Gamma Functions, Beta Functions, and Related
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Incomplete gamma function
Digamma function
BETA (programming language)
Symbol (formal)
Beta function (physics)
Digamma function
Incomplete gamma function
Reflection
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In this paper, some properties for the v-analogue of Gamma and digamma functions are investigated. Also, a celebrated Bohr-Mollerup type theorem related to the v-analogue of Gamma function is given. Furthermore, an expression for the v-digamma function is presented by using the v-analogue of beta function. Also, some limits for the v-analogue of Gamma and beta functions are given.
Digamma function
Incomplete gamma function
BETA (programming language)
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In this paper, two completely monotonic functions involving the q-gamma and the q-trigamma functions where q is a positive real, are introduced and exploited to derive sharp bounds for the q-gamma function in terms of the q-trigamma function. These results, when letting q → 1, are shown to be new. Also, sharp bounds for the q-digamma function in terms of the q-tetragamma function are derived. Furthermore, an infinite class of inequalities for the q-polygamma function is established.
Digamma function
Incomplete gamma function
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We apply our simultaneous contour integral method to an infinite sum in Prudnikov et al. and use it to derive the infinite sum of the Incomplete gamma function in terms of the Hurwitz zeta function. We then evaluate this formula to derive new series in terms of special functions and fundamental constants. All the results in this work are new.
Incomplete gamma function
Digamma function
Polylogarithm
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The main result of this article is to establish new and accurate approximations of the gamma function in terms of the digamma function, that improve a result of Alzer and Batir [Appl. Math. Lett. 20(2007):778–781].
Digamma function
Incomplete gamma function
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Abstract A variety of integral representations for some special functions have been developed. Here we aim at presenting certain (new or known) integral representations for , B(α, β), and by using some of the known integral representations of the Hurwitz (or generalized) Zeta function ζ(s, a). As a by-product of our main formulas, several integral representations for the Glaisher–Kinkelin constant A and the Psi (or Digamma) function ψ(a) are also given. Relevant connections of some of the results presented here with those obtained in earlier works are indicated. We also indicate the potential for the usefulness of these results. Keywords: Euler–Mascheroni constantGlaisher–Kinkelin constantGamma functionDouble Gamma functionWeierstrass factorization theoremBeta functionPsi (or Digamma) functionRiemann Zeta functionHurwitz (or generalized) Zeta functionHurwitz–Lerch Zeta functionWeierstrass canonical productdeterminants of the Laplacians 2000 Mathematics Subject Classification : Primary: 11M0633B1533B99Secondary: 33C05
Digamma function
Polylogarithm
Incomplete gamma function
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Abstract This paper is devoted to derive some properties and expansions associated with the q-digamma function. The Newton series which is consisting of terms of forward difference operator, is established for the q-digamma function. The maltiplication formula of the q-gamma function is used to present some recurrence relations for the q-digamma function. The q-analogue of well-known results in the theory of the digamma function are investigated. One of the most important identities of q-beta integrals is used to introduce some integral representations of the q-digamma function.
Digamma function
Beta function (physics)
Mittag-Leffler function
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Digamma function
Incomplete gamma function
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Digamma function
Incomplete gamma function
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Background. The article is dedicated to studies of the main properties of new generalized gamma-functions, generalized incomplete gamma-functions, generalized digamma-functions for their best applications in applied sciences, for calculations of integrals which are absent in scientific literature.Objective. Introduction and study of the basic properties of the new generalized gamma-functions, generalized incomplete gamma-functions, generalized digamma-functions and their applications.Methods. We apply the following methods: the methods of the theory of functions of the real variable, the theory of the special functions, the theory of the mathematical physics, the methods of applied analysis.Results. Some new forms of generalized gamma-functions, incomplete gamma-functions, digamma-functions are introduced. The main properties of these generalized special functions are explored. Examples of application of new generalized gamma-functions are given.Conclusions. With the help of the r-generalized confluent hypergeometric functions the new generalization of gamma-functions, incomplete gamma-functions, digamma-functions are introduced. The main properties of the new generalized special functions are explored, examples of application of these functions are given.
Digamma function
Special functions
Generalized function
Incomplete gamma function
Elementary function
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