An analogy of Jacobi's formula and its applications

TL;DR

This paper establishes an analogy of Jacobi's formula connecting hypergeometric functions and theta constants, leading to a new transformation formula and applications in expressing limits of sequences defined by mean iterations.

Contribution

It introduces a novel analogy of Jacobi's formula involving hypergeometric functions and theta constants, and derives a new transformation formula with applications to sequence limits.

Findings

Derived a transformation formula for a specific hypergeometric function

Expressed sequence limits using hypergeometric functions

Established an analogy connecting hypergeometric functions and theta constants

Abstract

We give an analogy of Jacobi's formula, which relates the hypergeometric function with parameters $(1/4,1/4,1)$ and theta constants. By using this analogy and twice formulas of theta constants, we obtain a transformation formula for this hypergeometric function. As its application, we express the limit of a pair of sequences defined by a mean iteration by this hypergeometric function.