Integral Representation of Some Functions Related to the Gamma Function

The renewal process generated by the uniform distribution, when interpreted as a transformation of the uniform distribution into a discrete distribution, gives rise to the question of uniqueness of the inverse image. The paper deals with a particular problem from the described domain, that arose in the construction of a complex stochastic test intended to evaluate pseudo-random number generators. The connection of the treated problem with the question of a unique integral representation of Gamma-function is also mentioned.

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Regularized Integral Representations of the Reciprocal Gamma Function

Fractal and Fractional ◽

10.3390/fractalfract3010001 ◽

2019 ◽

Vol 3 (1) ◽

pp. 1 ◽

Cited By ~ 3

Author(s):

Dimiter Prodanov

Keyword(s):

Integral Representation ◽

Computer Algebra ◽

Gamma Function ◽

Real Line ◽

Computer Algebra System ◽

Integral Representations ◽

Complex Representation ◽

Hypersingular Integral ◽

The Real ◽

Two Parameter

This paper establishes a real integral representation of the reciprocal Gamma function in terms of a regularized hypersingular integral along the real line. A regularized complex representation along the Hankel path is derived. The equivalence with the Heine’s complex representation is demonstrated. For both real and complex integrals, the regularized representation can be expressed in terms of the two-parameter Mittag-Leffler function. Reference numerical implementations in the Computer Algebra System Maxima are provided.

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Asymptotic formulas for the triple Gamma function Γ3 by means of its integral representation

Applied Mathematics and Computation ◽

10.1016/j.amc.2011.08.002 ◽

2011 ◽

Vol 218 (6) ◽

pp. 2631-2640 ◽

Cited By ~ 4

Author(s):

Junesang Choi ◽

H.M. Srivastava

Keyword(s):

Integral Representation ◽

Gamma Function ◽

Asymptotic Formulas

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New Two Parameter Gamma Function

10.20944/preprints202004.0537.v1 ◽

2020 ◽

Author(s):

Kuldeep Sing Gehlot

Keyword(s):

Integral Representation ◽

Gamma Function ◽

Pochhammer Symbol ◽

Two Parameter

In this paper we introduce the New/Generalized two parameter Gamma function and Pochhammer symbol. We named them, as Generalized p - k Gamma Function and Generalized p - k Pochhammer symbol and denoted as $ _{p}^{a}\Gamma_{k}(x) $ and $ _{p}^{a}(x)_{n,k} $ respectively. We prove the several identities for $ _{p}^{a}\Gamma_{k}(x) $ and $ _{p}^{a}(x)_{n,k} $ those satisfied by the classical Gamma function. Also we provide the integral representation for the $ _{p}^{a}\Gamma_{k}(x) $

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A DIRECT SOLUTION TO THE FOKKER–PLANCK EQUATION FOR EXPONENTIAL BROWNIAN FUNCTIONALS

Analysis and Applications ◽

10.1142/s0219530510001655 ◽

2010 ◽

Vol 08 (03) ◽

pp. 287-304 ◽

Cited By ~ 10

Author(s):

CAROLINE PINTOUX ◽

NICOLAS PRIVAULT

Keyword(s):

Integral Representation ◽

Gamma Function ◽

Planck Equation ◽

Heat Kernels ◽

Direct Solution ◽

Spectral Expansions ◽

Fokker Planck Equation ◽

Financial Application ◽

Exponential Brownian Functionals ◽

Fokker Planck

The solution of the Fokker–Planck equation for exponential Brownian functionals usually involves spectral expansions that are difficult to compute explicitly. In this paper, we propose a direct solution based on heat kernels and a new integral representation for the square modulus of the Gamma function. A financial application to bond pricing in the Dothan model is also presented.

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An Alternative perspective to Riemann Hypothesis

10.31219/osf.io/yxj3s ◽

2021 ◽

Author(s):

Jamal Salah

Keyword(s):

Functional Equation ◽

Integral Representation ◽

Zeta Function ◽

Gamma Function ◽

Riemann Zeta Function ◽

Riemann Hypothesis ◽

Additional Term ◽

Alternative Perspective ◽

Riemann Zeta

We review some main results of Riemann Zeta function; the Integral representation the analytic continuity and the first functional equation by the means of Gamma function and Hankel contour. We observe that an additional term is considered in both results. We justify the non-trivial location of Zeta non-trivial zeros subject to an approximation.

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Regularized Integral Representations of the Reciprocal Gamma Function

10.20944/preprints201812.0310.v1 ◽

2018 ◽

Author(s):

Dimiter Prodanov

Keyword(s):

Integral Representation ◽

Gamma Function ◽

Integral Representations ◽

Complex Representation ◽

Hypersingular Integral

This paper establishes a real integral representation of the reciprocal $\Gamma$ function in terms of a regularized hypersingular integral. The equivalence with the usual complex representation is demonstrated. A regularized complex representation along the usual Hankel path is derived.

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