𝑏(𝑞)𝑏2,(3.3)112𝑛=1𝑛𝑞𝑛1𝑞𝑛8𝑛𝑞2𝑛1𝑞2𝑛+9𝑛𝑞3𝑛1𝑞3𝑛=(𝑞;𝑞)12𝑞6;𝑞62𝑞2;𝑞26𝑞3;𝑞34=𝑏4(𝑞)𝑏2𝑞2,(3.4)1+3𝑛=12𝑛𝑞𝑛1𝑞𝑛𝑛𝑞2𝑛1𝑞2𝑛9𝑛𝑞6𝑛1𝑞6𝑛=𝑞2;𝑞212𝑞3;𝑞32(𝑞;𝑞)6𝑞6;𝑞64=𝑏4𝑞2𝑏2,(𝑞)(3.5)1+3𝑛=1𝑛𝑞𝑛1𝑞𝑛4𝑛𝑞2𝑛1𝑞2𝑛9𝑛𝑞3𝑛1𝑞3𝑛+4𝑛𝑞4𝑛1𝑞4𝑛+36𝑛𝑞6𝑛1𝑞6𝑛36𝑛𝑞12𝑛1𝑞12𝑛=𝑞2;𝑞212𝑞3;𝑞3𝑞12;𝑞12(𝑞;𝑞)3𝑞4;𝑞43𝑞6;𝑞64=𝑏4𝑞2𝑞𝑏(𝑞)𝑏4,(3.6)1+3𝑛=05𝑛𝑞2𝑛1𝑞2𝑛𝑛𝑞𝑛1𝑞𝑛3𝑛𝑞4𝑛1𝑞4𝑛9𝑛𝑞12𝑛1𝑞12𝑛=(𝑞;𝑞)3𝑞4;𝑞49𝑞6;𝑞62𝑞3;𝑞3𝑞12;𝑞123𝑞2;𝑞26=𝑏(𝑞)𝑏3𝑞4𝑏2𝑞2,(3.7)1+3𝑛=120𝑛𝑞2𝑛1𝑞2𝑛3𝑛𝑞𝑛1𝑞𝑛9𝑛𝑞3𝑛1𝑞3𝑛16𝑛𝑞4𝑛1𝑞4𝑛=(𝑞;𝑞)9𝑞4;𝑞43𝑞6;𝑞62𝑞3;𝑞33𝑞12;𝑞12𝑞2;𝑞26=𝑏3𝑞(𝑞)𝑏4𝑏2𝑞2,(3.8)107+3322𝑛=111𝑛3𝑞𝑛1𝑞𝑛16𝑛3𝑞2𝑛1𝑞2𝑛+540𝑛3𝑞6𝑛1𝑞6𝑛3325+24𝑛=1𝑛𝑞𝑛1𝑞𝑛6𝑛𝑞6𝑛1𝑞6𝑛2=(𝑞;𝑞)6𝑞2;𝑞26𝑞3;𝑞32𝑞6;𝑞62=𝑏2(𝑞)𝑏2𝑞2,(3.9)11+3644𝑛=1𝑛3𝑞𝑛1𝑞𝑛2𝑛3𝑞2𝑛1𝑞2𝑛54𝑛3𝑞6𝑛1𝑞6𝑛+3645+24𝑛=1𝑛𝑞𝑛1𝑞𝑛6𝑛𝑞6𝑛1𝑞6𝑛2=𝑞2;𝑞224𝑞3;𝑞34(𝑞;𝑞)12𝑞6;𝑞68=𝑏8𝑞2𝑏4,(𝑞)(3.10)7424𝑛=131𝑛3𝑞𝑛1𝑞𝑛128𝑛3𝑞2𝑛1𝑞2𝑛243𝑛3𝑞3𝑛1𝑞3𝑛+1080𝑛3𝑞6𝑛1𝑞6𝑛+35+24𝑛=1𝑛𝑞𝑛1𝑞𝑛6𝑛𝑞6𝑛1𝑞6𝑛2=(𝑞;𝑞)24𝑞6;𝑞64𝑞2;𝑞212𝑞3;𝑞38=𝑏8(𝑞)𝑏4𝑞2,(3.11)1+2𝑛=1𝑛𝑞𝑛1𝑞𝑛𝑛𝑞3𝑛1𝑞3𝑛+4𝑛𝑞6𝑛1𝑞6𝑛16𝑛𝑞12𝑛1𝑞12𝑛=𝑞3;𝑞36𝑞6;𝑞63𝑞4;𝑞4(𝑞;𝑞)2𝑞2;𝑞2𝑞12;𝑞123=𝑐2𝑞(𝑞)𝑐2𝑞9𝑐4,(3.12)𝑛=1𝑛𝑞