Asymptotic Behavior of the Pollaczek Polynomials and Their Zeros

Abstract

In 1954, A. Novikoff studied the asymptotic behavior of the Pollaczek polynomials Pn(x; a, b) when , where t > 0 is fixed. He divided the positive t‐axis into two regions, 0 < t < (a + b)1/2 and t > (a + b)1/2, and derived an asymptotic formula in each of the two regions. Furthermore, he found an asymptotic formula for the zeros of these polynomials. Recently M. E. H. Ismail (1994) reconsidered this problem in an attempt to prove a conjecture of R. A. Askey and obtained a two‐term expansion for these zeros. Here we derive an infinite asymptotic expansion for , which holds uniformly for 0 < ε ≤ t ≤ M < ∞, and show that Ismail's result is incorrect.