A question by Chihara about shell polynomials and indeterminate moment problems

TL;DR

This paper investigates shell polynomials related to indeterminate moment problems, focusing on the generalized Stieltjes--Wigert polynomials, and provides explicit parameter sequences and measures, addressing a question posed by Chihara.

Contribution

It introduces a discrete measure nd computes the associated orthonormal polynomials and parameter sequences, extending understanding of shell polynomials in indeterminate moment problems.

Findings

Explicit formulas for orthonormal polynomials nd parameter sequences

Identification of the measure nd its mass points

Resolution of Chihara's 2001 question in the case p=q

Abstract

The generalized Stieltjes--Wigert polynomials depending on parameters 0\le p<1 and 0<q<1 are discussed. By removing the mass at zero of the N-extremal solution concentrated in the zeros of the D-function from the Nevanlinna parametrization, we obtain a discrete measure \mu^M which is uniquely determined by its moments. We calculate the coefficients of the corresponding orthonormal polynomials (P^M_n). As noticed by Chihara, these polynomials are the shell polynomials corresponding to the maximal parameter sequence for a certain chain sequence. We also find the minimal parameter sequence, as well as the parameter sequence corresponding to the generalized Stieltjes--Wigert polynomials, and compute the value of related continued fractions. The mass points of \mu^M have been studied in recent papers of Hayman, Ismail--Zhang and Huber. In the special case of p=q, the maximal parameter sequence is constant and the determination of \mu^M and (P^M_n) gives an answer to a question posed by Chihara in 2001