3𝑛1𝑞3𝑛𝑛𝑞4𝑛1𝑞4𝑛𝑛𝑞6𝑛1𝑞6𝑛+𝑛𝑞12𝑛1𝑞12𝑛=𝑞3𝑞12;𝑞126𝑞6;𝑞63(𝑞;𝑞)𝑞4;𝑞42𝑞2;𝑞2𝑞3;𝑞33=𝑐2𝑞4𝑐𝑞2,9𝑐(𝑞)(3.13)𝑛=1𝑛𝑞𝑛1𝑞𝑛2𝑛𝑞2𝑛1𝑞2𝑛𝑛𝑞3𝑛1𝑞3𝑛+2𝑛𝑞6𝑛1𝑞6𝑛=𝑞𝑞3;𝑞33𝑞6;𝑞63(𝑞;𝑞)𝑞2;𝑞2=𝑞𝑐(𝑞)𝑐29,(3.14)𝑛=1𝑛𝑞2𝑛1𝑞2𝑛2𝑛𝑞3𝑛1𝑞3𝑛+𝑛𝑞6𝑛1𝑞6𝑛=𝑞2𝑞6;𝑞612(𝑞;𝑞)2𝑞2;𝑞24𝑞3;𝑞36=𝑐4𝑞29𝑐2,(𝑞)(3.15)1+4𝑛=1𝑛𝑞𝑛1𝑞𝑛+𝑛𝑞3𝑛1𝑞3𝑛8𝑛𝑞6𝑛1𝑞6𝑛=𝑞3;𝑞312𝑞2;𝑞22(𝑞;𝑞)4𝑞6;𝑞66=𝑐4(𝑞)9𝑐2𝑞2,(3.16)𝑛=1𝑛𝑞𝑛1𝑞𝑛4𝑛𝑞2𝑛1𝑞2𝑛𝑛𝑞3𝑛1𝑞3𝑛+4𝑛𝑞4𝑛1𝑞4𝑛+4𝑛𝑞6𝑛1𝑞6𝑛4𝑛𝑞12𝑛1𝑞12𝑛𝑞=𝑞6;𝑞612(𝑞;𝑞)𝑞4;𝑞4𝑞2;𝑞2𝑞3;𝑞33𝑞12;𝑞123=𝑐4𝑞2𝑞9𝑐(𝑞)𝑐4,(3.17)𝑛=1𝑛𝑞3𝑛1𝑞3𝑛+𝑛𝑞4𝑛1𝑞4𝑛5𝑛𝑞6𝑛1𝑞6𝑛+3𝑛𝑞12𝑛1𝑞12𝑛=𝑞3𝑞2;𝑞22𝑞3;𝑞33𝑞12;𝑞129(𝑞;𝑞)𝑞4;𝑞43𝑞6;𝑞66=𝑐(𝑞)𝑐3𝑞49𝑐2𝑞2,(3.18)𝑛=13𝑛𝑞3𝑛1𝑞3𝑛+𝑛𝑞𝑛1𝑞𝑛20𝑛𝑞6𝑛1𝑞6𝑛+16𝑛𝑞12𝑛1𝑞12𝑛𝑞=𝑞2;𝑞22𝑞3;𝑞39𝑞12;𝑞123(𝑞;𝑞)3𝑞4;𝑞4𝑞6;𝑞66=𝑐3𝑞(𝑞)𝑐49𝑐2𝑞2,1(3.19)18𝑛=15𝑛3𝑞𝑛1𝑞𝑛12𝑛3𝑞3𝑛1𝑞3𝑛+132𝑛3𝑞6𝑛1𝑞6𝑛18645+24𝑛=1𝑛𝑞𝑛1𝑞𝑛6𝑛𝑞6𝑛1𝑞6𝑛2+25=𝑞8642𝑞3;𝑞36𝑞6;𝑞66(𝑞;𝑞)2𝑞2;𝑞22=𝑐2(𝑞)𝑐2𝑞2,81(3.20)251172836𝑛=15𝑛3𝑞𝑛1𝑞𝑛18𝑛3𝑞2𝑛1𝑞2𝑛48𝑛3𝑞3𝑛1𝑞3𝑛+186𝑛3𝑞6𝑛1𝑞6𝑛+117285+24𝑛=1𝑛𝑞𝑛1𝑞𝑛6𝑛𝑞6𝑛1𝑞6𝑛2=𝑞4𝑞6;𝑞624(𝑞;𝑞)4𝑞2;𝑞28𝑞3;𝑞312=𝑐8𝑞281𝑐4,2(𝑞)(3.21)8279𝑛=1𝑛3𝑞𝑛1𝑞𝑛+3𝑛3𝑞3𝑛1𝑞3𝑛24𝑛3𝑞6𝑛1𝑞6