American Journal of Mathematics and Statistics

p-ISSN: 2162-948X    e-ISSN: 2162-8475

2012;  2(1): 25-28

doi: 10.5923/j.ajms.20120201.06

Identities for Analogous Ramanujan’s Functions by Jacobi’s Triple Product Identity

Bhaskar Srivastava

Department of Mathematics and Astronomy, University of Lucknow, Lucknow, India

Correspondence to: Bhaskar Srivastava , Department of Mathematics and Astronomy, University of Lucknow, Lucknow, India.

Email:

Copyright © 2012 Scientific & Academic Publishing. All Rights Reserved.

Abstract

From a continued fraction analogous to Rogers-Ramanujan continued fraction we have theta functions analogous to Ramanujan’s functions G(q) and H(q). We find certain identities for these analogous theta functions using Jacobi’s triple product identity, thus having a unified approach. We also find some modular equations.

Keywords: q-Hypergeometric Series, Jacobi’s Triple Product Identity, Theta Functions

Cite this paper: Bhaskar Srivastava , "Identities for Analogous Ramanujan’s Functions by Jacobi’s Triple Product Identity", American Journal of Mathematics and Statistics, Vol. 2 No. 1, 2012, pp. 25-28. doi: 10.5923/j.ajms.20120201.06.

Article Outline

1. Introduction
2. Basic Notation
4. Modular Equation

1. Introduction

In[5] considering the continued fraction for
(1.1)
(1.2)
(1.3)
which, I think, is analogous to the famous celebrated Rogers-Ramanujan continued fraction
(1.4)
(1.5)
(1.6)
I thought of defining the theta functions:
(1.7)
And
(1.8)
which are analogous to Ramanujan’s theta functions
(1.9)
and
(1.10)

2. Basic Notation

The following standard q-hypergeometric notations will be used:
For and for n ,
(2.1)
We also have the Ramanujan’s functions
(2.2)
(2.3)
(2.4)
(2.5)
and Jacobi’s triple product identity
(2.6)
We shall be needing the following two Lemmas of Ramanujan [2, Entry 29, p 45]
(2.7)
and
(2.8)
Also [2, Entry 30 (ii)-(iii), p 46]
(2.9)
and
(2.10)
Theorem 1
(3.1)
and
(3.2)
Proof of Theorem 1
Proof of (3.1)
where we have used (2.9) in the last step to prove (3.1).
Proof of (3.2)
Proof is exactly the same as in (3.1) only we have used (2.10) in the last step to obtain (3.2).
Theorem 2
(3.3)
(3.4)
(3.5)
(3.6)
Proof of Theorem 2
By definition
and
Proof of (3.3) and (3.4)
Put in (3.1) and (3.2) to get (3.3) and (3.4), respectively.
Proof of (3.5) and (3.6)
Put x=q2 in (3.1) and (3.2) to get (3.5) and (3.6), respectively.
Theorem 3
(3.7)
(3.8)
(3.9)
(3.10)
Proof of Theorem 3
Proof of (3.7)
we have used (2.7) in the penultimate step to obtain (3.7).
Proof of (3.8)
The proof is exactly the same as of (3.7) only we use (2.8) in penultimate step to obtain (3.8).
Proof of (3.9)
we have used (2.7) in the penultimate step to obtain (3.9).
Proof of (3.10)
The proof is exactly the same as in (3.9) only we have used (2.8) to obtain (3.10).
Theorem 4
(3.11)
(3.12)
Proof of Theorem 4
Proof of (3.11)
we have used (2.7) in the penultimate step to obtain (3.11).
Proof of (3.12)
The proof is exactly the same, only we use (2.8) in the penultimate step to obtain (3.12).
Theorem 5
(3.13)
(3.14)
(3.15)
Proof of Theorem 5
Proof of (3.13)
we have used (2.7) in the penultimate step to obtain (3.13).
Proof of (3.14)
The proof is same only in the penultimate step we have used (2.8) to obtain (3.14).
Proof of (3.15)
Multiply (3.13) and (3.14) to have (3.15).

4. Modular Equation

We prove the following :
Theorem 6
(4.1)
Proof of Theorem 6
Writing (4.1) as
(4.2)
Corollary 1
Put in (4.2), to get
Corollary 2
Put in (4.2), to get
Corollary 3
Put in (4.2), to get
Corollary 4
Put in (4.2), to get

References

[1]  G. E. Andrews, Application of basic hypergeometric functions, SIAM Rev.16 (1974), 441-484
[2]  B.C. Berndt, Ramanujan’s Notebooks, Part III, Springer-Verlag, New York, 1991
[3]  W. Chu and Q. Yan, Jacobi’s triple product identity and theta function identities, Math. Commun. 16 (2011), 191-203
[4]  G. Gasper and M. Rahman, Basic Hypergeometric Series, Cambridge University Press, Cambridge (1990)
[5]  Bhaskar Srivastava, on modular equations and Lambert series for a continued fraction of Ramanujan, Bull. Math. Anal. Appl. 3 (2011), 148-155