Inner bounds for the extreme zeros of 3F2 hypergeometric polynomials

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.The first author would like to thank Alexander von Humboldt Foundation and TWAS for rewarding an AGNES Grant for Junior Researchers 2014, as well as TWAS and DFG for sponsoring a research visit at the Institute of Mathematics of the University of Kassel in 2016 (reference KO 1122/12-1). The second author would like to thank TWAS and DFG for sponsoring a research visit at the Institute of Mathematics of the University of Kassel in 2015 (reference 3240278140).http://www.tandfonline.com/loi/gitr202018-09-30hb2017Mathematics and Applied Mathematic