𝑛+1275+24𝑛=1𝑛𝑞𝑛1𝑞𝑛6𝑛𝑞6𝑛1𝑞6𝑛2=𝑞3;𝑞324𝑞2;𝑞24(𝑞;𝑞)8𝑞6;𝑞612=𝑐8(𝑞)81𝑐4𝑞2,1(3.22)10811+24𝑛=1𝑛𝑞𝑛1𝑞𝑛12𝑛𝑞12𝑛1𝑞12𝑛212165+24𝑛=1𝑛𝑞𝑛1𝑞𝑛6𝑛𝑞6𝑛1𝑞6𝑛2+29𝑛=1𝑛3𝑞𝑛1𝑞𝑛6𝑛3𝑞3𝑛1𝑞3𝑛+96𝑛3𝑞6𝑛1𝑞6𝑛96𝑛3𝑞12𝑛1𝑞12𝑛1=𝑞2163;𝑞312𝑞6;𝑞66𝑞4;𝑞42(𝑞;𝑞)4𝑞2;𝑞22𝑞12;𝑞126=𝑐4(𝑞)𝑐2𝑞281𝑐2𝑞4.(3.23)

Remark 3.2. The identities (1.16) and (1.17) in [8] contain typos, they should be (3.1) and (3.5), respectively.

Proof. We first prove the formula (3.1) by our method. We assume that 𝐶1𝑞2𝐿2𝐿(𝑞)+𝐶2𝑞3𝐿3𝐿(𝑞)+𝐶3𝑞4𝐿4𝐿(𝑞)+𝐶4𝑞6𝐿6𝐿(𝑞)+𝐶5𝑞12𝐿12=𝑏𝐿(𝑞)2𝑞(𝑞)𝑏2𝑏𝑞4.(3.24) Equating the coefficients of 𝑘2, 𝑝𝑘2, 𝑝2𝑘2, 𝑝3𝑘2, and 𝑝4𝑘2 on both sides of (3.24), we obtain the following five equations: 𝐶1+2𝐶2+3𝐶3+5𝐶4+11𝐶5=1,14𝐶1+16𝐶2+18𝐶3+22𝐶4+34𝐶5=1,24𝐶1+36𝐶2+36𝐶3+36𝐶4+36𝐶5=3,14𝐶1+16𝐶2+24𝐶3+22𝐶4+16𝐶5𝐶=5,1+2𝐶2+5𝐶4+2𝐶5=2.(3.25) Solving the above five equations, we obtain 𝐶11=2,𝐶23=4,𝐶3=1,𝐶4=0,𝐶5=0.(3.26) Substituting the above values into (3.24), from (1.2) and (1.7), we obtain (3.1).
