Recurrence equations and their classical orthogonal polynomial solutions on a quadratic or a q-quadratic lattice

ABSTRACT

If $\left(p_n\right(x){)}_{n\ge 0}$ is an orthogonal polynomial system, then $p_n\left(x\right)$ satisfies a three-term recurrence relation of type $p_{n+1}\left(x\right)=(A_{n}x+B_n)p_n\left(x\right)-C_n{p}_{n-1}\left(x\right)(n=0,1,2,\dots ,p_{-1}\equiv 0),$ with $C_n{A}_n\ne 0$. On the other hand, Favard's theorem states that the converse is true. A general method to derive the coefficients $A_n$, $B_n$, $C_n$ in terms of the polynomial coefficients of the divided-difference equations satisfied by orthogonal polynomials on a quadratic or q-quadratic lattice is recalled. If a three-term recurrence relation is given as input, the Maple implementations rec2ortho of Koorwinder and Swarttouw or retode of Koepf and Schmersau can identify its solution which is a (linear transformation of a) classical orthogonal polynomial system of a continuous, a discrete or a q-discrete variable, if applicable. The two implementations rec2ortho and retode do not handle classical orthogonal polynomials on a quadratic or q-quadratic lattice. Motivated by an open problem, submitted by Alhaidari during the 14th International Symposium on Orthogonal Polynomials, Special Functions and Applications, which will serve as application, the Maple implementation retode of Koepf and Schmersau is extended to cover classical orthogonal polynomial solutions on quadratic or q-quadratic lattices of three-term recurrence relations.

KEYWORDS:

• Classical orthogonal polynomials
• three-term recurrence equation
• divided-difference equation
• retode

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 65Q10
• 65Q30
• 33C45
• 33D45
• 33F99

Acknowledgements

The author is grateful to the referees for the valuable comments and suggestions which considerably improved the paper. The author thanks Wolfram Koepf and W. Van Assche, both for bringing his attention to the open problem by Alhaidari and for helpful discussions to improve the Maple implementation.

Disclosure statement

No potential conflict of interest was reported by the author.

Additional information

Funding

This work has been supported by the Institute of Mathematics of the University of Kassel, Germany.