# Explicit logarithmic formulas of special values of hypergeometric functions 3F2

## TL;DR

This paper develops explicit formulas for special values of the hypergeometric function 3F2, extending previous work by providing a method to explicitly describe these values in terms of logarithms of algebraic numbers.

## Contribution

It introduces a new method to explicitly compute special values of 3F2 hypergeometric functions, building on prior results about their algebraic and logarithmic properties.

## Key findings

- Derived explicit logarithmic formulas for 3F2 special values
- Extended previous techniques to obtain explicit descriptions
- Provided a systematic method for calculating these values

## Abstract

In a joint paper [4] by Otsubo, Terasoma and the first author, we proved that the special value 3F2(a,b,q;a+b,q;1) of the generalized hypergeometric function is a linear combination of log of algebraic numbers if the triplet (a,b,q) of rational numbers satisfies a certain numerical condition. However there remains a question how to obtain explicit descriptions of the values. In this paper, we give a method to do this, which is a further development of the technique in [4].