scholarly journals Partial differential equations for hypergeometric functions of two argument matrices

1972 ◽  
Vol 2 (3) ◽  
pp. 332-338 ◽  
Author(s):  
A.G. Constantine ◽  
R.J. Muirhead
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Hypergeometric Functions ◽  
Partial Differential

scholarly journals Linearly Independent Solutions and Integral Representations for Certain Quadruple Hypergeometric Function

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Anvar Hasanov ◽  
Jihad Younis ◽  
Hassen Aydi
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Hypergeometric Function ◽  
Hypergeometric Functions ◽  
Existence Of Solutions ◽  
Integral Representations ◽  
Partial Differential ◽  
Linearly Independent

Recently, hypergeometric functions of four variables are investigated by Bin-Saad and Younis. In this manuscript, our goal is to initiate a new quadruple hypergeometric function denoted by X 84 4 , and then, we ensure the existence of solutions of systems of partial differential equations for this function. We also establish some integral representations involving the quadruple hypergeometric function X 84 4 .

scholarly journals Decomposition formulas for some quadruple hypergeometric series

2020 ◽  
Vol 100 (4) ◽  
pp. 43-54
Author(s):  
A.S. Berdyshev ◽  
◽  
A. Hasanov ◽  
A.R. Ryskan ◽  
◽  
...  
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Value Problems ◽  
Hypergeometric Functions ◽  
Hypergeometric Series ◽  
Boundary Value ◽  
Second Order ◽  
Partial Differential

In the present work, the authors obtained operator identities and decomposition formulas for second order Gauss hypergeometric series of four variables into products containing simpler hypergeometric functions. A Choi–Hasanov method based on the inverse pairs of symbolic operators is used. The obtained expansion formulas for the hypergeometric functions of four variables will allow us to study the properties of these functions. These decompositions are used to study the solvability of boundary value problems for degenerate multidimensional partial differential equations.

XXXIX.—Transformations of Hypergeometric Functions of Two Variables

1948 ◽  
Vol 62 (3) ◽  
pp. 378-385 ◽  
Author(s):  
A. Erdélyi
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Hypergeometric Functions ◽  
Hypergeometric Series ◽  
Double Series ◽  
Limited Range ◽  
Transformation Theory ◽  
Partial Differential ◽  
Functions Of Two Variables ◽  
Infinite Sum

I. There are several methods for obtaining transformations of hypergeometric functions of two variables.Firstly, by transformation of the hypergeometric series. When the double series is rewritten as an infinite sum of hypergeometric functions of one variable, the known transformation theory of such functions can be applied to each term. This method is quite simple and, in a limited range, very effective for discovering transformations as well as proving them.Secondly, by transformation of the systems of partial differential equations satisfied by the hypergeometric functions. This method, though simple in theory, is rather laborious in practice and not very useful for discovering new transformations.

Appell Hypergeometric Functions

2017 ◽  
Author(s):  
Paula Tretkoff
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Hypergeometric Functions ◽  
Hypergeometric Series ◽  
Monodromy Group ◽  
Complex Variables ◽  
Partial Differential ◽  
Line Arrangement ◽  
Monodromy Groups ◽  
Complete Quadrilateral

This chapter discusses the complete quadrilateral line arrangement, and especially its relationship with the space of regular points of the system of partial differential equations defining the Appell hypergeometric function. Appell introduced four series F1, F2, F3, F4 in two complex variables, each of which generalizes the classical Gauss hypergeometric series and satisfies its own system of two linear second order partial differential equations. The solution spaces of the systems corresponding to the series F2, F3, F4 all have dimension 4, whereas that of the system corresponding to the series F1 has dimension 3. This chapter focuses on the F1-system whose monodromy group, under certain conditions, acts on the complex 2-ball. It first considers the action of S5 on the blown-up projective plane before turning to Appell hypergeometric functions, arithmetic monodromy groups, and an invariant known as the signature.

scholarly journals DECOMPOSITION FORMULAS WITH OPERATORS H FOR SECOND-ORDER GAUSS HYPERGEOMETRIC SERIES OF FOUR VARIABLES

2020 ◽  
Vol 71 (3) ◽  
pp. 91-97
Author(s):  
A.R. Ryskan ◽  
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Value Problems ◽  
Hypergeometric Functions ◽  
Hypergeometric Series ◽  
Boundary Value ◽  
Second Order ◽  
Partial Differential

In this paper decomposition formulas and operator identities for second-order Gauss hypergeometric series of four variables in products of simpler known hypergeometric functions were obtained. The Choi - Hasanov method is used, based on inverse pairs of symbolic operators H(a,c) and H¯¯¯¯¯(a,c) introduced in 2011 in the article of Junesang Choi, Anvar Hasanov «Applications of the operator H(a,c) to the Humbert double hypergeometric functions». The obtained expansion formulas for hypergeometric functions of four variables will allow us to study the properties of these functions. By means of these expansions we can investigate the solvability of some boundary value problems for partial differential equations.

scholarly journals Linear Independent Solutions and Operational Representations for Hypergeometric Functions of Four Variables

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Maged G. Bin-Saad ◽  
Anvar Hasanov
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Value Problems ◽  
Hypergeometric Function ◽  
Hypergeometric Functions ◽  
Applied Mathematics ◽  
Integral Operators ◽  
Partial Differential ◽  
Finite Series ◽  
Linearly Independent

In investigation of boundary-value problems for certain partial differential equations arising in applied mathematics, we often need to study the solution of system of partial differential equations satisfied by hypergeometric functions and find explicit linearly independent solutions for the system. Here we choose the Exton function K2 among his 21 functions to show how to find the linearly independent solutions of partial differential equations satisfied by this function K2. Based upon the classical derivative and integral operators, we introduce a new operational images for hypergeometric function K2. By means of these operational images, a number of finite series and decomposition formulas are then found.

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