ON GENERAL SERIES-PRODUCT IDENTITIES

2009 ◽  
Vol 05 (06) ◽  
pp. 1129-1148 ◽  
Author(s):  
SIN-DA CHEN ◽  
SEN-SHAN HUANG
Keyword(s):  
Quintuple Product Identity ◽  
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.

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 Two Simple Proofs of Winquist's Identity

10.37236/388 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Chutchai Nupet ◽  
Sarachai Kongsiriwong
Keyword(s):  
Triple Product ◽  
Roots Of Unity ◽  
Basic Properties ◽  
Product Identity ◽  
Jacobi Triple Product Identity ◽  
Winquist’S Identity ◽  
Winquist's Identity

We give two new proofs of Winquist's identity. In the first proof, we use basic properties of cube roots of unity and the Jacobi triple product identity. The latter does not use the Jacobi triple product identity.

scholarly journals Bijections and Congruences for Generalizations of Partition Identities of Euler and Guy

10.37236/1796 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
James A. Sellers ◽  
Andrew V. Sills ◽  
Gary L. Mullen
Keyword(s):  
Famous Theorem ◽  
Partition Identities ◽  
Product Identity ◽  
Powers Of 2 ◽  
Pentagonal Number Theorem ◽  
Series Product

In 1958, Richard Guy proved that the number of partitions of $n$ into odd parts greater than one equals the number of partitions of $n$ into distinct parts with no powers of 2 allowed, which is closely related to Euler's famous theorem that the number of partitions of $n$ into odd parts equals the number of partitions of $n$ into distinct parts. We consider extensions of Guy's result, which naturally lead to a new algorithm for producing bijections between various equivalent partition ideals of order 1, as well as to two new infinite families of parity results which follow from Euler's Pentagonal Number Theorem and a well-known series-product identity of Jacobi.

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 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.

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