Results 141 to 150 of about 2,821,192 (174)

Inequalities for the Polygamma Functions

SIAM Journal on Mathematical Analysis, 1998
Let F_n(x; c) = (\Psi^{(n)}(x))^2 - c \Psi^{(n - 1)}(x) \Psi^{(n + 1)}(x) \quad (x > 0), where $\Psi$ denotes the logarithmic derivative of the gamma function, $n \geq 2$ is an integer, and c is a real number. The authors prove that the function $x \mapsto F_n (x; \alpha)$ is strictly completely monotonic on $(0, \infty)$ if and only if $\alpha \leq
Alzer, Horst, Wells, Jim
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Exponential, Gamma and Polygamma Functions: Simple Proofs of Classical and New Inequalities

Journal of Mathematical Analysis and Applications, 2013
Some inequalities for the exponential, gamma and polygamma functions are established by means of the mean value theorem of differential calculus. In particular, the cases of the ratio of gamma and polygamma functions are considered.
Pierpaolo Natalini
exaly   +2 more sources

SPECIAL VALUES OF THE POLYGAMMA FUNCTIONS

International Journal of Number Theory, 2009
Let q be a natural number and [Formula: see text]. We consider the Dirichlet series ∑n ≥ 1 f(n)/ns and relate its value when s is a natural number, to the special values of the polygamma function. For certain types of functions f, we evaluate the special value explicitly and use this to study linear independence over ℚ of L(k,χ) as χ ranges over ...
Murty, M. Ram, Saradha, N.
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Turán-Type Inequalities for Generalized Polygamma Function

International Journal of Scientific Research in Science and Technology, 2022
Inspired by the work of C. Mortici [1] and A. Laforgia et. al [2] we have established some new Turán-type inequalities for k-polygamma function and p-k-polygamma function.
exaly   +4 more sources

Polygamma Functions of Arbitrary Order

SIAM Journal on Mathematical Analysis, 1976
Functions $\psi ^{(\nu)} (x)$ are defined for every complex $\nu$. They are entire functions of $\nu$ and generalize the well-known polygamma functions. Some of their properties are derived and their relation to another possible generalization is described.
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Inequalities for the Gamma and Polygamma Functions

Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 1998
Author offers an extension, with new proof, of a result by \textit{G. D. Anderson} and \textit{S.-L. Qiu} [Proc. Am. Math. Soc. 125, No. 11, 3355-3362 (1997; Zbl 0881.33001)] of the strict increasing of \(\log (\Gamma(x+1))/(x\log x),\) from \(]1,\infty[\) to \(]0,\infty[,\) furthermore inequalities for \(\psi^{(n)}(x+1)-\psi^{(n)}(x+s) (\psi=\Gamma ...
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Computation of the polygamma functions

Communications in Statistics - Simulation and Computation, 1984
An algorithm is presented for computing the psi(digamma) and other polygamma functions. 12 references, 2 tables.
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Very accurate estimates of the polygamma functions

Asymptotic Analysis, 2010
The aim of this paper is to give some refinements of an estimation for the polygamma functions, stated in J. Math. Anal. Approx. Theory 1(2) (2006), 124–134. Finally, a general method to establish increasingly accurate estimations is given.
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Sharp inequalities for the digamma and polygamma functions

form, 2004
Several inequalities for the digamma function \(\psi=\Gamma'/\Gamma\) and the polygamma functions \(\psi'\), \(\psi''\), \(\psi'''\), \(\ldots\) are given. The new results improve and extend several classical and recently published results.
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Values of the polygamma functions at rational arguments

Journal of Physics A: Mathematical and Theoretical, 2007
Gauss in 1812, in his celebrated memoir on the hypergeometric series, presented a remarkable formula for the psi (or digamma) function, ?(z), at rational arguments z, which can be expressed in terms of elementary functions. Davis in 1935 extended Gauss's result to the polygamma functions by using a known series representation of ?(n)(z) in an ...
Junesang Choi, Djurdje Cvijović
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