# Proof of the $(\alpha,\beta)$--inversion formula conjectured by Hsu and Ma

## TL;DR

This paper proves the conjectured $(eta,eta)$-inversion formula by Hsu and Ma, and demonstrates its applications in deriving known and new matrix inversions involving elliptic sequences and theta functions.

## Contribution

It establishes the $(eta,eta)$-inversion formula conjecture and applies it to obtain new matrix inversions related to elliptic divisibility sequences and theta functions.

## Key findings

- Proved the $(eta,eta)$-inversion formula conjecture.
- Recovered known matrix inversions using the new inversion formula.
- Derived three new matrix inversions involving elliptic sequences and theta functions.

## Abstract

In light of the well-known fact that the $n$th divided difference of any polynomial of degree $m$ must be zero while $m<n$,the present paper proves the $(\alpha,\beta)$-inversion formula conjectured by Hsu and Ma [J. Math. Res. $\&$ Exposition 25(4) (2005) 624]. As applications of $(\alpha,\beta)$-inversion, we not only recover some known matrix inversions due to Gasper, Schlosser, and Warnaar, but also fin three new matrix inversions related to elliptic divisibility sequence and theta functions.