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 Investigation of Extended k-Hypergeometric Functions and Associated Fractional Integrals

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Mohamed Abdalla ◽  
Muajebah Hidan ◽  
Salah Mahmoud Boulaaras ◽  
Bahri-Belkacem Cherif
Keyword(s):  
Probability Theory ◽  
Hypergeometric Functions ◽  
Fractional Integral ◽  
Special Functions ◽  
Integral Operators ◽  
Integral Representations ◽  
Fractional Integrals ◽  
Convergence Properties ◽  
Fractional Integral Operators ◽  
Derivative Formulas

Hypergeometric functions have many applications in various areas of mathematical analysis, probability theory, physics, and engineering. Very recently, Hidan et al. (Math. Probl. Eng., ID 5535962, 2021) introduced the (p, k)-extended hypergeometric functions and their various applications. In this line of research, we present an expansion of the k-Gauss hypergeometric functions and investigate its several properties, including, its convergence properties, derivative formulas, integral representations, contiguous function relations, differential equations, and fractional integral operators. Furthermore, the current results contain several of the familiar special functions as particular cases, and this extension may enrich the theory of special functions.

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>

On the Stresses in a Rotating Disk of Variable Thickness

1952 ◽  
Vol 19 (3) ◽  
pp. 263-266
Author(s):  
Ti-Chiang Lee
Keyword(s):  
Variable Thickness ◽  
Stress Function ◽  
Hypergeometric Functions ◽  
Analytic Solution ◽  
Radial Distance ◽  
Rotating Disk ◽  
Numerical Examples ◽  
Confluent Hypergeometric Functions ◽  
Method Of Solution ◽  
Two Parameters

Abstract This paper presents an analytic solution of the stresses in a rotating disk of variable thickness. By introducing two parameters, the profile of the disk is assumed to vary exponentially with any power of the radial distance from the center of the disk. In some respects this solution may be considered as a generalization of Malkin’s solution, but it differs essentially from the latter in the method of solution. Here, the stresses are solved through a stress function instead of being solved directly. The required stress function is expressed in terms of confluent hypergeometric functions. Numerical examples are also shown for illustration.

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.

The oblique reflexion of long wireless waves from the ionosphere at the places where the Earth's magnetic field is regarded as vertical

1952 ◽  
Vol 244 (887) ◽  
pp. 469-503 ◽  
Keyword(s):  
Magnetic Field ◽  
Differential Equations ◽  
Electromagnetic Waves ◽  
Hypergeometric Functions ◽  
Collision Frequency ◽  
Path Difference ◽  
Stratified Medium ◽  
Graphical Form ◽  
Confluent Hypergeometric Functions ◽  
Ionization Density

The calculation of reflexion coefficients for long wireless waves incident obliquely on the ionosphere requires an exact solution of the differential equations governing the propagation of electromagnetic waves in the ionosphere. Equations are developed for the electromagnetic field in a horizontally stratified medium of varying electron density, the presence of a vertical external magnetic field and also the collision frequency of the electrons with neutral molecules being taken into account. Provided certain inequalities hold amongst these ionospheric characteristics, the ionosphere splits up effectively into two regions, in each of which the differential equations of wave propagation approximate to simpler forms. If a model ionosphere is chosen in which the ionization density increases exponentially with height/and the collision frequency is assumed constant over the range of height responsible for reflexion, the equations for the two regions can be solved exactly. The solution for the lower region is expressed in terms of hypergeometric functions, and that for the upper region in terms of generalized confluent hypergeometric functions. Exact expressions in terms of factorial functions can then be deduced for the reflexion coefficients of both regions separately. Moreover, these coefficients can be combined, with due allowance for the path difference between the two regions, to give the overall reflexion coefficients for the effect of the ionosphere as a whole on an incident wave. A suitable definition is given for the apparent height of reflexion in terms of the phase of the reflected wave. The results of the theory are illustrated in graphical form for a particular model ionosphere approximating to the 'tail’ of a Chapman region, and a brief comparison with experimental observations concludes the paper.

scholarly journals Investigation of the k-Analogue of Gauss Hypergeometric Functions Constructed by the Hadamard Product

