# Ultraspherical moments on a set of disjoint intervals

## TL;DR

This paper computes ultraspherical moments on disjoint intervals with gaps, expressing them via hypergeometric functions, and shows they interpolate between full and half-range ultraspherical moments.

## Contribution

It introduces a new method to evaluate moments on disjoint intervals for generalized ultraspherical weights using hypergeometric functions.

## Key findings

- Moments are expressed explicitly in terms of hypergeometric functions.
- Identifies a deformation parameter connecting full and half-range ultraspherical moments.
- Provides a framework for studying non-classical orthogonal polynomial systems.

## Abstract

Moment evaluations are important for the study of non-classical orthogonal polynomial systems for which explicit representations are not known. In this paper we compute, in terms of the hypergeometric function, the moments associated with a generalized ultraspherical weight on a collection of intervals with two symmetric gaps. These moments, parametrized by the endpoints of the gaps, are identified as a one parameter deformation between the full range ultraspherical moments and the half range ultraspherical moments.