Similarly, utilizing the same method, we also derive the following formulas: 𝑏2𝑞4𝑏𝑞2𝑏=1(𝑞)𝑞162𝐿21𝐿(𝑞)𝑞324𝐿4+3𝐿(𝑞)𝑞3212𝐿12,𝑞𝐿(𝑞)𝑏(𝑞)𝑏21=8𝑞2𝐿23𝐿(𝑞)8𝑞3𝐿3+3𝐿(𝑞)8𝑞6𝐿6,𝑏𝐿(𝑞)4(𝑞)𝑏2𝑞2𝑞=22𝐿2+3𝐿(𝑞)2𝑞3𝐿3,𝑏𝐿(𝑞)4𝑞2𝑏2=1(𝑞)𝑞162𝐿2+3𝐿(𝑞)𝑞166𝐿6,𝑏𝐿(𝑞)4𝑞2𝑞𝑏(𝑞)𝑏4=14𝑞2𝐿2+3𝐿(𝑞)8𝑞3𝐿31𝐿(𝑞)8𝑞4𝐿43𝐿(𝑞)4𝑞6𝐿6+3𝐿(𝑞)8𝑞12𝐿12,𝐿(𝑞)𝑏(𝑞)𝑏3𝑞4𝑏2𝑞25=𝑞162𝐿2+3𝐿(𝑞)𝑞324𝐿4+3𝐿(𝑞)𝑞3212𝐿12,𝑏𝐿(𝑞)3𝑞(𝑞)𝑏4𝑏2𝑞25=4𝑞2𝐿2+3𝐿(𝑞)8𝑞3𝐿3+1𝐿(𝑞)2𝑞4𝐿4,𝑏𝐿(𝑞)2(𝑞)𝑏2𝑞2=11𝑀1160(𝑞)𝑀𝑞102+278𝑀𝑞63𝑞326𝐿6𝐿(𝑞)2,𝑏8𝑞2𝑏4=1(𝑞)1320𝑀(𝑞)𝑀𝑞160227𝑀𝑞1606+3𝑞646𝐿6𝐿(𝑞)2,𝑏8(𝑞)𝑏4𝑞2=31𝑀10(𝑞)+645𝑀𝑞2+243𝑀𝑞103𝑞108𝑀6𝑞+36𝐿6𝐿(𝑞)2,𝑐2𝑞(𝑞)𝑐2𝑞9𝑐4=1𝑞363𝐿31𝐿(𝑞)𝑞186𝐿6+1𝐿(𝑞)9𝑞12𝐿12,𝑐𝐿(𝑞)2𝑞4𝑐𝑞219𝑐(𝑞)=𝑞723𝐿3+1𝐿(𝑞)𝑞964𝐿4+1𝐿(𝑞)𝑞1446𝐿61𝐿(𝑞)𝑞28812𝐿12,𝑞𝐿(𝑞)𝑐(𝑞)𝑐29=1𝑞242𝐿2+1𝐿(𝑞)𝑞723𝐿31𝐿(𝑞)𝑞726𝐿6,𝑐𝐿(𝑞)4𝑞29𝑐21(𝑞)=𝑞482𝐿2+1𝐿(𝑞)𝑞363𝐿31𝐿(𝑞)𝑞1446𝐿6,𝑐𝐿(𝑞)4(𝑞)9𝑐2𝑞21=𝑞183𝐿3+2𝐿(𝑞)9𝑞6𝐿6,𝑐𝐿(𝑞)4𝑞2𝑞9𝑐(𝑞)𝑐4=34𝑞2𝐿2+1𝐿(𝑞)8𝑞3𝐿33𝐿(𝑞)8𝑞4𝐿41𝐿(𝑞)4𝑞6𝐿6+1𝐿(𝑞)8𝑞12𝐿12,𝐿(𝑞)𝑐(𝑞)𝑐3𝑞49𝑐2𝑞21=8𝑞3𝐿33𝐿(𝑞)𝑞324𝐿4+5𝐿(𝑞)𝑞166𝐿63𝐿(𝑞)𝑞3212𝐿12,𝑐𝐿(𝑞)3𝑞(𝑞)𝑐49𝑐2𝑞23=8𝑞3𝐿3+5𝐿(𝑞)4𝑞6𝐿61𝐿(𝑞)2𝑞12𝐿12,𝑐𝐿(𝑞)2(𝑞)𝑐2𝑞2=381932𝑀(𝑞)𝑀𝑞403+99𝑀𝑞4063𝑞326𝐿6𝐿(𝑞)