Results 141 to 150 of about 5,444,686 (168)

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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HÖLDER-TYPE INEQUALITIES FOR THE POLYGAMMA FUNCTIONS

Analysis and Applications, 2008
The nth polygamma function is defined by ψ(n) = (Γ′/Γ)(n), where Γ denotes Euler's gamma function. We prove: let n ≥ 1 be an integer and let p, q be non-zero real numbers. (i) The inequality [Formula: see text] holds for all positive numbers x, y if and only if p > 0, q > 0, 1/p + 1/q ≥ 1.
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Some Integrals of Gamma, Polygamma and Volterra Functions

IMA Journal of Applied Mathematics, 1985
Defining the generalized Volterra function \[ \mu (z,\beta,\alpha)=(1/\Gamma (\beta +1))\int^{\infty}_{0}\frac{z^{\alpha +t}t^{\beta}dt}{\Gamma (\alpha +t+1)}\quad (Re \alpha >-1,\quad Re \beta >-1) \] the author discuses (mainly by using Laplace transforms), a number of integrals, both definite and indefinite, which can either be expressed in terms of
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MONOTONICITY RESULTS FOR FUNCTIONS INVOLVING THE q-POLYGAMMA FUNCTIONS

Rocky Mountain Journal of Mathematics
zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Yang, Zhen-Hang, Tian, Jing-Feng
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Corrigendum to “A harmonic mean inequality for the polygamma function"

Mathematical Inequalities & Applications
Summary: We point out that the proof of Lemma 5 in the article ``A harmonic mean inequality for the polygamma function'' by \textit{S. Das} and \textit{A. Swaminathan} [ibid. 23, No. 1, 71--76 (2020; Zbl 1437.33002)] contains a small mistake. Giving a new and proper proof of Lemma 5 is the main aim of this paper.
Zhang, Ju-Mei, Yin, Li
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Sums in terms of polygamma functions

2010
Summary: We consider sums involving the product of reciprocal binomial coefficient and polynomial terms and develop some double integral identities. In particular cases it is possible to express the sums in closed form, give some general results, recover some known results in \textit{M. W. Coffey} [J. Comput. Appl. Math. 183, No.
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IRRATIONALITY OF ZEROS OF POLYGAMMA FUNCTIONS

Bulletin of the Australian Mathematical Society
Abstract Our work owes its origin to a recent note of Ram Murty [‘Irrationality of zeros of the digamma function’, Number Theory in Memory of Eduard Wirsing (eds. H. Maier, R. Steuding and J.
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Mean-value inequalities for the polygamma functions

Aequationes Mathematicae, 2001
The polygamma functions are derivatives of the logarithmic derivative \(\psi\) of the gamma function. As one of the main results necessary and sufficient conditions are offered for the inequality \(|\psi^{(k)} [M^r (\mathbf{x},\mathbf{p})]|\leq M^s (|\psi^{(k)}(\mathbf{x)}|,\mathbf{p})\) to hold for all \(\mathbf{x}=(x_1,\dots ,x_n)\in ]0,\infty[^n\), \
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