Kerstin Heidrun Jordaan

Interlacing of the zeros of Jacobi polynomials with different parameters

We prove results for the interlacing of zeros of Jacobi polynomials of the same or adjacent degree as one or both of the parameters are shifted continuously within a certain range. Numerical examples are given to illustrate situations where interlacing fails to occur.

Interlacing of zeros of shifted sequences of one-parameter orthogonal polynomials

We study the interlacing property of zeros of Laguerre polynomials of adjacent degree, where the free parameters differ by an integer, and of the same degree, where the free parameter is shifted continuously. Similar interlacing results are proven for the positive zeros of Gegenbauer polynomials.

The Relationship Between Semiclassical Laguerre Polynomials and the Fourth Painlevé Equation

We discuss the relationship between the recurrence coefficients of orthogonal polynomials with respect to a semiclassical Laguerre weight and classical solutions of the fourth Painlevé equation. We show that the coefficients in these recurrence relations can be expressed in terms of Wronskians of parabolic cylinder functions that arise in the description of special function solutions of the fourth Painlevé equation.

Full length article: Bounds for extreme zeros of some classical orthogonal polynomials

We use mixed three term recurrence relations typically satisfied by classical orthogonal polynomials from sequences corresponding to different parameters to derive upper (lower) bounds for the smallest (largest) zeros of Jacobi, Laguerre and Gegenbauer polynomials.

Convergence of ray sequences of padé approximants for 2 F 1 ( a , 1; c ; z ), ( c 〉 a 〉 0)

The Padé table of 2 F 1(a, 1; c; z) is normal for c > a > 0 (cf. [4]). For m ≥ n - 1 and c ∉ Z-, the denominator polynomial Q mn (z) in the [m/n] Padé approximant P mn (z)/Q mn (z) for 2 F 1(a, 1; c; z) and the remainder term Q mn (z)2 F 1(a, 1; c; z)-Pmn (z) were explicitly evaluated by Padé (cf. [2], [6] or [9]). We show that for c > a > 0 and m ≥ n - 1, the poles of Pmn (z)/Qmn (z) lie on the cut (1,∞). We deduce that the sequence of approximants Pmn (z)/Qmn (z) converges to 2 F 1(a, 1; c; z) as m → ∞, n/m → ρ with 0 < ρ ≤ 1, uniformly on compact subsets of the unit disc |z| < 1 for c > a > 0.

Convexity of the zeros of some orthogonal polynomials and related functions

We study convexity properties of the zeros of some special functions that follow from the convexity theorem of Sturm. We prove results on the intervals of convexity for the zeros of Laguerre, Jacobi and ultraspherical polynomials, as well as functions related to them, using transformations under which the zeros remain unchanged. We give upper as well as lower bounds for the distance between consecutive zeros in several cases.

Asymptotic zero distribution of a class of 3F2 hypergeometric functions

Abstract We study the asymptotic behavior of the zeros of certain families of 3F2 functions. Classical tools are used to analyse the asymptotic behavior of the zeros of the polynomial 3 F 2 ( −n, n+1, 1 2 b+n+1, 1−b−n ;z) In addition, families of 3F2 functions that are connected in a formulaic sense with Gauss hypergeometric polynomials of the form 2 F 1 ( −n, kn+1 kn+2 ;z), k > 0 and 2 F 1 ( −n,b −2n ;z) b > 0 are investigated. Numerical evidence of the clustering o zeros on certain curves is generated by Mathematica.

A generalized Freud weight

The London Mathematical Society for support through a “Research in Pairs” grant, as well as the National Research Foundation of South Africa.

Zeros of \(_3 F_2 (_{d,e}^{ - n,b,c} ;z) \) Polynomials

We establish the location of the zeros of several classes of 3F2 hypergeometric polynomials that admit representations as various kinds of products involving 2F1 polynomials. We categorise the 3F2 polynomials considered here according to whether they are well-poised or k-balanced. Our results include and extend those obtained in [5].

Bounds for zeros of Meixner and Kravchuk polynomials

The zeros of certain different sequences of orthogonal polynomials interlace in a well-defined way. The study of this phenomenon and the conditions under which it holds lead to a set of points that can be applied as bounds for the extreme zeros of the polynomials. We consider different sequences of the discrete orthogonal Meixner and Kravchuk polynomials and use mixed three-term recurrence relations, satisfied by the polynomials under consideration, to identify bounds for the extreme zeros of Meixner and Kravchuk polynomials.