2,𝑐8𝑞281𝑐4(=𝑞)27𝑀𝑞16023964𝑀(𝑞)+𝑀𝑞203279𝑀𝑞1606+3𝑞646𝐿6𝐿(𝑞)2,𝑐8(𝑞)81𝑐4𝑞23=𝑀910(𝑞)𝑀𝑞103+365𝑀𝑞6𝑞+36𝐿6𝐿(𝑞)2,𝑐4(𝑞)𝑐2𝑞281𝑐2𝑞4=3940𝑀(𝑞)𝑀𝑞203+365𝑀𝑞6365𝑀𝑞1238𝑞6𝐿6𝐿(𝑞)2+34𝑞12𝐿12𝐿(𝑞)2.(3.27) From the above identities and (1.2), (1.3), (1.7) and (1.8), we may derive other formulas.

4. Convolution Sums 𝑖+6𝑗=𝑛𝜎(𝑖)𝜎(𝑗) and 𝑖+12𝑗=𝑛𝜎(𝑖)𝜎(𝑗)

Let 𝑛,𝑘. The divisor function 𝜎𝑘(𝑛) is defined by 𝜎𝑘(𝑛)=𝑑|𝑛𝑑𝑘,(4.1) where 𝑑 runs through the positive divisors of 𝑛. If 𝑛 is not a positive integer, set 𝜎𝑖(𝑛)=0. As usual, we write 𝜎(𝑛) for 𝜎1(𝑛). For all 𝑛, the convolution 𝑖+𝑘𝑗=𝑛𝜎(𝑖)𝜎(𝑗) has been evaluated explicitly for 𝑘=1,2,3,4,5,6,7,8,9,12,16,18, and 24, see [1019]. In this section, we also derive the representations for the convolution sums 𝑖+6𝑗=𝑛𝜎(𝑖)𝜎(𝑗) and 𝑖+12𝑗=𝑛𝜎(𝑖)𝜎(𝑗) from the identities in Theorem 3.1. Our representations are different from those derived in [11, 12]. In fact, we can derive many formulas for the convolution sums 𝑖+6𝑗=𝑛𝜎(𝑖)𝜎(𝑗) and 𝑖+12𝑗=𝑛𝜎(𝑖)𝜎(𝑗), and here we just list two of them.

