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)?.

scholarly journals A generalisation of the quintuple product identity

1988 ◽  
Vol 44 (1) ◽  
pp. 42-45 ◽  
Author(s):  
M. D. Hirschhorn
Keyword(s):  
General Identity ◽  
Quintuple Product Identity ◽  
Product Identity

AbstractThe quintuple product identity has appeared many times in the literature. Indeed, no fewer than 12 proofs have been given. We establish a more general identity from which the quintuple product identity follows in two ways.

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 Hirschhorn's identities

1999 ◽  
Vol 60 (1) ◽  
pp. 73-80 ◽  
Author(s):  
Paul Hammond ◽  
Richard Lewis ◽  
Zhi-Guo Liu
Keyword(s):  
Power Series ◽  
General Identity ◽  
Quintuple Product Identity ◽  
Product Identity ◽  
Winquist’S Identity ◽  
Winquist's Identity

We prove a general identity between power series and use this identity to give proofs of a number of identities proposed by M.D. Hirschhorn. We also use the identity to give proofs of a well-known result of Jacobi, the quintuple-product identity and Winquist's 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AbstractOne of Ramanujan's unpublished, unproven identities has excited considerable interest over the years.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.

Short and Easy Computer Proofs of the Rogers-Ramanujan Identities and of Identities of Similar Type

10.37236/1190 ◽  
1994 ◽  
Vol 1 (1) ◽  
Author(s):  
Peter Paule
Keyword(s):  
Short Proof ◽  
Similar Type ◽  
Quintuple Product Identity ◽  
Product Identity ◽  
Computer Proofs ◽  
Creative Telescoping ◽  
Ramanujan Identity

New short and easy computer proofs of finite versions of the Rogers-Ramanujan identities and of similar type are given. These include a very short proof of the first Rogers-Ramanujan identity that was missed by computers, and a new proof of the well-known quintuple product identity by creative telescoping.

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