On a sequence transformation with integral coefficients for Euler's constant, II

✍ Scribed by Carsten Elsner

• Publisher: Elsevier Science
• Year: 2007
• Tongue: English
• Weight: 134 KB
• Volume: 124
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1016/j.jnt.2006.10.002

✦ Synopsis

Let s n = 1 + 1/2 + • • • + 1/(n -1)log n. In 1995, the author has found a series transformation of the type n k=0 μ n,k,τ s k+τ with integer coefficients μ n,k,τ , from which geometric convergence to Euler's constant γ for τ = O(n) results. In recently published papers T. Rivoal and Kh. & T. Hessami Pilehrood have generalized this result. In this paper we introduce a series transformation S := n k=0 μ n,k,τ 1 s k+τ 2 with two parameters τ 1 and τ 2 and integer coefficients μ n,k,τ 1 . By applying the analysis of the ψ-function, we prove a sharp upper bound for |Sγ |. A similar result holds for generalized Stieltjes constants.