# A solution to the Al-Salam--Chihara moment problem

## TL;DR

This paper solves the Al-Salam--Chihara indeterminate moment problem by analyzing a $q$-hypergeometric difference operator, leading to a spectral analysis and an explicit inverse transform for a $q$-analog of the Jacobi function.

## Contribution

It provides a spectral analysis of the $q$-hypergeometric operator and explicitly solves the moment problem using a $q$-analog of the Jacobi function transform.

## Key findings

- Explicit inverse of the $q$-analog Jacobi function transform
- Spectral analysis of the $q$-hypergeometric difference operator
- Solution to the Al-Salam--Chihara moment problem

## Abstract

We study the $q$-hypergeometric difference operator $L$ on a particular Hilbert space. In this setting $L$ can be considered as an extension of the Jacobi operator for $q^{-1}$-Al-Salam--Chihara polynomials. Spectral analysis leads to unitarity and an explicit inverse of a $q$-analog of the Jacobi function transform. As a consequence a solution of the Al-Salam--Chihara indeterminate moment problem is obtained.