Abstract and Applied Analysis

Volume 2013 (2013), Article ID 826472, 5 pages

http://dx.doi.org/10.1155/2013/826472

Research Article

A Note on Certain Modular Equations about Infinite Products of Ramanujan

Hong-Cun Zhai

Mathematics College, LuoYang Normal University, LuoYang 471022, China

Received 17 October 2013; Revised 8 November 2013; Accepted 8 November 2013

Academic Editor: Ming Li

Copyright © 2013 Hong-Cun Zhai. 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.

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.

1. Introduction

theory is undoubtedly one of the most famous and useful mathematical theorems, such as Andrews-Askey type integral [1]

Askey-Roy type integral [2]

Moment integrals [3]

(where),-Fractional Calculus Equations [4] and-Calculus [5]. For more information, please refer to [1–5].

The theta functions are very useful tool in researching -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:

where

sometimes written as

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]:

By definition of Ramanujan theta functions one can easily verify the following identities [8, page 45]:

From (4), if, we have

Thus settingand, we find that

Thus when, we have

Similarly we have that

The special case of these identities can be written as the following form by using Jacobian theta function [6, 7]:

The authors of [6, 7] give simple proofs and very important use of it.

In the above two identities, puttingand, we easily obtain

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,

in whichand. In the same way, we are able to give the simple and elementary proofs of the following identities of Ramanujan [8, 11, 12]:

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,

Proof. Note thatand. By (9), we get that

From (10), we have

Dividing by, respectively, and then applying (25), we derive

Multiplying by, respectively, we complete the proofs of (24).

Proof of (19). Let; then it is easy to know that and.

One has

Then we obtain that

In (16), letandthen we have that

Dividing by, respectively, we arrive at

Multiplying (31), combining with (33), and then multiplyed by, we are able to obtain (19).

Theorem 2. For,

Proof. First we recall that, and . Using (9), we have

Then we know easily that

In (13) and (14), setting, , , and, we get that

Dividing the above two equations by, respectively, and then combining with (37), we obtain that

Multiplied by, the identities (39) and (40) become (34) and (35).

Multiplying the two refinements in Theorems 1 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 ,

Acknowledgments

This work was supported by the National Science Foundation of China, Project nos. 11071107, 11371184, and U1304103. The author would like to thank the referee and editor for many valuable comments and suggestions.

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