[Paper Review] Continuous q-Hermite polynomials: An elementary approach

Johann Cigler|arXiv (Cornell University)|Jul 1, 2013

Advanced Mathematical Identities17 references3 citations

TL;DR

This paper presents an elementary approach to continuous q-Hermite polynomials, establishing their connections to Fibonacci, Lucas, and Chebyshev polynomials through q-analogues. It derives their properties using combinatorial identities and generating functions, revealing deep structural links within orthogonal polynomial families and providing a self-contained foundation for further study in q-special functions.

ABSTRACT

This overview article gives an elementary approach to continuous q-Hermite polynomials. We stress their relation to Fibonacci, Lucas and Chebyshev polynomials and to some q-analogues of these polynomials.

Motivation & Objective

• To provide a self-contained, elementary introduction to continuous q-Hermite polynomials without advanced analysis.
• To clarify the structural relationships between continuous q-Hermite polynomials and classical orthogonal polynomials such as Fibonacci, Lucas, and Chebyshev polynomials.
• To explore q-analogues of these classical sequences and their polynomial generalizations.
• To establish generating functions and recurrence relations using combinatorial and algebraic techniques.
• To unify various q-polynomial families under a common framework through elementary methods.

Proposed method

• Derives the continuous q-Hermite polynomials using a generating function approach based on q-exponential series.
• Establishes recurrence relations through direct algebraic manipulation and q-identity applications.
• Uses q-integer and q-factorial notation to express polynomial coefficients and identities.
• Demonstrates connections to Fibonacci and Lucas polynomials via q-analogue identities and generating functions.
• Relates continuous q-Hermite polynomials to Chebyshev polynomials through trigonometric and q-trigonometric substitutions.
• Employs combinatorial interpretations of coefficients to support structural insights and recurrence patterns.

Experimental results

Research questions

• RQ1 How can continuous q-Hermite polynomials be derived using only elementary algebraic and combinatorial tools?
• RQ2 What are the precise algebraic relationships between continuous q-Hermite polynomials and q-analogues of Fibonacci and Lucas polynomials?
• RQ3 In what way do continuous q-Hermite polynomials generalize classical Chebyshev polynomials in the q-setting?
• RQ4 How do generating functions for continuous q-Hermite polynomials reflect their structural and recursive properties?
• RQ5 What role do q-integer identities and q-factorials play in the coefficient structure of these polynomials?

Key findings

• The continuous q-Hermite polynomials are shown to satisfy a second-order linear recurrence relation with q-polynomial coefficients.
• A generating function for continuous q-Hermite polynomials is derived in terms of q-exponential functions and q-Pochhammer symbols.
• The polynomials are explicitly linked to q-Fibonacci and q-Lucas polynomials through functional identities and coefficient comparisons.
• The connection to Chebyshev polynomials is established via the limit as q approaches 1, recovering classical orthogonal polynomial behavior.
• Combinatorial interpretations of polynomial coefficients are provided using q-integer partitions and weighted lattice paths.
• The paper demonstrates that the q-Hermite polynomials form a natural bridge between classical orthogonal polynomials and their q-analogues.