scholarly journals Triple product identity, Quintuple product identity and Ramanujan's differential equations for the classical Eisenstein series

2007 ◽  
Vol 135 (07) ◽  
pp. 1987-1993 ◽  
Author(s):  
Heng Huat Chan
Keyword(s):  
Differential Equations ◽  
Eisenstein Series ◽  
Triple Product ◽  
Quintuple Product Identity ◽  
Product Identity

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 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 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.

CONSTRUCTION OF EISENSTEIN SERIES FOR Γ0(p)

2009 ◽  
Vol 05 (05) ◽  
pp. 765-778 ◽  
Author(s):  
SHAUN COOPER
Keyword(s):  
Eisenstein Series ◽  
Congruence Subgroup ◽  
Gauss Sums ◽  
Triple Product ◽  
Simple Construction ◽  
Dedekind Eta Function ◽  
Modular Transformation ◽  
Product Identity ◽  
Jacobi Triple Product Identity ◽  
Extra Difficulty

A simple construction of Eisenstein series for the congruence subgroup Γ0(p) is given. The construction makes use of the Jacobi triple product identity and Gauss sums, but does not use the modular transformation for the Dedekind eta-function. All positive integral weights are handled in the same way, and the conditionally convergent cases of weights 1 and 2 present no extra difficulty.

scholarly journals Truncated Series with Nonnegative Coefficients from the Jacobi Triple Product

2022 ◽  
Vol Volume 44 - Special... ◽  
Author(s):  
Liuquan Wang
Keyword(s):  
Analogous Result ◽  
Triple Product ◽  
Combinatorial Proof ◽  
Jacobi’S Triple Product Identity ◽  
Product Identity ◽  
Pentagonal Number Theorem ◽  
Nonnegative Coefficients ◽  
Jacobi's Triple Product Identity

Andrews and Merca investigated a truncated version of Euler's pentagonal number theorem and showed that the coefficients of the truncated series are nonnegative. They also considered the truncated series arising from Jacobi's triple product identity, and they conjectured that its coefficients are nonnegative. This conjecture was posed by Guo and Zeng independently and confirmed by Mao and Yee using different approaches. In this paper, we provide a new combinatorial proof of their nonnegativity result related to Euler's pentagonal number theorem. Meanwhile, we find an analogous result for a truncated series arising from Jacobi's triple product identity in a different manner.

Sign in / Sign up

Export Citation Format

Share Document