A generalisation of the quintuple 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.

A new form of the quintuple product identity and its application

Filomat ◽

10.2298/fil1707869s ◽

2017 ◽

Vol 31 (7) ◽

pp. 1869-1873

Author(s):

Bhaskar Srivastava

Keyword(s):

Simple Proof ◽

Continued Fractions ◽

Direct Application ◽

General 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)?.

Hirschhorn's identities

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700033347 ◽

1999 ◽

Vol 60 (1) ◽

pp. 73-80 ◽

Cited By ~ 7

Author(s):

Paul Hammond ◽

Richard Lewis ◽

Zhi-Guo Liu

Keyword(s):

Power Series ◽

General 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.

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

Proceedings of the American Mathematical Society ◽

10.1090/s0002-9939-07-08723-0 ◽

2007 ◽

Vol 135 (07) ◽

pp. 1987-1993 ◽

Cited By ~ 7

Author(s):

Heng Huat Chan

Keyword(s):

Differential Equations ◽

Eisenstein Series ◽

Triple Product ◽

A simple proof of the quintuple product identity

Proceedings of the American Mathematical Society ◽

10.1090/s0002-9939-1972-0289316-2 ◽

1972 ◽

Vol 32 (1) ◽

pp. 42-42 ◽

Cited By ~ 25

Author(s):

L. Carlitz ◽

M. V. Subbarao

Keyword(s):

Simple Proof ◽

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

Tamkang Journal of Mathematics ◽

10.5556/j.tkjm.33.2002.285-288 ◽

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 ◽

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

The Electronic Journal of Combinatorics ◽

10.37236/1190 ◽

1994 ◽

Vol 1 (1) ◽

Cited By ~ 12

Author(s):

Peter Paule

Keyword(s):

Short Proof ◽

Similar Type ◽

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.

Another simple proof of the quintuple product identity

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/ijmms.2005.2511 ◽

2005 ◽

Vol 2005 (15) ◽

pp. 2511-2515 ◽

Cited By ~ 4

Author(s):

Hei-Chi Chan

Keyword(s):

Simple Proof ◽

Triple Product ◽

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.

The quintuple product identity and shifted partition functions

Journal of Computational and Applied Mathematics ◽

10.1016/0377-0427(95)00251-0 ◽

1996 ◽

Vol 68 (1-2) ◽

pp. 3-13 ◽

Cited By ~ 4

Author(s):

Krishnaswami Alladi

Keyword(s):

Partition Functions ◽

A BIJECTIVE PROOF OF THE QUINTUPLE PRODUCT IDENTITY

International Journal of Number Theory ◽

10.1142/s1793042110002909 ◽

2010 ◽

Vol 06 (02) ◽

pp. 247-256 ◽

Cited By ~ 3

Author(s):

SUN KIM

Keyword(s):

Recurrence Relation ◽

Triple Product ◽

Jacobi’S Triple Product Identity ◽

Bijective Proof ◽

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.

ON GENERAL SERIES-PRODUCT IDENTITIES

International Journal of Number Theory ◽

10.1142/s1793042109002596 ◽

2009 ◽

Vol 05 (06) ◽

pp. 1129-1148 ◽

Cited By ~ 1

Author(s):

SIN-DA CHEN ◽

SEN-SHAN HUANG

Keyword(s):

General Series ◽

Product Identity ◽

Winquist’S Identity ◽

Series Product ◽

Winquist's Identity

We derive the general series-product identities from which we deduce several applications, including an identity of Gauss, the generalization of Winquist's identity by Carlitz and Subbarao, an identity for [Formula: see text], the quintuple product identity, and the octuple product identity.