Theorem 4.1. Let 𝑛 be a positive integer. One has 𝑖+6𝑗=𝑛1𝜎(𝑖)𝜎(𝑗)=𝜎10831(𝑛)+𝜎273𝑛2+1𝑛24𝜎(𝑛)+16𝑛𝜎𝑛246+𝐴(𝑛),648(4.2)𝑖+12𝑗=𝑛𝜎(𝑖)𝜎(𝑗)=11𝜎86431(𝑛)+𝜎1083𝑛21𝜎963𝑛37𝜎483𝑛6+73𝜎3𝑛+122𝑛48𝜎(𝑛)6𝑛1𝜎𝑛24+12𝐴(𝑛)2592𝐵(𝑛),128(4.3) where 1+𝑛=1𝐴(𝑛)𝑞𝑛=(𝑞;𝑞)6𝑞2;𝑞26𝑞3;𝑞32𝑞6;𝑞62,(4.4)1+𝑛=1𝐵(𝑛)𝑞𝑛=𝑞3;𝑞312𝑞6;𝑞66𝑞4;𝑞42(𝑞;𝑞)4𝑞2;𝑞22𝑞12;𝑞126.(4.5)

Remark. Alaca and Williams [11] derived the representation for 𝑖+6𝑗=𝑛𝜎(𝑖)𝜎(𝑗)𝑖+6𝑗=𝑛𝜎𝜎(𝑖)𝜎(𝑗)=3(𝑛)+𝜎1203(𝑛/2)+303𝜎3(𝑛/3)+403𝜎3(𝑛/6)+101𝑛24𝜎(𝑛)+16𝑛𝜎𝑛246𝑐6(𝑛),120(4.6) where 𝑛=1𝑐6(𝑛)𝑞𝑛=𝑞(𝑞;𝑞)2𝑞2;𝑞22𝑞3;𝑞32𝑞6;𝑞62.(4.7)

Alaca et al. [12] also derived the representation for 𝑖+12𝑗=𝑛𝜎(𝑖)𝜎(𝑗)𝑖+12𝑗=𝑛𝜎𝜎(𝑖)𝜎(𝑗)=3(𝑛)+𝜎4803(𝑛/2)+1603𝜎3(𝑛/3)+𝜎1603(𝑛/4)+309𝜎3(𝑛/6)+1603𝜎3(𝑛/12)+102𝑛48𝜎(𝑛)+16𝑛𝜎𝑛241211𝑐4801,12(𝑛),(4.8) where 11𝑛=1𝑐1,12(𝑛)𝑞𝑛𝑞=10𝑞2;𝑞22𝑞3;𝑞33𝑞4;𝑞43𝑞6;𝑞62(𝑞;𝑞)𝑞12;𝑞12𝑞+𝑞2;𝑞28𝑞6;𝑞68(𝑞;𝑞)2𝑞3;𝑞32𝑞4;𝑞42𝑞12;𝑞122.(4.9)

Proof. From (3.9), we have (𝑞;𝑞)6𝑞2;𝑞26𝑞3;𝑞32𝑞6;𝑞62=1+332𝑛=1𝜎3(𝑛)𝑞𝑛24𝑛=1𝜎3(𝑛)𝑞2𝑛+810𝑛=1𝜎3(𝑛)𝑞6𝑛+135𝑛=1𝜎(𝑛)𝑞6𝑛452𝑛=1𝜎(𝑛)𝑞𝑛1944𝑛=1𝜎(𝑛)𝑞6𝑛2+648𝑛=1𝜎(𝑛)𝑞6𝑛𝑛=1𝜎(𝑛)𝑞𝑛54𝑛=1𝜎(𝑛)𝑞𝑛2.(4.10)
For 𝑛, equating the coefficients of 𝑞𝑛 on both sides of (4.10), we obtain 𝐴(𝑛)=332𝜎3(𝑛)24𝜎3𝑛2+810𝜎3𝑛6𝑛+135𝜎6452𝜎(𝑛)1944𝑖+𝑗=𝑛/6𝜎(𝑖)𝜎(𝑗)+648𝑖+6𝑗=𝑛𝜎(𝑖)𝜎(𝑗)54𝑖+𝑗=𝑛𝜎(𝑖)𝜎(𝑗),(4.11) where 𝐴(𝑛) is defined by (4.4). From (4.11), utilizing the convolution sum 𝑖+𝑗=𝑛5𝜎(𝑖)𝜎(𝑗)=𝜎123𝑛(𝑛)2𝜎(𝑛)+𝜎(𝑛),12(4.12) we can derive the formula (4.2).
