Properties and zeros of 3F2 hypergeometric functions

• Main Author: Johnston, Sarah Jane
• Format: Others
• Language: en
• Published: 2006
• Subjects: hypergeometric functions; hypergeometric polynomials; euler integral representation; asymptotics; orthoganality; quasi-orthoganality
• Online Access: http://hdl.handle.net/10539/1510

Student Number : 9606114D PhD Thesis School of Mathematics Faculty of Science === In 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.