The generating condition for the extension of the classical Gauss series-product identity

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.

Download Full-text

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):

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.

Download Full-text

AN EXTENSION OF THE CLASSICAL GAUSS SERIES-PRODUCT IDENTITY BY BOSON–FERMIONIC REALIZATION OF THE AFFINE ALGEBRA $\widehat{\mathfrak{gl}}_n$

Journal of Algebra and Its Applications ◽

10.1142/s0219498810003793 ◽

2010 ◽

Vol 09 (01) ◽

pp. 123-133

Author(s):

TOMISLAV ŠIKIĆ

Keyword(s):

Lie Algebra ◽

Affine Algebra ◽

Affine Lie Algebra ◽

Product Identity ◽

Series Product

The main results of this paper are two infinite families of series-product identities which are based on a classical Gauss identity and two different interpretations of characters of fundamental modules for the affine Kac–Moody Lie algebra [Formula: see text], one of them stemming from a boson–fermionic realization of the affine Lie algebra [Formula: see text].

Download Full-text