High precision computation and a new asymptotic formula for the generalized Stieltjes constants

TL;DR

This paper introduces an efficient numerical method using DE quadrature for computing generalized Stieltjes constants with high precision, especially for large indices, and derives a new asymptotic formula for these constants.

Contribution

It presents a novel integral-based numerical approach and a highly accurate asymptotic formula for generalized Stieltjes constants, improving computational efficiency and accuracy.

Findings

Method achieves high accuracy for large n

Asymptotic formula provides precise approximations

Efficient evaluation near saddle points

Abstract

We provide an efficient method to evaluate the generalized Stieltjes constants $\gamma_n(a)$ numerically to arbitrary accuracy for large $n$ and $n \gg |a|$ values. The method uses an integral representation for the constants and evaluates the integral by applying the double exponential (DE) quadrature method near the saddle points of the integrands. Further, we provide a highly accurate asymptotic formula for the generalized Stieltjes constants.