4,386 research outputs found
Eight interesting identities involving the exponential function, derivatives, and Stirling numbers of the second kind
In the paper, the author establishes some identities which show that the
functions and the derivatives can be expressed each other by linear combinations with
coefficients involving the combinatorial numbers and the Stirling numbers of
the second kind, where and .Comment: 9 page
Some bounds for the complete elliptic integrals of the first and second kinds
In the article, the complete elliptic integrals of the first and second kinds
are bounded by using the power series expansions of some functions, the
celebrated Wallis' inequality, and an integral inequality due to R. P. Agarwal,
P. Cerone, S. S. Dragomir and F. Qi.Comment: 9 page
A class of completely monotonic functions involving divided differences of the psi and polygamma functions and some applications
A class of functions involving the divided differences of the psi function
and the polygamma functions and originating from Kershaw's double inequality
are proved to be completely monotonic. As applications of these results, the
monotonicity and convexity of a function involving ratio of two gamma functions
and originating from establishment of the best upper and lower bounds in
Kershaw's double inequality are derived, two sharp double inequalities
involving ratios of double factorials are recovered, the probability integral
or error function is estimated, a double inequality for ratio of the volumes of
the unit balls in and respectively is
deduced, and a symmetrical upper and lower bounds for the gamma function in
terms of the psi function is generalized.Comment: 11 page
An alternative proof of Elezovi\'c-Giordano-Pe\v{c}ari\'c's theorem
In the present note, an alternative proof is supplied for Theorem~1 in [N.
Elezovi\'c, C. Giordano and J. Pe\v{c}ari\'c, \textit{The best bounds in
Gautschi's inequality}, Math. Inequal. Appl. \textbf{3} (2000),
239\nobreakdash--252.].Comment: 5 page
Some properties of extended remainder of Binet's first formula for logarithm of gamma function
In the paper, we extend Binet's first formula for the logarithm of the gamma
function and investigate some properties, including inequalities, star-shaped
and sub-additive properties and the complete monotonicity, of the extended
remainder of Binet's first formula for the logarithm of the gamma function and
related functions.Comment: 8 page
A completely monotonic function involving the tri- and tetra-gamma functions
The psi function is defined by
and for
denote the polygamma functions, where is the gamma function. In
this paper we prove that a function involving the difference between
and a proper fraction of is completely monotonic
on .Comment: 10 page
A refinement of a double inequality for the gamma function
In the paper, we present a monotonicity result of a function involving the
gamma function and the logarithmic function, refine a double inequality for the
gamma function, and improve some known results for bounding the gamma function.Comment: 8 page
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