Rational approximations for the quotient of gamma values

TL;DR

This paper develops new rational approximation methods for the quotient of Gamma function values at rational points, extending previous work on Euler's constant and providing explicit formulas for approximants.

Contribution

It introduces rational approximations for Gamma function quotients using Jacobi-Laguerre polynomials, generalizing Aptekarev's approximants to Euler's constant.

Findings

Derived explicit formulas for numerator and denominator of approximants.

Extended rational approximation techniques to Gamma function quotients.

Connected new approximations to existing Euler's constant approximants.

Abstract

In this paper, we continue to study properties of rational approximations to Euler's constant and values of the Gamma function defined by linear recurrences, which were found recently by A. I. Aptekarev and T. Rivoal. Using multiple Jacobi-Laguerre orthogonal polynomials we present rational approximations to the quotient of values of the Gamma function at rational points. As a limit case of our result, we obtain new explicit formulas for numerators and denominators of the Aptekarev approximants to Euler's constant.