Results Math 72 (2017), 225–238
2016 Springer International Publishing
published online November 29, 2016
Results in Mathematics
Some Asymptotic Formulas and Inequalities
About Gamma and Psi Functions Based
on Continued Fractions
Dawei Lu, Lixin Song, and Junying Wang
Abstract. In this paper, based on continued fractions, some faster as-
ymptotic formulas of gamma, psi and psi’s derivative functions are given.
Then, we provide some relevant inequalities. Finally, in order to demon-
strate that our new asymptotic formulas have superiority, some numerical
computations are also given.
Mathematics Subject Classiﬁcation. 33B15, 11A55, 41A60.
Keywords. Gamma function, Psi function, continued fraction,
rate of convergence, inequalities.
Two functions, the classical Euler gamma function
and its logarithmic derivative named digamma or psi function
are concerned by us, playing an essential role not only in kinds of mathematical
branches but also in natural science, like physics and even engineering.
Many literatures were published on ﬁnding asymptotic formulas of Γ , ψ,
as well as providing interesting inequalities of them. For instance,
Abramowitz  presented some asymptotic expansions,