Combinatorial Proofs of Ismail's Identities on Al-Salam--Chirara Polynomials

TL;DR

The paper provides purely combinatorial proofs of two Ismail-type $q$-series identities by translating them into partition-generating-function statements. It develops a framework based on partitions, Durfee squares, and $q$-binomial coefficients, using 2-modular diagrams and Sylvester-type bijections to connect minimal configurations with Ferrers diagrams and to establish equal generating counts. The first proof (Theorem 1) equates counts of partition pairs under a linear constraint with counts of minimal configurations; the second proof (Theorem 2) leverages 2-modular constructions to realize Ramanujan-type identities via explicit bijections and $q$-binomial generating functions, including a combinatorial treatment of Ramanujan’s function. Together, these results deepen the link between $q$-series identities, partition theory, and orthogonal polynomials, and point to further refinements and algorithmic approaches (e.g., $q$-Zeilberger) for related identities.

Abstract

We present new proofs of two identities arising in the work of Mourad Ismail using partition theoretic generating function interpretations.

Theorems & Definitions (8)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Corollary 1.5
• Theorem 3.1
• Theorem 5.1
• Theorem 5.2