Properties of the zeros of generalized hypergeometric polynomials

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:419

Article

Bihun, Oksana¹  Calogero, Francesco^2,3 

[1] Concordia Coll, Dept Math, Moorhead, MN 56562 USA

[2] Univ Roma La Sapienza, Dept Phys, I-00185 Rome, Italy

[3] Ist Nazl Fis Nucl, Sez Roma, Milan, Italy

关键词: Hypergeometric polynomials;    Diophantine properties;    Jacobi polynomials;    Isospectral matrices;    Special functions;   

DOI  :  10.1016/j.jmaa.2014.05.023

来源: Elsevier

【 摘 要 】

We define the generalized hypergeometric polynomial of degree N as follows: Here N is an arbitrary positive integer, p and q are arbitrary nonnegative integers, the p q parameters aj and /3k are arbitrary (generic, possibly complex) numbers, (a)n, is the Pochhammer symbol and p_FiFq (ao, al, oh,; is the generalized hypergeometric function. In this paper we obtain a set of N nonlinear algebraic equations satisfied by the N zeros Cn of this polynomial. We moreover manufacture an N x N matrix L in terms of the 1 p q parameters N, aj, f3e characterizing this polynomial, and of its N zeros (n, and we show that it features the N eigenvalues An, = m ru=,(-)3, +1- m), m = 1, N. These N eigenvalues depend only on the q parameters Pe, implying that the N x N matrix L is isospectral for variations of the p parameters aj; and they clearly are integer (or rational) numbers if the q parameters are themselves integer (or rational) numbers: a nontrivial Diophantine property. 2014 Elsevier Inc. All rights reserved.

【 授权许可】

Free