scholarly journals Ramanujan's remarkable summation formula as a $2$-papameter generalization of the quintuple product identity

2002 ◽  
Vol 33 (3) ◽  
pp. 285-288
Author(s):  
S. Bhargava ◽  
Chandrashekar Adiga ◽  
M. S. Mahadeva Naika
Keyword(s):  
Summation Formula ◽  
Triple Product ◽  
Binomial Theorem ◽  
Suitable Transformation ◽  
Quintuple Product Identity ◽  
Product Identity

It is well known that `Ramanujan's remarkable summation formula' unifies and generalizes the $q$-binomial theorem and the triple product identity and has numerous applications. In this note we will demonstrate how, after a suitable transformation of the series side, it can be looked upon as a $2$-parameter generalization of the quintuple product identity also.

scholarly journals QUINTUPLE PRODUCT IDENTITY AS A SPECIAL CASE OF RAMANUJAN'S 1ψ1 SUMMATION FORMULA

2011 ◽  
Vol 04 (01) ◽  
pp. 31-34
Author(s):  
S. Bhargava ◽  
Chandrashekar Adiga ◽  
M. S. Mahadeva Naika
Keyword(s):  
Summation Formula ◽  
Triple Product ◽  
Binomial Theorem ◽  
Interesting Fact ◽  
Quintuple Product Identity ◽  
Product Identity ◽  
Jacobi Triple Product Identity ◽  
Special Case

In this note we observe an interesting fact that the well-known quintuple product identity can be regarded as a special case of the celebrated 1ψ1 summation formula of Ramanujan which is known to unify the Jacobi triple product identity and the q -binomial theorem.

scholarly journals Another simple proof of the quintuple product identity

2005 ◽  
Vol 2005 (15) ◽  
pp. 2511-2515 ◽  
Author(s):  
Hei-Chi Chan
Keyword(s):  
Simple Proof ◽  
Triple Product ◽  
Quintuple Product Identity ◽  
Product Identity

We give a simple proof of the well-known quintuple product identity. The strategy of our proof is similar to a proof of Jacobi (ascribed to him by Enneper) for the triple product identity.

A BIJECTIVE PROOF OF THE QUINTUPLE PRODUCT IDENTITY

2010 ◽  
Vol 06 (02) ◽  
pp. 247-256 ◽  
Author(s):  
SUN KIM
Keyword(s):  
Recurrence Relation ◽  
Triple Product ◽  
Jacobi’S Triple Product Identity ◽  
Bijective Proof ◽  
Quintuple Product Identity ◽  
Product Identity ◽  
Jacobi's Triple Product Identity

We give a bijective proof of the quintuple product identity using bijective proofs of Jacobi's triple product identity and Euler's recurrence relation.

scholarly journals A Note on Four-Variable Reciprocity Theorem

2009 ◽  
Vol 2009 ◽  
pp. 1-9
Author(s):  
Chandrashekar Adiga ◽  
P. S. Guruprasad
Keyword(s):  
Summation Formula ◽  
Reciprocity Theorem ◽  
Triple Product ◽  
Jacobi’S Triple Product Identity ◽  
Product Identity ◽  
Jacobi's Triple Product Identity

We give new proof of a four-variable reciprocity theorem using Heine's transformation, Watson's transformation, and Ramanujan's -summation formula. We also obtain a generalization of Jacobi's triple product identity.

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