The modular properties and the integral representations of the multiple elliptic gamma functions

The article represents the elementary and general introduction of some characterizations of the extended gamma and beta Functions and their important properties with various representations. This paper provides reviews of some of the new proposals to extend the form of basic functions and some closed-form representation of more integral functions is described. Some of the relative behaviors of the extended function, the special cases resulting from them when fixing the parameters, the decomposition equation, the integrative representation of the proposed general formula, the correlations related to the proposed formula, the frequency relationships, and the differentiation equation for these basic functions were investigated. We also investigated the asymptotic behavior of some special cases, known formulas, the basic decomposition equation, integral representations, convolutions, recurrence relations, and differentiation formula for these target functions by studying. Applications of these functions have been presented in the evaluation of some reversible Laplace transforms to the complex of definite integrals and the infinite series of related basic functions.

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The incomplete Srivastava’s triple hypergeometric functions γHB and ΓHB

Filomat ◽

10.2298/fil1607779c ◽

2016 ◽

Vol 30 (7) ◽

pp. 1779-1787 ◽

Cited By ~ 3

Author(s):

Junesang Choi ◽

Rakesh Parmar ◽

Purnima Chopra

Keyword(s):

Recurrence Relation ◽

Hypergeometric Functions ◽

Integral Representations ◽

Reduction Formula ◽

Gamma Functions ◽

Fundamental Properties ◽

Special Cases ◽

Incomplete Gamma Functions ◽

Derivative Formula

Recently Srivastava et al. [26] introduced the incomplete Pochhammer symbols by means of the incomplete gamma functions ?(s,x) and ?(s,x), and defined incomplete hypergeometric functions whose a number of interesting and fundamental properties and characteristics have been investigated. Further, ?etinkaya [6] introduced the incomplete second Appell hypergeometric functions and studied many interesting and fundamental properties and characteristics. In this paper, motivated by the abovementioned works, we introduce two incomplete Srivastava?s triple hypergeometric functions ?HB and ?HB by using the incomplete Pochhammer symbols and investigate certain properties, for example, their various integral representations, derivative formula, reduction formula and recurrence relation. Various (known or new) special cases and consequences of the results presented here are also considered.

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Extended incomplete version of hypergeometric functions

Filomat ◽

10.2298/fil2002653o ◽

2020 ◽

Vol 34 (2) ◽

pp. 653-662

Author(s):

Mehmet Özarslan ◽

Ceren Ustaoğlu

Keyword(s):

Hypergeometric Functions ◽

Fractional Derivatives ◽

Integral Representations ◽

Elementary Functions ◽

Gamma Functions ◽

Mellin Transforms ◽

Incomplete Gamma Functions ◽

Derivative Formulas ◽

Transformation Formulas

Recently, the incomplete Pochhammer ratios are defined in terms of incomplete beta and gamma functions [10]. In this paper, we introduce the extended incomplete version of Pochhammer symbols in terms of the generalized incomplete gamma functions. With the help of this extended incomplete version of Pochhammer symbols we introduce the extended incomplete version of Gauss hypergeometric and Appell?s functions and investigate several properties of them such as integral representations, derivative formulas, transformation formulas, Mellin transforms and log convex properties. Furthermore, we investigate incomplete fractional derivatives for extended incomplete version of some elementary functions.

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ON SUMS OF INDEPENDENT GENERALIZED PARETO RANDOM VARIABLES WITH APPLICATIONS TO INSURANCE AND CAT BONDS

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964817000055 ◽

2017 ◽

Vol 32 (2) ◽

pp. 296-305 ◽

Cited By ~ 1

Author(s):

Saralees Nadarajah ◽

Yuanyuan Zhang ◽

Tibor K. Pogány

Keyword(s):

Random Variables ◽

Exact Distribution ◽

Integral Representations ◽

Gamma Functions ◽

Catastrophe Bonds ◽

Generalized Pareto ◽

Cat Bonds ◽

Incomplete Gamma Functions ◽

Single Integral

We derive single integral representations for the exact distribution of the sum of independent generalized Pareto random variables. The integrands involve the incomplete and complementary incomplete gamma functions. Applications to insurance and catastrophe bonds are described.

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MORE SPECIAL FUNCTIONS TRAPPED

International Journal of Modern Physics C ◽

10.1142/s0129183113500046 ◽

2013 ◽

Vol 24 (02) ◽

pp. 1350004

Author(s):

CHARLES SCHWARTZ

Keyword(s):

Hypergeometric Function ◽

Bessel Functions ◽

Special Functions ◽

Mathematical Physics ◽

Confluent Hypergeometric Function ◽

Trapezoidal Rule ◽

Integral Representations ◽

Gamma Functions ◽

Incomplete Gamma Functions

We extend the technique of using the trapezoidal rule for efficient evaluation of the special functions of mathematical physics given by integral representations. This technique was recently used for Bessel functions, and here we treat incomplete gamma functions and the general confluent hypergeometric function.

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Differentiation of the Mittag-Leffler Functions with Respect to Parameters in the Laplace Transform Approach

Mathematics ◽

10.3390/math8050657 ◽

2020 ◽

Vol 8 (5) ◽

pp. 657

Author(s):

Alexander Apelblat

Keyword(s):

Laplace Transform ◽

Integral Representations ◽

The Other ◽

Gamma Functions ◽

Inverse Transform ◽

Convolution Integrals ◽

The Laplace Transform ◽

Infinite Power ◽

Derivatives Of ◽

Two Parameter

In this work, properties of one- or two-parameter Mittag-Leffler functions are derived using the Laplace transform approach. It is demonstrated that manipulations with the pair direct–inverse transform makes it far more easy than previous methods to derive known and new properties of the Mittag-Leffler functions. Moreover, it is shown that sums of infinite series of the Mittag-Leffler functions can be expressed as convolution integrals, while the derivatives of the Mittag-Leffler functions with respect to their parameters are expressible as double convolution integrals. The derivatives can also be obtained from integral representations of the Mittag-Leffler functions. On the other hand, direct differentiation of the Mittag-Leffler functions with respect to parameters produces an infinite power series, whose coefficients are quotients of the digamma and gamma functions. Closed forms of these series can be derived when the parameters are set to be integers.

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Integral representations of the de Branges functional for univalent functions

Bulletin de la Classe des sciences ◽

10.3406/barb.2001.28182 ◽

2001 ◽

Vol 12 (1) ◽

pp. 121-135

Author(s):

Pavel G. Todorov

Keyword(s):

Univalent Functions ◽

Integral Representations

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ON p – q DUALITY AND EXPLICIT SOLUTIONS IN c ≤ 1 2D GRAVITY MODELS

International Journal of Modern Physics A ◽

10.1142/s0217751x95000577 ◽

1995 ◽

Vol 10 (08) ◽

pp. 1219-1236 ◽

Cited By ~ 17

Author(s):

S. KHARCHEV ◽

A. MARSHAKOV

Keyword(s):

String Theory ◽

Exact Solutions ◽

2D Gravity ◽

Integral Formula ◽

Integral Representations ◽

Gravity Models ◽

Explicit Solutions ◽

String Equation ◽

Duality Transformation

We study the role of integral representations in the description of nonperturbative solutions to c ≤ 1 string theory. A generic solution is determined by two functions, W(x) and Q(x), which behave at infinity like xp and xq respectively. The integral formula for arbitrary (p, q) models is derived, which explicitly realizes a duality transformation between (p, q) and (q, p) 2D gravity solutions. We also discuss the exact solutions to the string equation and reduction condition and present several explicit examples.

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