scholarly journals On a generalized three-parameter wright function of Le Roy type

2017 ◽  
Vol 20 (5) ◽  
Author(s):  
Roberto Garrappa ◽  
Sergei Rogosin ◽  
Francesco Mainardi
Keyword(s):  
Asymptotic Expansion ◽  
Power Series ◽  
Laplace Transform ◽  
Analytic Continuation ◽  
Special Function ◽  
Special Functions ◽  
Integral Representations ◽  
Radius Of Convergence ◽  
Wright Function ◽  
Finite Radius

AbstractRecently S. Gerhold and R. Garra – F. Polito independently introduced a new function related to the special functions of the Mittag-Leffler family. This function is a generalization of the function studied by É. Le Roy in the period 1895-1905 in connection with the problem of analytic continuation of power series with a finite radius of convergence. In our note we obtain two integral representations of this special function, calculate its Laplace transform, determine an asymptotic expansion of this function on the negative semi-axis (in the case of an integer third parameter

scholarly journals The a-points of Faber polynomials for a special function

1970 ◽  
Vol 11 (1) ◽  
pp. 1-6
Author(s):  
Hassoon S. Al-Amiri
Keyword(s):  
Analytic Function ◽  
Power Series ◽  
Analytic Continuation ◽  
Special Function ◽  
Laurent Expansion ◽  
Faber Polynomials ◽  
Analytic Element ◽  
Formal Expansion ◽  
Image Position

Let f(ζ) be a power series of the formwhere lim sup |an|1/n < ∞. The Faber polynomials {fn(ζ)} (n = 0, 1, 2, …) are the polynomial parts of the formal expansion of (f(ζ))n about ζ = ∞. Series (1) defines an analytic element of an analytic function which we designate as w = f(ζ). Since at ζ = ∞ the analytic element is univalent in some neighborhood of infinity; thus the inverse of this element is uniquely determined in some neighborhood of w= ∞, and has a Laurent expansion of the formwhere lim sup |bn|1/n = p < ∞. Let ζ = g(w) be this single-valued function defined by (2) in |w| > p. No analytic continuation of g(w) will be considered.

A generalization of the Hurwitz composition theorem to irregular power series

1951 ◽  
Vol 47 (3) ◽  
pp. 477-482 ◽  
Author(s):  
H. G. Eggleston
Keyword(s):  
Power Series ◽  
Radius Of Convergence ◽  
Finite Radius ◽  
Image Position ◽  
Composition Theorem

When two functions are given, each with a finite radius of convergence, a theorem due independently to Hurwitz and Pincherle (1, 2) provides information about the position of the singularities of the functionin terms of the positions of the singularities of f(z) and g(z).

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.

scholarly journals On a solution of a fractional hyper-Bessel differential equation by means of a multi-index special function

2021 ◽  
Vol 24 (5) ◽  
pp. 1559-1570
Author(s):  
Riccardo Droghei
Keyword(s):  
Differential Equation ◽  
Fractional Derivatives ◽  
Special Function ◽  
Special Functions ◽  
Wright Function ◽  
Fractional Partial Differential Equation ◽  
Caputo Fractional Derivatives ◽  
Multi Index ◽  
Multiple Parameters ◽  
Bessel Differential Equation

Abstract In this paper we introduce a new multiple-parameters (multi-index) extension of the Wright function that arises from an eigenvalue problem for a case of hyper-Bessel operator involving Caputo fractional derivatives. We show that by giving particular values to the parameters involved in this special function, this leads to some known special functions (as the classical Wright function, the α-Mittag-Leffler function, the Tricomi function, etc.) that on their turn appear as cases of the so-called multi-index Mittag-Leffler functions. As an application, we mention that this new generalization Wright function nis an isochronous solution of a nonlinear fractional partial differential equation.

