Topics in special functions. II

Anderson, G. D.; Vamanamurthy, M. K.; Вуоринен, Матти

doi:10.1090/s1088-4173-07-00168-3

Cited by 19 publications

(18 citation statements)

References 52 publications

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“…(−1) k k! (k − 4)!B 3,k w 3−k + R 3,m (w), (1) where the triple Bernoulli polynomials B 3,k (x) are defined by Here, w > 0 and m ≥ 3. See [18, (3.13) and (3.14)].…”

Section: Introduction and Resultsmentioning

confidence: 99%

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On the asymptotic expansion of the logarithm of Barnes triple gamma function II

Koumandos

Pedersen²

2017

Math. Scand.

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“…(−1) k k! (k − 4)!B 3,k w 3−k + R 3,m (w), (1) where the triple Bernoulli polynomials B 3,k (x) are defined by Here, w > 0 and m ≥ 3. See [18, (3.13) and (3.14)].…”

Section: Introduction and Resultsmentioning

confidence: 99%

“…For the case of power series the result has been known for a long time and has been used to obtain various inequalities for special functions. See for instance, [1], [2], [3], [4], [5], [6] and [10]. A particularly simple proof of this case can be found in [3].…”

Section: Proof Of Lemma 22mentioning

confidence: 92%

On the asymptotic expansion of the logarithm of Barnes triple gamma function II

Koumandos

Pedersen²

2017

Math. Scand.

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“…Batir offers a numerical table illustrating that his upper bound formula n n+1 e −n √ 2π/ n − 1/6 gives much better approximations to n! than does either (21) or (22).…”

Section: Factorials and Stirling's Formulamentioning

confidence: 90%

Topics in Special Functions III

Anderson

Вуоринен

Zhang

2014

Analytic Number Theory, Approximation Theory, and Special Functions

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The study of quasiconformal maps led the first two authors in their joint work with M. K. Vamanamurthy to formulate open problems or questions involving special functions [14,16]. During the past two decades, many authors have contributed to the solution of these problems. However, most of the problems posed in [14] are still open.The present paper is the third in a series of surveys by the first two authors, the previous papers [19,21] being written jointly with the late M. K. Vamanamurthy. The aim of this series of surveys is to review the results motivated by the problems in [14,16] and related developments during the past two decades. In the first of these we studied classical special functions, and in the next we focused on special functions occurring in the distortion theory of quasiconformal maps. Regretfully, Vamanamurthy passed away in 2009, and the remaining authors dedicate the present work as a tribute to his memory. For an update to the bibliographies of [19] and [21] the reader is referred to [23].In 1993 the following monotone rule was derived [17, Lemma 2.2]. Though simple to state and easy to prove by means of the Cauchy Mean Value Theorem, this l'Hôpital Monotone Rule (LMR) has had wide application to special functions by many authors. Vamanamurthy was especially skillful in the application of this rule. We here quote the rule as it was restated in [20, Theorem 2].

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“…The Gamma function (x) = ∞ 0 t x−1 e −t dt, x > 0 is a generalization of the factorial function n!, and it has important applications in various branches of mathematics Anderson et al (2007); Wang et al (2016Wang et al ( , 2017. It is a long history that the gamma function and its wide applications have been studied.…”

Section: Introductionmentioning

confidence: 99%

Approximations for the Ratio of the Gamma Functions via the Digamma Function and Its Applications

Han

Zhang

You

et al. 2020

Bull Braz Math Soc, New Series

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In this paper, the authors establish some asymptotic formulas and two-sided inequalities for the ratio of the gamma function in terms of the digamma function. Considering the application of the gamma function to central Bernoulli coefficients, the authors give some better approximations and two-sided inequalities of central Bernoulli coefficients. Finally, for demonstrating the superiority of our results, some numerical computations are given. Keywords Digamma function • Ratio of the gamma function • Approximation • Power series expansion • Continued fraction • Central Bernoulli coefficients • Two-sided inequalities Mathematics Subject Classification 33B15 • 26D15 • 41A25

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Topics in special functions. II

Cited by 19 publications

References 52 publications

On the asymptotic expansion of the logarithm of Barnes triple gamma function II

On the asymptotic expansion of the logarithm of Barnes triple gamma function II

Topics in Special Functions III

Approximations for the Ratio of the Gamma Functions via the Digamma Function and Its Applications

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