scholarly journals Expressions of the Laguerre polynomial and some other special functions in terms of the generalized Meijer $ G $-functions

2021 ◽  
Vol 6 (11) ◽  
pp. 11631-11641
Author(s):  
Syed Ali Haider Shah ◽  
◽  
Shahid Mubeen
Keyword(s):  
Hypergeometric Functions ◽  
Laguerre Polynomial ◽  
Special Functions ◽  
Laguerre Polynomials ◽  
Exponential Functions ◽  
Confluent Hypergeometric Functions ◽  
Logarithmic Functions

<abstract><p>In this paper, we investigate the relation of generalized Meijer $ G $-functions with some other special functions. We prove the generalized form of Laguerre polynomials, product of Laguerre polynomials with exponential functions, logarithmic functions in terms of generalized Meijer $ G $-functions. The generalized confluent hypergeometric functions and generalized tricomi confluent hypergeometric functions are also expressed in terms of the generalized Meijer $ G $-functions.</p></abstract>

scholarly journals Distributions of the Ratio and Product of Two Independent Weibull and Lindley Random Variables

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
N. J. Hassan ◽  
A. Hawad Nasar ◽  
J. Mahdi Hadad
Keyword(s):  
Hypergeometric Functions ◽  
Special Functions ◽  
Random Variables ◽  
Distribution Functions ◽  
Cumulative Distribution ◽  
Parabolic Cylinder ◽  
Generalized Hypergeometric Functions ◽  
Confluent Hypergeometric Functions ◽  
Parabolic Cylinder Functions ◽  
The Moment

In this paper, we derive the cumulative distribution functions (CDF) and probability density functions (PDF) of the ratio and product of two independent Weibull and Lindley random variables. The moment generating functions (MGF) and the k-moment are driven from the ratio and product cases. In these derivations, we use some special functions, for instance, generalized hypergeometric functions, confluent hypergeometric functions, and the parabolic cylinder functions. Finally, we draw the PDF and CDF in many values of the parameters.

Two singular integral equations involving confluent hypergeometric functions

1969 ◽  
Vol 66 (1) ◽  
pp. 71-89 ◽  
Author(s):  
Tilak Raj Prabhakar
Keyword(s):  
Integral Equations ◽  
Singular Integral ◽  
Hypergeometric Functions ◽  
Laguerre Polynomial ◽  
Singular Integral Equations ◽  
Whittaker Functions ◽  
Confluent Hypergeometric Functions ◽  
The Laplace Transform ◽  
Similar Transform ◽  
Image Position

Widder(1) obtained an inversion of the convolution transformby the method of the Laplace transform, Ln(x) being the Laguerre polynomial. Buschman (2) inverted a similar transform with a generalized Laguerre polynomial as kernel and also solved (3) the singular integral equationusing Mikusinski operators. Srivastava(4, 4a) solved singular integral equations with kernels involving and Whittaker functions Mk,μ(x).

Conditionally Specified Bivariate Kummer-Gamma Distribution

2021 ◽  
Vol 20 ◽  
pp. 196-206
Author(s):  
Daya K. Nagar ◽  
Edwin Zarrazola ◽  
Alejandro Roldán-Correa
Keyword(s):  
Gamma Distribution ◽  
Hypergeometric Functions ◽  
Special Functions ◽  
Bimodal Distribution ◽  
Bivariate Distribution ◽  
Product Moments ◽  
Confluent Hypergeometric Functions ◽  
Conditional Moments ◽  
Bivariate Models

The Kummer-gamma distribution is an extension of gamma distribution and for certain values of parameters slides to a bimodal distribution. In this article, we introduce a bivariate distribution with Kummer-gamma conditionals and call it the conditionally specified bivariate Kummer-gamma distribution/bivariate Kummer-gamma conditionals distribution. Various representations are derived for its product moments, marginal densities, marginal moments, conditional densities, and conditional moments. We also discuss several important properties including, entropies, distributions of sum, and quotient. Most of these representations involve special functions such as the Gauss and the confluent hypergeometric functions. The bivariate Kummer-gamma conditionals distribution studied in this article may serve as an alternative to many existing bivariate models with support on (0, ∞) × (0, ∞).