VI.—On Solvability of Some Two-Parameter Eigenvalue Problems in Hilbert Space

1968 ◽  
Vol 68 (1) ◽  
pp. 83-93
Author(s):  
Z. Bohte
Keyword(s):  
Hilbert Space ◽  
Power Series ◽  
Eigenvalue Problem ◽  
Eigenvalue Problems ◽  
Radius Of Convergence ◽  
Finite Radius ◽  
Image Position ◽  
Adjoint Operators ◽  
Two Parameter

SynopsisThis paper studies two particular cases of the general 2-parameter eigenvalue problem namelywhere A, B, B1, B2, C, C1, C2 are self-adjoint operators in Hilbert space, all except A being bounded. The disposable parameters λ and μ have to be determined so that the equations have non-trivial solutions x, y.On the assumption that the solution is known for ∊ = o, solutions are constructed in the form of series for λ, μ, x, y as power series in ∊ with finite radius of convergence.

scholarly journals Integral Representations and Algebraic Decompositions of the Fox-Wright Type of Special Functions

2018 ◽  
Author(s):  
Dimiter Prodanov
Keyword(s):  
Hypergeometric Functions ◽  
Special Functions ◽  
Integral Operators ◽  
Integral Representations ◽  
Wright Function ◽  
Fractional Differential

The manuscript surveys the special functions of the Fox-Wright type. These functions are generalizations of the hypergeometric functions. Notable representatives of the type are the Mittag-Leffler functions and the Wright function. The integral representations of such functions are given and the conditions under which these function can be represented by simpler functions are demonstrated. The connection with generalized fractional differential and integral operators is demonstrated and discussed.

scholarly journals A study on certain properties of generalized special functions defined by Fox-Wright function

2020 ◽  
Vol 5 (1) ◽  
pp. 147-162
Author(s):  
Enes Ata ◽  
İ. Onur Kıymaz
Keyword(s):  
Hypergeometric Function ◽  
Special Functions ◽  
Confluent Hypergeometric Function ◽  
Integral Representations ◽  
Gauss Hypergeometric Function ◽  
Wright Function ◽  
Summation Formulas ◽  
Frequent Use ◽  
Derivative Formulas ◽  
Transformation Formulas

AbstractIn this study, motivated by the frequent use of Fox-Wright function in the theory of special functions, we first introduced new generalizations of gamma and beta functions with the help of Fox-Wright function. Then by using these functions, we defined generalized Gauss hypergeometric function and generalized confluent hypergeometric function. For all the generalized functions we have defined, we obtained their integral representations, summation formulas, transformation formulas, derivative formulas and difference formulas. Also, we calculated the Mellin transformations of these functions.

scholarly journals Convergent Power Series of sech⁡(x) and Solutions to Nonlinear Differential Equations

2018 ◽  
Vol 2018 ◽  
pp. 1-10 ◽  
Author(s):  
U. Al Khawaja ◽  
Qasem M. Al-Mdallal
Keyword(s):  
Power Series ◽  
Differential Equations ◽  
Series Expansion ◽  
Taylor Series ◽  
Nonlinear Differential Equations ◽  
Series Representation ◽  
Power Series Expansion ◽  
Detailed Comparison ◽  
Radius Of Convergence ◽  
Finite Radius

It is known that power series expansion of certain functions such as sech⁡(x) diverges beyond a finite radius of convergence. We present here an iterative power series expansion (IPS) to obtain a power series representation of sech⁡(x) that is convergent for all x. The convergent series is a sum of the Taylor series of sech⁡(x) and a complementary series that cancels the divergence of the Taylor series for x≥π/2. The method is general and can be applied to other functions known to have finite radius of convergence, such as 1/(1+x2). A straightforward application of this method is to solve analytically nonlinear differential equations, which we also illustrate here. The method provides also a robust and very efficient numerical algorithm for solving nonlinear differential equations numerically. A detailed comparison with the fourth-order Runge-Kutta method and extensive analysis of the behavior of the error and CPU time are performed.

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.

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