q-Rational and q-Real Binomial Coefficients

2023/01/19 by John Machacek, Machacek, John, Nicholas Ovenhouse +1
Mathematics · Physics and Astronomy · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Combinatorics (math.CO) #FOS: Mathematics #Nonlinear Waves and Solitons #Quantum Algebra (math.QA)

paper · pdf · doi:10.48550/arxiv.2301.08185

openalex publication_date 2023/01/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider q-binomial coefficients built from the q-rational and q-real numbers defined by Morier-Genoud and Ovsienko in terms of continued fractions. We establish versions of both the q-Pascal identity and the q-binomial theorem in this setting. These results are then used to find more identities satisfied by the q-analogues of Morier-Genoud and Ovsienko, including a Chu--Vandermonde identity and q-Gamma function identities.

Related

• Two new families of q-positive integers
• Higher q-Continued Fractions
• Identities for the q-harmonic numbers and q-binomial coefficients
• Multiple analogues of binomial coefficients and related families of special numbers
• Algebraic identities on q-harmonic numbers and q-binomial coefficients