The q-Binomial Coefficient for Negative Arguments and Some q-Binomial Summation Identities

2022/11/15 by Kronenburg, M. J.
#Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2211.08256

Abstract

Using a property of the q-shifted factorial, an identity for q-binomial coefficients is proved, which is used to derive the formulas for the q-binomial coefficient for negative arguments. The result is in agreement with an earlier paper about the normal binomial coefficient for negative arguments. Some new q-binomial summation identities are derived, and the formulas for negative arguments transform some of these summation identities into each other. One q-binomial summation identity is transformed into a new q-binomial summation identity.

Related

• The Binomial Coefficient for Negative Arguments
• The negative q-binomial
• The "Shape" of q-Binomial Coefficients
• The q-binomial formula and the Rogers dilogarithm identity
• A generalization of the Gaussian formula and a q-analog of Fleck's congruence