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Completely monotonic degrees of remainders of asymptotic expansions of the digamma function

Feng Qi, Mansour Mahmoud

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Abstract

Motivated by several conjectures posed in the paper [F. Qi and A.-Q. Liu, Completely monotonic degrees for a difference between the logarithmic and psi functions, J. Comput. Appl. Math. 361 (2019), 366--371; available online at https://doi.org/10.1016/j.cam.2019.05.001], the authors calculate some completely monotonic degrees of remainders of the asymptotic expansion of the logarithm of the gamma function, compute some completely monotonic degrees of remainders of asymptotic expansions of the digamma function, and confirm some conjectures on completely monotonic degrees of remainders of the asymptotic expansion of the logarithm of the gamma function and on completely monotonic degrees of remainders of asymptotic expansions of polygamma functions.

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What this paper is about

Motivated by several conjectures posed in the paper [F. Qi and A.-Q. Liu, Completely monotonic degrees for a difference between the logarithmic and psi functions, J. Comput. Appl. Math. 361 (2019), 366--371; available online at https://doi.org/10.1016/j.cam.2019.05.001], the authors calculate some completely monotonic degrees of remainders of the asymptotic expansion of the logarithm of the gamma function, compute some completely monotonic degrees of remainders of asymptotic expansions of the digamma function, and confirm some conjectures on completely monotonic degrees of remainders of the asymptotic expansion of the logarithm of the gamma function and on completely monotonic degrees of remainders of asymptotic expansions of polygamma functions.

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Available abstract

Motivated by several conjectures posed in the paper [F. Qi and A.-Q. Liu, Completely monotonic degrees for a difference between the logarithmic and psi functions, J. Comput. Appl. Math. 361 (2019), 366--371; available online at https://doi.org/10.1016/j.cam.2019.05.001], the authors calculate some completely monotonic degrees of remainders of the asymptotic expansion of the logarithm of the gamma function, compute some completely monotonic degrees of remainders of asymptotic expansions of the digamma function, and confirm some conjectures on completely monotonic degrees of remainders of the asymptotic expansion of the logarithm of the gamma function and on completely monotonic degrees of remainders of asymptotic expansions of polygamma functions.

Key concepts: Monotonic function, Digamma function, Logarithm, Mathematics, Gamma function, Asymptotic expansion, Function (biology), Asymptotic analysis

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