Search

Abstract: We study the asymptotic distribution of the zeros of certain classes of Gauss hypergeometric polynomials. Classical analytic methods such as Watson’s lemma and the method of steepest descents are used to analyze the large-n behavior of the zeros of the Gauss hypergeometric polynomials:where n is a nonnegative integer, the constants k, a, τ, λ >0, p >1, and . Owing to the connection between Jacobi polynomials and Gauss hypergeometric polynomials, we prove a special case of a conjecture made by Martínez–Finkelshtein, Martínez–González, and Orive. Numerical evidence and graphical illustrations of the clustering of zeros on certain curves are generated by Mathematica (Version 4.0). [Copyright &y& Elsevier]

In this paper, we investigate the uniform asymptotic behavior of the single variable Bell polynomials on the negative real axis, to which all zeros belong. It is found that there exists an ascending sequence {Zk}1∞⊂(−e,0) such that the polynomials are represented by a finite sum of infinite asymptotic series, each in terms of the Airy function and its derivative, and the number of series under this sum is 1 in the interval (−∞,Z1) and k+1 in [Zk,Zk+1), k⩾1. Furthermore, it is shown that an asymptotic expansion, also in terms of Airy function and its derivative, completed with error bounds, holds uniformly in (−∞,−δ] for positive δ. [Copyright &y& Elsevier]