From (3.23), we have 𝑞3;𝑞312𝑞6;𝑞66𝑞4;𝑞42(𝑞;𝑞)4𝑞2;𝑞22𝑞12;𝑞1262=1+9𝑛=1𝑛3𝑞𝑛1𝑞𝑛43𝑛=1𝑛3𝑞3𝑛1𝑞3𝑛+643𝑛=1𝑛3𝑞6𝑛1𝑞6𝑛643𝑛=1𝑛3𝑞12𝑛1𝑞12𝑛+203𝑛=1𝑛𝑞6𝑛1𝑞6𝑛+349𝑛=1𝑛𝑞𝑛1𝑞𝑛96𝑛=1𝑛𝑞6𝑛1𝑞6𝑛2+32𝑛=1𝑛𝑞6𝑛1𝑞6𝑛𝑛=1𝑛𝑞𝑛1𝑞𝑛+83𝑛=1𝑛𝑞6𝑛1𝑞6𝑛21763𝑛=1𝑛𝑞12𝑛1𝑞12𝑛+768𝑛=1𝑛𝑞12𝑛1𝑞12𝑛2128𝑛=1𝑛𝑞12𝑛1𝑞12𝑛𝑛=1𝑛𝑞𝑛1𝑞𝑛.(4.13)
For 𝑛, equating the coefficients of 𝑞𝑛 on both sides of (4.13), we obtain 2𝐵(𝑛)=9𝜎34(𝑛)3𝜎3𝑛3+643𝜎3𝑛6643𝜎3𝑛+12203𝜎𝑛6+349𝜎(𝑛)1763𝜎𝑛1296𝑖+𝑗=𝑛/6𝜎(𝑖)𝜎(𝑗)+32𝑖+6𝑗=𝑛+8𝜎(𝑖)𝜎(𝑗)3𝑖+𝑗=𝑛𝜎(𝑖)𝜎(𝑗)+768𝑖+𝑗=𝑛/12𝜎(𝑖)𝜎(𝑗)128𝑖+12𝑗=𝑛𝜎(𝑖)𝜎(𝑗),(4.14) where 𝐵(𝑛) is defined by (4.5). From (4.2), (4.12), and (4.14), we derive the formula (4.3).

The advantage of the formulas of Alaca et al. is that the values of 𝑐6(𝑛) and 𝑐1,12(𝑛) are often very small. Numerical evidence suggests that |𝑐6(𝑛)/120|<|𝐴(𝑛)/648| and |11𝑐1,12(𝑛)/480|<|𝐴(𝑛)/2592𝐵(𝑛)/128|. The advantage of our formulas is that the signs of 𝐴(𝑛) and 𝐴(𝑛)/2592𝐵(𝑛)/128 appear to have periodicity. Numerical evidence suggests that for 𝑛3, 𝐴(𝑛)>0 when 3𝑛 and 𝐴(𝑛)<0 when, 3𝑛 and (𝐴(𝑛)/2592)(𝐵(𝑛)/128)<0. Therefore, we conjecture that, for 𝑛3, we have 𝑖+6𝑗=𝑛1𝜎(𝑖)𝜎(𝑗)>𝜎10831(𝑛)+𝜎273𝑛2+1𝑛24𝜎(𝑛)+16𝑛𝜎𝑛246,3𝑛,𝑖+6𝑗=𝑛1𝜎(𝑖)𝜎(𝑗)<𝜎10831(𝑛)+𝜎273𝑛2+1𝑛24𝜎(𝑛),3𝑛.(4.15)

Acknowledgments

The authors would like to thank the anonymous referee very much for valuable suggestions, corrections, and comments which resulted in a great improvement of the original paper. This work was supported by the National Natural Science Foundation of China and the Jiangsu University Foundation Grants 11JDG035 and 11JDG036.

References

  1. J. M. Borwein and P. B. Borwein, “A cubic counterpart of Jacobi's identity and the AGM,” Transactions of the American Mathematical Society, vol. 323, no. 2, pp. 691–701, 1991. View at Publisher · View at Google Scholar
  2. B. C. Berndt, Number Theory in the Spirit of Ramanujan, vol. 34, American Mathematical Society, Providence, RI, USA, 2006.
