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February 2022 A series representation of Euler–Stieltjes constants and an identity of Ramanujan
Tapas Chatterjee, Suraj Singh Khurana
Rocky Mountain J. Math. 52(1): 49-64 (February 2022). DOI: 10.1216/rmj.2022.52.49

Abstract

We derive a series representation of the generalized Stieltjes constants which arise in the Laurent series expansion of partial zeta function at the point s=1. In the process, we introduce a generalized gamma function and deduce its properties such as functional equation, Weierstrass product and reflection formulas along the lines of the study of a generalized gamma function introduced by Dilcher in 1994. These properties are used to obtain a series representation for the k-th derivative of Dirichlet series with periodic coefficients at the point s=1. Another application involves evaluation of a class of infinite products of which a special case is an identity of Ramanujan.

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Tapas Chatterjee. Suraj Singh Khurana. "A series representation of Euler–Stieltjes constants and an identity of Ramanujan." Rocky Mountain J. Math. 52 (1) 49 - 64, February 2022. https://doi.org/10.1216/rmj.2022.52.49

Information

Received: 21 January 2021; Revised: 22 May 2021; Accepted: 24 May 2021; Published: February 2022
First available in Project Euclid: 19 April 2022

Digital Object Identifier: 10.1216/rmj.2022.52.49

Subjects:
Primary: 11M06 , 30B50 , 30D05
Secondary: 11Y60 , 33B15

Keywords: Dirichlet L-series , functional equation , Gamma function , generalized Euler constants , Weierstrass product formula

Rights: Copyright © 2022 Rocky Mountain Mathematics Consortium

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Vol.52 • No. 1 • February 2022
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