TWO-SIDED ESTIMATES OF GAMMA-FUNCTION ON THE REAL SEMIAXIS
https://doi.org/10.22405/2226-8383-2017-18-2-205-221
Abstract
In this paper we present new two-sided estimates of gamma-function Γ(x + 1) on the real semiaxis x > 0. Based on this result, we improve well-known estimates for the factorial n!, which hold for all n ≥ 1. Some of obtained estimates of gamma-function Γ(x+1) hold only for x ≥ 1/2 and some only for x ≥ 1. The main estimates are connected to the notion of alternation round of a function by asymptotic series in the strong sense. However such a strong alternation is proved only for several partial sums. We have a conjecture that the asymptotic series alternates round a logarithm of gamma-function in strong sense. Similary we propose new inequalities for the number of n-combination from 2n. These considerations indicate that next investigation is promissing and give a method for construction of new two-sided estimates for functions having alternating asymptotic series.
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Review
For citations:
Popov A.Yu. TWO-SIDED ESTIMATES OF GAMMA-FUNCTION ON THE REAL SEMIAXIS. Chebyshevskii Sbornik. 2017;18(2):205-221. (In Russ.) https://doi.org/10.22405/2226-8383-2017-18-2-205-221
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