On zeros of hypergeometric polynomials

Our focus, in this thesis, is on zeros of hypergeometric polynomials. Several problems in various areas of science can be seen in terms of the search of zeros of functions; and this search can be reduced to finding the zeros of approximating polynomials, since under some conditions, functions can be approximated by polynomials. In this thesis, we consider the zeros of a specific polynomial, namely the hypergeometric polynomial. We review some work done on the zero location and the asymptotic zero distribution of Gauss hypergeometric polynomials with real parameters. We extend some contiguous relations of 2F1 functions, and then we deduce the zero location for some classes of Gauss polynomials with non-real parameters. We study the asymptotic zero distribution of some classes of 3F2polynomials that extend results in the literature.Dissertation (MSc (Applied Mathematics))--University of Pretoria, 2007.Mathematics and Applied MathematicsMScunrestricte

Real zeros of 2F1 hypergeometric polynomials

We use a method based on the division algorithm to determine all the values of the real parameters b and c for which the hypergeometric polynomials 2F1(−n, b; c; z) have n real, simple zeros. Furthermore, we use the quasi-orthogonality of Jacobi polynomials to determine the intervals on the real line where the zeros are located.Research by the first author was supported by a Humboldt Research Fellowship for Experienced Researchers from the Alexander von Humboldt Foundation. Research by the third author was partially supported by the National Research Foundation under grant number 2054423.http://www.elsevier.com/locate/camhb201

Properties and zeros of 3F2 hypergeometric functions

Student Number : 9606114D PhD Thesis School of Mathematics Faculty of ScienceIn this thesis, our primary interest lies in the investigation of the location of the zeros and the asymptotic zero distribution of hypergeometric polynomials. The location of the zeros and the asymptotic zero distribution of general hy- pergeometric polynomials are linked with those of the classical orthogonal polynomials in some cases, notably 2F1 and 1F1 hypergeometric polynomials which have been extensively studied. In the case of 3F2 polynomials, less is known about their properties, including the location of their zeros, because there is, in general, no direct link with orthogonal polynomials. Our intro- duction in Chapter 1 outlines known results in this area and we also review recent papers dealing with the location of the zeros of 2F1 and 1F1 hyperge- ometric polynomials. In Chapter 2, we consider two classes of 3F2 hypergeometric polynomials, each of which has a representation in terms of 2F1 polynomials. Our first result proves that the class of polynomials 3F2(−n, a, b; a−1, d; x), a, b, d 2 R, n 2 N is quasi-orthogonal of order 1 on an interval that varies with the values of the real parameters b and d. We deduce the location of (n−1) of its zeros and dis- cuss the apparent role played by the parameter a with regard to the location of the one remaining zero of this class of polynomials. We also prove re- sults on the location of the zeros of the classes 3F2(−n, b, b−n 2 ; b−n, b−n−1 2 ; x), b 2 R, n 2 N and 3F2 (−n, b, b−n 2 + 1; b − n, b−n+1 2 ; x), n 2 N, b 2 R by using the orthogonality and quasi-orthogonality of factors involved in its representation. We use Mathematica to plot the zeros of these 3F2 hypergeometric polynomials for different values of n as well as for different ranges of the pa- rameters. The numerical data is consistent with the results we have proved. The Euler integral representation of the 2F1 Gauss hypergeometric function is well known and plays a prominent role in the derivation of transformation identities and in the evaluation of 2F1(a, b; c; 1), among other applications (cf. [1], p.65). The general p+kFq+k hypergeometric function has an integral repre- sentation (cf. [37], Theorem 38) where the integrand involves pFq. In Chapter 3, we give a simple and direct proof of an Euler integral representation for a special class of q+1Fq functions for q >= 2. The values of certain 3F2 and 4F3 functions at x = 1, some of which can be derived using other methods, are deduced from our integral formula. In Chapter 4, we prove that the zeros of 2F1 (−n, n+1 2 ; n+3 2 ; z) asymptotically approach the section of the lemniscate {z : |z(1 − z)2| = 4 27 ;Re(z) > 1 3} as n ! 1. In recent papers (cf. [31], [32], [34], [35]), Mart´ınez-Finkelshtein and Kuijlaars and their co-authors have used Riemann-Hilbert methods to derive the asymptotic distribution of Jacobi polynomials P(an,bn) n when the limits A = lim n!1 an n and B = lim n!1 Bn n exist and lie in the interior of certain specified regions in the AB-plane. Our result corresponds to one of the transitional or boundary cases for Jacobi polynomials in the Kuijlaars Mart´ınez-Finkelshtein classification

Mellin transforms with only critical zeros: generalized Hermite functions

We consider the Mellin transforms of certain generalized Hermite functions based upon certain generalized Hermite polynomials, characterized by a parameter $\mu>-1/2$. We show that the transforms have polynomial factors whose zeros lie all on the critical line. The polynomials with zeros only on the critical line are identified in terms of certain $_2F_1(2)$ hypergeometric functions, being certain scaled and shifted Meixner-Pollaczek polynomials. Other results of special function theory are presented.Comment: 17 pages, no figure