scholarly journals INEQUALITIES FOR SOME SPECIAL FUNCTIONS-II

1995 ◽  
Vol 26 (3) ◽  
pp. 235-242
Author(s):  
S. K. BISSU ◽  
C. M. JOSHI
Keyword(s):  
Hypergeometric Functions ◽  
Bessel Functions ◽  
Special Functions ◽  
Upper Bounds ◽  
Modified Bessel Functions ◽  
Lower And Upper Bounds ◽  
Confluent Hypergeometric Functions

Some inequalities for Bessel functions, modified Bessel functions of the first kind and of their ratios involving both lower and upper bounds are given. The inequalities improve the results of earlier authours. Also incorporated in the discussion are some inequalities for the ratios of confluent hypergeometric functions of one variable.

scholarly journals ON A NEW GENERALIZED BETA FUNCTION DEFINED BY THE GENERALIZED WRIGHT FUNCTION AND ITS APPLICATIONS

2021 ◽  
Vol 6 (2) ◽  
pp. 852
Author(s):  
UMAR MUHAMMAD ABUBAKAR ◽  
Soraj Patel
Keyword(s):  
Hypergeometric Functions ◽  
Special Functions ◽  
Beta Function ◽  
Integral Representations ◽  
Wright Function ◽  
Confluent Hypergeometric Functions ◽  
Summation Formulas ◽  
Generalized Wright Function ◽  
Beta Error ◽  
Extended Beta Function

Various extensions of classical gamma, beta, Gauss hypergeometric and confluent hypergeometric functions have been proposed recently by many researchers. In this paper, we further generalized extended beta function with some of its properties such as symmetric properties, summation formulas, integral representations, connection with some other special functions such as classical beta, error, Mittag – Leffler, incomplete gamma, hypergeometric, classical Wright, Fox – Wright, Fox H and Meijer G – functions. Furthermore, the generalized beta function is used to generalize classical and other extended Gauss hypergeometric, confluent hypergeometric, Appell’s and Lauricella’s functions.

scholarly journals Some results on (p, k)-extension of the hypergeometric functions

2021 ◽  
Author(s):  
Mohamed Abdalla ◽  
H Hidan
Keyword(s):  
Differential Equations ◽  
Hypergeometric Functions ◽  
Special Functions ◽  
Natural Extension ◽  
Convergence Properties ◽  
Basic Properties ◽  
Confluent Hypergeometric Functions ◽  
Derivative Formulas ◽  
Two Parameters

Abstract In this study, we investigate a new natural extension of hypergeometric functions with the two parameters p and k which is so called (p, k)-extended hypergeometric functions”. In particular, we introduce the (p, k)-extended Gauss and Kummer (or confluent) hypergeometric functions. The basic properties of the (p, k)-extended Gauss and Kummer hypergeometric functions, including convergence properties, integral and derivative formulas, contiguous function relations and differential equations. Since the latter functions contain many of the familiar special functions as sub-cases, this extension is enriches theory of k-special functions.

scholarly journals Properties and Applications of Extended Hypergeometric Functions

2014 ◽  
Vol 10 (19) ◽  
pp. 11-31 ◽  
Author(s):  
Daya K. Nagar ◽  
Raúl Alejandro Morán-Vásquez ◽  
Arjun K. Gupta
Keyword(s):  
Hypergeometric Functions ◽  
Special Functions ◽  
Distribution Theory ◽  
Confluent Hypergeometric Functions

In this article, we study several properties of extended Gauss hypergeometric and extended confluent hypergeometric functions. We derive several integrals, inequalities and establish relationship between these and other special functions. We also show that these functions occur naturally instatistical distribution theory.

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