Series representations of the remainders in the expansions for certain functions with applications

Chao-Ping Chen; Richard B. Paris

doi:10.1007/s00025-016-0612-1

Series representations of the remainders in the expansions for certain functions with applications

Chao-Ping Chen, Richard B. Paris

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Abstract

We present a summary of the series representations of the remainders in the expansions in ascending powers of t of 2/(et+1), sech t and coth t and establish simple bounds for these remainders when t>0. Several applications of these expansions are given which enable us to deduce some inequalities and completely monotonic functions associated with the ratio of two gamma functions. In addition, we derive a (presumably new) quadratic recurrence relation for the Bernoulli numbers B_n.

Original language	English
Pages (from-to)	1443–1457
Number of pages	15
Journal	Results in Mathematics
Volume	71
Issue number	3-4
Early online date	12 Oct 2016
DOIs	https://doi.org/10.1007/s00025-016-0612-1
Publication status	Published - Jun 2017

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abstract = "We present a summary of the series representations of the remainders in the expansions in ascending powers of t of 2/(et+1), sech t and coth t and establish simple bounds for these remainders when t>0. Several applications of these expansions are given which enable us to deduce some inequalities and completely monotonic functions associated with the ratio of two gamma functions. In addition, we derive a (presumably new) quadratic recurrence relation for the Bernoulli numbers Bn.",

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AB - We present a summary of the series representations of the remainders in the expansions in ascending powers of t of 2/(et+1), sech t and coth t and establish simple bounds for these remainders when t>0. Several applications of these expansions are given which enable us to deduce some inequalities and completely monotonic functions associated with the ratio of two gamma functions. In addition, we derive a (presumably new) quadratic recurrence relation for the Bernoulli numbers Bn.

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