کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
---|---|---|---|---|
6418400 | 1339326 | 2014 | 19 صفحه PDF | دانلود رایگان |
We define the generalized hypergeometric polynomial of degree N as follows:PN(α1,...,αp;β1,...,βq;z)=âm=0N[(âN)m(α1)mâ â â (αp)mzNâmm!(β1)mâ â â (βq)m]=zNFqp+1(âN,α1,...,αp;β1,...,βq;1/z). Here N is an arbitrary positive integer, p and q are arbitrary nonnegative integers, the p+q parameters αj and βk are arbitrary (“generic”, possibly complex) numbers, (α)m is the Pochhammer symbol and Fqp+1(α0,α1,...,αp;β1,...,βq;z) is the generalized hypergeometric function. In this paper we obtain a set of N nonlinear algebraic equations satisfied by the N zeros ζn of this polynomial. We moreover manufacture an NÃN matrix L̲ in terms of the 1+p+q parameters N, αj, βk characterizing this polynomial, and of its N zeros ζn, and we show that it features the N eigenvalues λm=mâk=1q(âβk+1âm), m=1,...,N. These N eigenvalues depend only on the q parameters βk, implying that the NÃN matrix L̲ is isospectral for variations of the p parameters αj; and they clearly are integer (or rational) numbers if the q parameters βk are themselves integer (or rational) numbers: a nontrivial Diophantine property.
Journal: Journal of Mathematical Analysis and Applications - Volume 419, Issue 2, 15 November 2014, Pages 1076-1094