The Experts below are selected from a list of 132 Experts worldwide ranked by ideXlab platform
Cristinel Mortici - One of the best experts on this subject based on the ideXlab platform.
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A survey on recent extensions of the Stirling Formula
arXiv: Classical Analysis and ODEs, 2013Co-Authors: Sorinel Dumitrescu, Cristinel MorticiAbstract:We present a survey on recent results about Stirling's Formula. More exactly, we reffer to a method using a form of Cesaro-Stolz lemma firstly introduced in [C. Mortici Product approximations via asymptotic integration Amer. Math. Monthly 117 (5) (2010) 434-441]. As an example we improve a result obtained in [C. Mortici A substantial improvement of the Stirling Formula Appl. Math. Lett. 24 (2011) no. 8 1351-1354]. Finally, some numerical computations are made.
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The best rational remainders in the Stirling Formula
Integral Transforms and Special Functions, 2012Co-Authors: Cristinel MorticiAbstract:The aim of this paper is to improve the asymptotic Stirling series to a new series which is faster than Stiletjes’ continued fraction. We prove that our series and consequently the Stiletjes’ continued fraction is optimal in some sense. Finally some inequalities involving the factorial function are given.
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A substantial improvement of the Stirling Formula
Applied Mathematics Letters, 2011Co-Authors: Cristinel MorticiAbstract:Abstract We introduce the approximation Formula n ! ∼ 2 π n ( n e + 1 12 e n ) n for the factorial function. Finally, some numerical computations are made to prove the superiority over other well-known Formulas.
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On the monotonicity and convexity of the remainder of the Stirling Formula
Applied Mathematics Letters, 2011Co-Authors: Cristinel MorticiAbstract:Abstract Shi, Liu and Hu [X. Shi, F. Liu, M. Hu, A new asymptotic series for the Gamma function, J. Comput. Appl. Math. 195 (2006) 134–154] proved that the function θ ( x ) defined by Γ ( x + 1 ) = 2 π ( x / e ) x e θ ( x ) / 12 x is strictly increasing for x ≥ 1 . The aim of our work is to prove that − x − 1 θ ‴ ( x ) is strictly completely monotonic on ( 0 , ∞ ) . As direct consequences, we show that θ is strictly convex on ( 0 , ∞ ) , and then we prove that θ is strictly decreasing on ( 0 , β ) , and strictly increasing on ( β , ∞ ) , where β = 0.34142 … .
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New improvements of the Stirling Formula
Applied Mathematics and Computation, 2010Co-Authors: Cristinel MorticiAbstract:The aim of this paper is to propose some improvements of the Stirling Formula for approximation of the factorial function.
Ahmed Salem - One of the best experts on this subject based on the ideXlab platform.
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Three Classes of The Stirling Formula for The q-factorial Function
2016Co-Authors: Ahmed SalemAbstract:[[abstract]]In this paper, q-analogues of the Stirling Formula for the q-factorial function are derived and expressed as innite integral, innite series and double innite series
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Some completely monotonic functions associated with the q-gamma and the q-polygamma functions
Acta Mathematica Scientia, 2015Co-Authors: Ahmed Salem, Eid S. KamelAbstract:Abstract In this paper, the q-analogue of the Stirling Formula for the q-gamma function (Moak Formula) is exploited to prove the complete monotonicity properties of some functions involving the q-gamma and the q-polygamma functions for all real number q > 0. The monotonicity of these functions is used to establish sharp inequalities for the q-gamma and the q-polygamma functions and the q-Harmonic number. Our results are shown to be a generalization of results which were obtained by Selvi and Batir [23].
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Complete monotonicity properties of functions involving q-gamma and q-digamma functions
Mathematical Inequalities & Applications, 2014Co-Authors: Ahmed SalemAbstract:In this paper, the q -analogue of the Stirling Formula (the Moak Formula) for the q gamma function is exploited to prove the complete monotonicity property of functions involving the q -gamma and the q -digamma functions. The monotonicity of these functions is used to establish sharp inequalities for the q -gamma and the q -polygamma functions and the q -Harmonic number. Mathematics subject classification (2010): 33D05, 26D07, 26A48.
