# Double Gegenbauer expansion of $|s - t|^\alpha$

## TL;DR

This paper derives a Gegenbauer polynomial expansion for the function |s - t|^α, providing a new analytical tool for representing this function in terms of ultraspherical polynomials with potential applications in mathematical analysis.

## Contribution

It introduces a Gegenbauer expansion of |s - t|^α in terms of ultraspherical polynomials, extending previous methods and exploring its generalizations and limits.

## Key findings

- Provides explicit Gegenbauer expansion for |s - t|^α
- Discusses generalizations and special cases of the expansion
- Analyzes limits and specializations of the expansion

## Abstract

We give a Gegenbauer expansion of the two variable function $| s - t |^{\alpha}$ in terms of the ultraspherical polynomials $C_l^{\lambda} (s)$ and $C^{\mu}_m (t)$. Generalization, specialization, and limits of the expansion are also discussed.