[Paper Review] The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue

Roelof Koekoek, René F. Swarttouw|arXiv (Cornell University)|Feb 20, 1996

Mathematical functions and polynomials368 references1,234 citations

TL;DR

This seminal paper systematically classifies hypergeometric and basic hypergeometric orthogonal polynomials within the Askey scheme and its q-analogue, providing comprehensive definitions, orthogonality relations, three-term recurrence relations, generating functions, and limit transitions between all classical families. The key contribution is a unified, exhaustive framework that establishes precise connections between all classical orthogonal polynomials and their q-analogues, enabling the derivation of one family from another via limiting processes or parameter specialization.

ABSTRACT

We list the so-called Askey-scheme of hypergeometric orthogonal polynomials. In chapter 1 we give the definition, the orthogonality relation, the three term recurrence relation and generating functions of all classes of orthogonal polynomials in this scheme. In chapeter 2 we give all limit relation between different classes of orthogonal polynomials listed in the Askey-scheme. In chapter 3 we list the q-analogues of the polynomials in the Askey-scheme. We give their definition, orthogonality relation, three term recurrence relation and generating functions. In chapter 4 we give the limit relations between those basic hypergeometric orthogonal polynomials. Finally in chapter 5 we point out how the `classical` hypergeometric orthogonal polynomials of the Askey-scheme can be obtained from their q-analogues.

Motivation & Objective

• To provide a complete, systematic classification of all classical hypergeometric orthogonal polynomials and their q-analogues.
• To unify the theory of classical orthogonal polynomials by establishing all possible limit relations between different families.
• To present a coherent framework for understanding the connections between the Askey scheme and its q-analogue through rigorous mathematical definitions and transformations.
• To serve as a foundational reference for researchers working with orthogonal polynomials by compiling all essential formulas and relations in one accessible report.
• To demonstrate how the classical hypergeometric polynomials can be derived as limiting cases of their q-analogues, thereby unifying the classical and q-theory perspectives.

Proposed method

• Define all classical hypergeometric orthogonal polynomials (e.g., Hermite, Laguerre, Jacobi, Meixner, Charlier) via their standard hypergeometric representations.
• Provide the three-term recurrence relation, orthogonality measure, and generating function for each polynomial family in the Askey scheme.
• Establish all possible limit relations between families in the Askey scheme by taking appropriate parameter limits (e.g., scaling parameters to zero or infinity).
• Introduce the q-analogue of each classical polynomial using q-hypergeometric functions and q-shifted factorials.
• Present the q-analogue families (e.g., q-Racah, q-Hahn, continuous q-Hermite) with their respective orthogonality relations, recurrence relations, and generating functions.
• Demonstrate that classical hypergeometric polynomials arise as limiting cases of their q-analogues by taking the limit q → 1−, thereby unifying the classical and q-theory frameworks.

Experimental results

Research questions

• RQ1 How are all classical hypergeometric orthogonal polynomials in the Askey scheme related through limiting processes?
• RQ2 What are the precise mathematical definitions, orthogonality measures, recurrence relations, and generating functions for each polynomial family in the Askey scheme?
• RQ3 How do the q-analogues of classical orthogonal polynomials relate to their classical counterparts via limiting procedures?
• RQ4 What are the complete sets of limit relations between the basic hypergeometric orthogonal polynomials in the q-Askey scheme?
• RQ5 Can the classical hypergeometric orthogonal polynomials be systematically derived as special cases of their q-analogues?

Key findings

• All classical hypergeometric orthogonal polynomials in the Askey scheme are interconnected via well-defined limit relations, such as Wilson → continuous dual Hahn → continuous Hahn → Jacobi.
• The q-analogue of the Askey scheme, including families like q-Racah, q-Hahn, and continuous q-Hermite, is fully characterized by their orthogonality measures, recurrence relations, and generating functions.
• The classical hypergeometric polynomials (e.g., Hermite, Laguerre, Jacobi) emerge as limiting cases of their q-analogues when q → 1−, confirming the consistency of the q-theory with the classical theory.
• The continuous q-Hermite polynomials are orthogonal on the interval [−1, 1] with respect to a weight function involving q-Pochhammer symbols and trigonometric parameters.
• The discrete q-Hermite I and II polynomials are related by a simple transformation involving imaginary units and parameter inversion, demonstrating deep symmetry in the q-theory.
• The Stieltjes–Wigert polynomials are orthogonal with respect to a log-normal weight function, and their moment problem is indeterminate, allowing for multiple weight functions, a key feature in q-orthogonal polynomial theory.