Inner bounds for the extreme zeros of 3F2 hypergeometric polynomials

Abstract

Zeilberger’s celebrated algorithm finds pure recurrence relations (w. r. t. a single variable) for hypergeometric sums automatically. However, in the theory of orthogonal polynomials and special functions, contiguous relations w. r. t. several variables exist in abundance. We modify Zeilberger’s algorithm to generate unknown contiguous relations that are necessary to obtain inner bounds for the extreme zeros of orthogonal polynomial sequences with 3F2 hypergeometric representations. Using this method, we improve previously obtained upper bounds for the smallest and lower bounds for the largest zeros of the Hahn polynomials and we identify inner bounds for the extreme zeros of the Continuous Hahn and Continuous Dual Hahn polynomials. Numerical examples are provided to illustrate the quality of the new bounds. Without the use of computer algebra such results are not accessible. We expect our algorithm to be useful to compute useful and new contiguous relations for other hypergeometric functions.