Estimates of generalized Stieltjes constants with a quasi-geometric rate of decay

Abstract

We approximate each generalized Stieltjes constant γ n ( a ) by means of a finite sum involving Bernoulli numbers. Such an approximation has a quasi-geometric rate of convergence, which improves as Re( a ) increases. A more detailed analysis, including numerical computations, is carried out for the constants γ 0 (1) and γ 1 (1). The key point in the proof is a probabilistic representation of the aforementioned constants, obtained as a consequence of a differential calculus concerning the gamma process.