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An infinite class of completely monotonic functions involving the q-gamma function
Journal of Mathematical Analysis and Applications, 2013Co-Authors: Ahmed SalemAbstract:Abstract In this paper, the q -analogue of the Stirling Formula for the q -gamma function is used to prove the complete monotonicity property for an infinite class of functions which are all closely related to the q -gamma function and its logarithmic derivatives ( q -polygamma functions). As an application of this result, the two-sided inequalities for the q -gamma and the q -polygamma functions are established.
Hassan Hassanabadi - One of the best experts on this subject based on the ideXlab platform.
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Polychronakos statistics and α-deformed Bose condensation of α-bosons
Modern Physics Letters B, 2018Co-Authors: Won Sang Chung, Hassan HassanabadiAbstract:In this paper, we consider the Polychronakos statistics for [Formula: see text]. We use the Stirling Formula for the [Formula: see text]-Gamma function to find the distribution function for the [Formula: see text]-bosons. As application, we discuss the [Formula: see text]-deformed Bose condensation for [Formula: see text]-boson gas.
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Polychronakos statistics and α-deformed Bose condensation of α-bosons
Modern Physics Letters B, 2018Co-Authors: Won Sang Chung, Hassan HassanabadiAbstract:In this paper, we consider the Polychronakos statistics for α < 0. We use the Stirling Formula for the α-Gamma function to find the distribution function for the α-bosons. As application, we discuss the α-deformed Bose condensation for α-boson gas.
Michitomo Nishizawa - One of the best experts on this subject based on the ideXlab platform.
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Multiple gamma functions and multiple q-gamma functions
Publications of The Research Institute for Mathematical Sciences, 1997Co-Authors: Kimio Ueno, Michitomo NishizawaAbstract:We give an asymptotic expansion (the higher Stirling Formula) and an infinite product representation (the Weierstrass canonical product representation) of the Vigneras multiple gamma functions by considering the classical limit of the multiple #-gamma functions. §
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The multiple gamma functions and the multiple q-gamma functions
arXiv: Quantum Algebra, 1996Co-Authors: Kimio Ueno, Michitomo NishizawaAbstract:We give an asymptotic expansion (the higher Stirling Formula) and an infinite product representation (the Weierstrass product representation) of the Vign\'{e}ras multiple gamma functions by considering the classical limit of the multiple q-gamma functions.
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the multiple gamma function and its q analogue
arXiv: Quantum Algebra, 1996Co-Authors: Kimio Ueno, Michitomo NishizawaAbstract:We give an asymptotic expansion (the higher Stirling Formula) and an infinite product representation (the Weierstrass product Formula) of the Vign\'{e}ras multiple gamma function by considering the classical limit of the multiple q-gamma function.
Malvina Vamvakari - One of the best experts on this subject based on the ideXlab platform.
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a q analogue of the Stirling Formula and a continuous limiting behaviour of the q binomial distribution numerical calculations
Methodology and Computing in Applied Probability, 2013Co-Authors: Andreas Kyriakoussis, Malvina VamvakariAbstract:In this article, we derive an asymptotic Formula for the q-factorial number of order n using the saddle point method. This Formula is a q-analogue, for 0 < q < 1, of the usual Stirling Formula for the factorial number of order n. Also, this Formula is used to provide a continuous limiting behaviour of the q-Binomial distribution in the sense of pointwise convergence. Specifically, the q-Binomial distribution converges to a continuous Stieltjes–Wigert distribution. Furthermore, we present some numerical calculations, using the computer program MAPLE, indicating a quite strong convergence.
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A q-Analogue of the Stirling Formula and a Continuous Limiting Behaviour of the q-Binomial Distribution—Numerical Calculations
Methodology and Computing in Applied Probability, 2011Co-Authors: Andreas Kyriakoussis, Malvina VamvakariAbstract:In this article, we derive an asymptotic Formula for the q-factorial number of order n using the saddle point method. This Formula is a q-analogue, for 0 < q < 1, of the usual Stirling Formula for the factorial number of order n. Also, this Formula is used to provide a continuous limiting behaviour of the q-Binomial distribution in the sense of pointwise convergence. Specifically, the q-Binomial distribution converges to a continuous Stieltjes–Wigert distribution. Furthermore, we present some numerical calculations, using the computer program MAPLE, indicating a quite strong convergence.