Kathy Driver

Normality in Nikishin systems

Abstract A Nikishin system of analytic functions is considered. For such a system, it is shown that certain sets of indices are normal, i.e. the simultaneous rational (Hermite-Pade type II) approximants are unique. In particular, an assertion of E.M. Nikishin is proved. In addition to approximants developed at a single point, we also consider the multipoint case. Associated with simultaneous rational approximation is a dual problem (Hermite-Pade type I), for which normality of indices is investigated in both the single and multipoint case.

Zeros of the hypergeometric polynomials F(−n, b; 2b; z)

Abstract Our purpose is to study the zeros of hypergeometric polynomials, especially those of F(−n,b; 2b; z), where b > − 1 2 . Although some properties are implicit in known connections with classical orthogonal polynomials, whose zeros are well understood, the implications for zeros of hypergeometric polynomials appear not to have been generally recognized. The results for polynomials F(−n,b; 2b; z) are applied to give information about the zeros of other systems of hypergeometric functions, including Jacobi polynomials and Legendre functions. In a subsequent paper [5], we will discuss the behaviour of the zeros as b descends below the critical value − 1 2 .

Zeros of the Hypergeometric Polynomials F(-n, b; -2n; z)

We investigate the location of the zeros of the hypergeometric polynomial F(-n, b; -2n; z) for b real. The Hilbert-Klein formulas are used to specify the number of real zeros in the intervals (-~, 0), (0, 1), or (1, ~). For b>0 we obtain the equation of the Cassini curve which the zeros of w^nF(-n, b; -2n; 1/w) approach as n->~ and thereby prove a special case of a conjecture made by Marti@?nez-Finkelshtein, Marti@?nez-Gonzalez, and Orive. We also present some numerical evidence linking the zeros of F with more general Cassini curves.

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 zeros of linear combinations of orthogonal polynomials

Let {pn} be a sequence of monic polynomials with pn of degree n, that are orthogonal with respect to a suitable Borel measure on the real line. Stieltjes showed that if m < n and x1,.....,xn are the zeros of pn with x1 < ... < xn then there are m distinct intervals f the form (xj, xj+1) each containing one zero of Pm. Our main theorem proves a similar result with Pm replaced by some linear combinations of p1,....,pm. The interlacing of the zeros of linear combinations of two and three adjacent orthogonal polynomials is also discussed.

Asymptotic zero distribution of hypergeometric polynomials

AbstractWe show that the zeros of the hypergeometric polynomials

On the size of lemniscates of polynomials in one and several variables

F\left( { - n,kn + 1;kn + 2;z} \right)