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 714
Author(s):  
Mohamed Abdalla ◽  
Muajebah Hidan
Keyword(s):  
Hypergeometric Functions ◽  
Hadamard Product ◽  
Special Function ◽  
Function Theory ◽  
Special Functions ◽  
Integral Transforms ◽  
Integral Representations ◽  
Convergence Properties ◽  
Pochhammer Symbol ◽  
Definition Of

Traditionally, the special function theory has many applications in various areas of mathematical physics, economics, statistics, engineering, and many other branches of science. Inspired by certain recent extensions of the k-analogue of gamma, the Pochhammer symbol, and hypergeometric functions, this work is devoted to the study of the k-analogue of Gauss hypergeometric functions by the Hadamard product. We give a definition of the Hadamard product of k-Gauss hypergeometric functions (HPkGHF) associated with the fourth numerator and two denominator parameters. In addition, convergence properties are derived from this function. We also discuss interesting properties such as derivative formulae, integral representations, and integral transforms including beta transform and Laplace transform. Furthermore, we investigate some contiguous function relations and differential equations connecting the HPkGHF. The current results are more general than previous ones. Moreover, the proposed results are useful in the theory of k-special functions where the hypergeometric function naturally occurs.

A family of the incomplete hypergeometric functions and associated integral transform and fractional derivative formulas

Filomat ◽  
2017 ◽  
Vol 31 (1) ◽  
pp. 125-140 ◽  
Author(s):  
Rekha Srivastava ◽  
Ritu Agarwal ◽  
Sonal Jain
Keyword(s):  
Hypergeometric Functions ◽  
Fractional Derivatives ◽  
Integral Transform ◽  
Special Functions ◽  
Applied Mathematics ◽  
Integral Transforms ◽  
Mathematical Physics ◽  
Natural Generalization ◽  
Derivative Formulas ◽  
Derivatives Of

Recently, Srivastava et al. [Integral Transforms Spec. Funct. 23 (2012), 659-683] introduced the incomplete Pochhammer symbols that led to a natural generalization and decomposition of a class of hypergeometric and other related functions as well as to certain potentially useful closed-form representations of definite and improper integrals of various special functions of applied mathematics and mathematical physics. In the present paper, our aim is to establish several formulas involving integral transforms and fractional derivatives of this family of incomplete hypergeometric functions. As corollaries and consequences, many interesting results are shown to follow from our main results.

scholarly journals On Value Distribution Theory of Second Order Periodic ODEs, Special Functions and Orthogonal Polynomials

2006 ◽  
Vol 58 (4) ◽  
pp. 726-767 ◽  
Author(s):  
Yik-Man Chiang ◽  
Mourad E. H. Ismail
Keyword(s):  
Differential Equations ◽  
Special Functions ◽  
Complete Characterization ◽  
Second Order ◽  
Value Distribution ◽  
Value Distribution Theory ◽  
Confluent Hypergeometric Functions ◽  
Number Of Zeros ◽  
Oscillation Of Solutions

AbstractWe show that the value distribution (complex oscillation) of solutions of certain periodic second order ordinary differential equations studied by Bank, Laine and Langley is closely related to confluent hypergeometric functions, Bessel functions and Bessel polynomials. As a result, we give a complete characterization of the zero-distribution in the sense of Nevanlinna theory of the solutions for two classes of the ODEs. Our approach uses special functions and their asymptotics. New results concerning finiteness of the number of zeros (finite-zeros) problem of Bessel and Coulomb wave functions with respect to the parameters are also obtained as a consequence. We demonstrate that the problem for the remaining class of ODEs not covered by the above “special function approach” can be described by a classical Heine problem for differential equations that admit polynomial solutions.

XXVI.—Some Confluent Hypergeometric Functions of Two Variables

1940 ◽  
Vol 60 (3) ◽  
pp. 344-361 ◽  
Author(s):  
A. Erdélyi
Keyword(s):  
Differential Equations ◽  
Hypergeometric Functions ◽  
Second Order ◽  
Linear Differential Equations ◽  
Confluent Hypergeometric Functions ◽  
Functions Of Two Variables ◽  
Partial Linear ◽  
Linear Differential ◽  
Image Position

1. This paper is the continuation of a former one (Erdélyi, 1939), and deals with the integration of the system of two partial linear differential equations of the second orderThe former paper will be referred to as I; all the notations of I will be retained without further explanation.

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