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.

scholarly journals A simple proof of an identity of Ramanujan

1983 ◽  
Vol 34 (1) ◽  
pp. 31-35 ◽  
Author(s):  
M. D. Hirschhorn
Keyword(s):  
Simple Proof ◽  
Triple Product ◽  
Jacobi’S Triple Product Identity ◽  
Product Identity ◽  
Jacobi's Triple Product Identity

AbstractOne of Ramanujan's unpublished, unproven identities has excited considerable interest over the years. Indeed, no fewer than four proofs have appeared in the literature. The object of this note is to present yet another proof, simpler than the others, relying only on Jacobi's triple product identity.

scholarly journals A simple proof of Jacobi's four-square theorem

1982 ◽  
Vol 32 (1) ◽  
pp. 61-67 ◽  
Author(s):  
M. D. Hirschhorn
Keyword(s):  
Positive Integer ◽  
Simple Proof ◽  
Triple Product ◽  
Jacobi’S Triple Product Identity ◽  
Product Identity ◽  
Four Square ◽  
Jacobi's Triple Product Identity

AbstractA celebrated result, due to Jacobi, says that the number of representations of the positive integer n as a sum of four squares is equal to eight times the sum of the divisors of n which are not divisible by 4. We give a new and simple proof of this result which depends only on Jacobi's triple 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 A new form of the quintuple product identity and its application

Filomat ◽  
2017 ◽  
Vol 31 (7) ◽  
pp. 1869-1873
Author(s):  
Bhaskar Srivastava
Keyword(s):  
Simple Proof ◽  
Continued Fractions ◽  
Direct Application ◽  
General Identity ◽  
Quintuple Product Identity ◽  
Product Identity ◽  
New Form

We give a new form of the quintuple product identity. As a direct application of this new form a simple proof of known identities of Ramanujan and also new identities for other well known continued fractions are given. We also give and prove a general identity for (q3m; q3m)?.

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.

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