  3. J. M. Borwein, P. B. Borwein, and F. G. Garvan, “Some cubic modular identities of Ramanujan,” Transactions of the American Mathematical Society, vol. 343, no. 1, pp. 35–47, 1994. View at Publisher · View at Google Scholar
  4. S. Ramanujan, Notebooks, Vols. 1, 2, Tata Institute of Fundamental Research, Mumbai, India, 1957.
  5. B. C. Berndt, S. Bhargava, and F. G. Garvan, “Ramanujan's theories of elliptic functions to alternative bases,” Transactions of the American Mathematical Society, vol. 347, no. 11, pp. 4163–4244, 1995. View at Publisher · View at Google Scholar
  6. H. H. Chan, “On Ramanujan's cubic transformation formula for 2F1(13,23;1;z),” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 124, no. 2, pp. 193–204, 1998. View at Publisher · View at Google Scholar
  7. Z.-G. Liu, “The Borweins' cubic theta function identity and some cubic modular identities of Ramanujan,” The Ramanujan Journal, vol. 4, no. 1, pp. 43–50, 2000. View at Publisher · View at Google Scholar
  8. Z.-G. Liu, “Some Eisenstein series identities,” Journal of Number Theory, vol. 85, no. 2, pp. 231–252, 2000. View at Publisher · View at Google Scholar
  9. A. Alaca, S. Alaca, and K. S. Williams, “On the two-dimensional theta functions of the Borweins,” Acta Arithmetica, vol. 124, no. 2, pp. 177–195, 2006. View at Publisher · View at Google Scholar
  10. A. Alaca, S. Alaca, and K. S. Williams, “Evaluation of the convolution sums l+18m=nσ(l)σ(m) and 2l+9m=nσ(l)σ(m),” International Mathematical Forum. Journal for Theory and Applications, vol. 2, no. 1–4, pp. 45–68, 2007. View at Google Scholar
  11. S. Alaca and K. S. Williams, “Evaluation of the convolution sums l+6m=nσ(l)σ(m) and 2l+3m=nσ(l)σ(m),” Journal of Number Theory, vol. 124, no. 2, pp. 491–510, 2007. View at Publisher · View at Google Scholar
  12. A. Alaca, S. Alaca, and K. S. Williams, “Evaluation of the convolution sums l+12m=nσ(l)σ(m) and 3l+4m=nσ(l)σ(m),” Advances in Theoretical and Applied Mathematics, vol. 1, no. 1, pp. 27–48, 2006. View at Google Scholar
  13. A. Alaca, S. Alaca, and K. S. Williams, “Evaluation of the convolution sums l+24m=nσ(l)σ(m) and 3l+8m=nσ(l)σ(m),” Mathematical Journal of Okayama University, vol. 49, pp. 93–111, 2007. View at Google Scholar
  14. A. Alaca, S. Alaca, and K. S. Williams, “The convolution sum m<n/16σ(m)σ(n-16m),” Canadian Mathematical Bulletin. Bulletin Canadien de Mathématiques, vol. 51, no. 1, pp. 3–14, 2008. View at Publisher · View at Google Scholar
  15. J. G. Huard, Z. M. Ou, B. K. Spearman, and K. S. Williams, “Elementary evaluation of certain convolution sums involving divisor functions,” in Number Theory for the Millennium, II, M. A. Bennett, B. C. Berndt, N. Boston, H. G. Diamond, A. J. Hildebrand, and W. Philipp, Eds., pp. 229–274, A K Peters, Natick, Mass, USA, 2002. View at Google Scholar
  16. M. Lemire and K. S. Williams, “Evaluation of two convolution sums involving the sum of divisors function,” Bulletin of the Australian Mathematical Society, vol. 73, no. 1, pp. 107–115, 2006. View at Publisher · View at Google Scholar
  17. E. Royer, “Evaluating convolution sums of the divisor function by quasimodular forms,” International Journal of Number Theory, vol. 3, no. 2, pp. 231–261, 2007. View at Publisher · View at Google Scholar
  18. K. S. Williams, “The convolution sum m<n/9σ(m)σ(n-9m),” International Journal of Number Theory, vol. 1, no. 2, pp. 193–205, 2005. View at Publisher · View at Google Scholar
  19. K. S. Williams, “The convolution sum m<n/8σ(m)σ(n-8m),” Pacific Journal of Mathematics, vol. 228, no. 2, pp. 387–396, 2006. View at Publisher · View at Google Scholar