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

Abstract

We approximate each generalized Stieltjes constant g 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 g 0 (1) and g 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.

Acceleration Methods for Series: A Probabilistic Perspective

“…In doing this, we need to compute forward differences of powers of logarithms ∆ j ψ k (0), and this can be achieved using Theorem 2.3 in Section 2. A similar result to Theorem 3.2 was obtained by Coffey [13], where each α k is computed at the geometric rate 1/3, with coefficients growing as a polynomial of degree k (see also [3] for slightly different computation formulas). Recently, Johansson [19] has given efficient algorithms to evaluate γ n based on the Euler-Mclaurin formula.…”

“…In this regard, differentiation formulas for linear operators represented by gamma processes have been used in [2] to obtain asymptotic estimates for Stieltjes constants, as well as to compute each Stieltjes constant in the classical spirit of Stieltjes and Berndt [9] (c.f. [3]). In many occasions, such a probabilistic approach gives short proofs of the results under consideration.…”

Rigorous high-precision computation of the Hurwitz zeta function and its derivatives

“…The best proven bounds for the Stieltjes constants appear to be very pessimistic (see for example [1]). In a recent paper, Knessl and Coffey [27] give an asymptotic approximation formula that seems to be very accurate even for small n, having the form γ n ≈ Bn −1/2 e An cos(an + b) (18) where A, B, a, b are functions that depend weakly on n. Notably, this formula captures both the asymptotic growth and the oscillation pattern of γ n .…”

“…Let B n denote the n-th Bernoulli number and letB n (t) = B n (t − t ) denote the n-th periodic Bernoulli polynomial. The Euler-Maclaurin summation formula (described in numerous works, such as [33]) states that U k=N f (k) = I + T + R (1) where…”

CERTAIN INTEGRAL REPRESENTATIONS OF GENERALIZED STIELTJES CONSTANTS γk(a)

“…Adell [1] approximated each generalized Stieltjes constants γ k (a) by means of a finite sum involving Bernoulli numbers. Kreminski [17] presented a new approach to high-precision approximation of γ k (a).…”