# On $q$-analogues Arising from Elliptic Integrals and the Arithmetic-Geometric Mean

## TL;DR

This paper develops $q$-analogues of classical elliptic integral identities and the arithmetic-geometric mean's functional equation, extending derivatives to complex orders and expressing results as infinite products.

## Contribution

It introduces new $q$-analogues for elliptic integrals and the AGM, connecting them through infinite product representations and complex extension of derivatives.

## Key findings

- Established $q$-analogues of elliptic integral identities.
- Extended derivatives of elliptic integrals to complex values.
- Expressed $q$-analogues as infinite products.

## Abstract

We prove $q$-analogues of identities that are equivalent to the functional equation of the arithmetic-geometric mean. We also present $q$-analogues of $F(\sqrt{k},\frac{\pi}{2})$, the complete elliptical integral of the first kind, and its derivatives evaluated at $k=\frac{1}{2}$. These $q$-analogues interpolate those $n$th derivative evaluations by extending $n$ to a complex variable $s$, and we prove that they can be expressed as an infinite product.