Askey scheme

Abstract: Foreword Preface 1. Preliminaries 2. Orthogonal polynomials 3. Differential equations, Discriminants and electrostatics 4. Jacobi polynomials 5. Some inverse problems 6. Discrete orthogonal polynomials 7. Zeros and inequalities 8. Polynomials orthogonal on the unit circle 9. Linearization, connections and integral representations 10. The Sheffer classification 11. q-series Preliminaries 12. q-Summation theorems 13. Some q-Orthogonal polynomials 14. Exponential and q-bessel functions 15. The Askey-Wilson polynomials 16. The Askey-Wilson operators 17. q-Hermite polynomials on the unit circle 18. Discrete q-orthogonal polynomials 19. Fractional and q-fractional calculus 20. Polynomial solutions to functional equations 21. Some indeterminate moment problems 22. The Riemann-Hilbert problem for orthogonal polynomials 23. Multiple orthogonal polynomials 24. Research problems Bibliography Index Author index.

Abstract: Let $K$ denote a field, and let $V$ denote a vector space over $K$ with finite positive dimension. We consider a pair of linear transformations $A:V\to V$ and $A^*:V\to V$ that satisfy the following two conditions: There exists a basis for $V$ with respect to which the matrix representing $A$ is irreducible tridiagonal and the matrix representing $A^*$ is diagonal. There exists a basis for $V$ with respect to which the matrix representing $A^*$ is irreducible tridiagonal and the matrix representing $A$ is diagonal. We call such a pair a Leonard pair on $V$. We give a correspondence between Leonard pairs and a class of orthogonal polynomials. This class coincides with the terminating branch of the Askey scheme and consists of the $q$-Racah, $q$-Hahn, dual $q$-Hahn, $q$-Krawtchouk, dual $q$-Krawtchouk, quantum $q$-Krawtchouk, affine $q$-Krawtchouk, Racah, Hahn, dual Hahn, Krawtchouk, Bannai/Ito, and orphan polynomials. We describe the above correspondence in detail. We show how, for the listed polynomials, the 3-term recurrence, difference equation, Askey-Wilson duality, and orthogonality can be expressed in a uniform and attractive manner using the corresponding Leonard pair. We give some examples that indicate how Leonard pairs arise in representation theory and algebraic combinatorics. We discuss a mild generalization of a Leonard pair called a tridiagonal pair. At the end we list some open problems. Throughout these notes our argument is elementary and uses only linear algebra. No prior exposure to the topic is assumed.