scholarly journals The generalized hypergeometric difference equation

2018 ◽  
Vol 51 (1) ◽  
pp. 62-75 ◽  
Author(s):  
Martin Bohner ◽  
Tom Cuchta
Keyword(s):  
Differential Equation ◽  
Difference Equation ◽  
Special Functions ◽  
Hypergeometric Differential Equation ◽  
Contiguous Relations

Abstract A difference equation analogue of the generalized hypergeometric differential equation is defined, its contiguous relations are developed, and its relation to numerous well-known classical special functions are demonstrated.

scholarly journals p-adic confluence of q-difference equations

2008 ◽  
Vol 144 (4) ◽  
pp. 867-919 ◽  
Author(s):  
Andrea Pulita
Keyword(s):  
Differential Equation ◽  
Difference Equation ◽  
Differential Equations ◽  
Difference Equations ◽  
Root Of Unity

AbstractWe develop the theory of p-adic confluence of q-difference equations. The main result is the fact that, in the p-adic framework, a function is a (Taylor) solution of a differential equation if and only if it is a solution of a q-difference equation. This fact implies an equivalence, called confluence, between the category of differential equations and those of q-difference equations. We develop this theory by introducing a category of sheaves on the disk D−(1,1), for which the stalk at 1 is a differential equation, the stalk at q isa q-difference equation if q is not a root of unity, and the stalk at a root of unity ξ is a mixed object, formed by a differential equation and an action of σξ.

scholarly journals k-Hypergeometric Series Solutions to One Type of Non-Homogeneous k-Hypergeometric Equations

Symmetry ◽  
2019 ◽  
Vol 11 (2) ◽  
pp. 262 ◽  
Author(s):  
Shengfeng Li ◽  
Yi Dong
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Hypergeometric Series ◽  
Second Order ◽  
Series Solutions ◽  
Frobenius Method ◽  
General Solutions ◽  
Hypergeometric Differential Equation ◽  
Polynomial Term ◽  
Hypergeometric Equations

In this paper, we expound on the hypergeometric series solutions for the second-order non-homogeneous k-hypergeometric differential equation with the polynomial term. The general solutions of this equation are obtained in the form of k-hypergeometric series based on the Frobenius method. Lastly, we employ the result of the theorem to find the solutions of several non-homogeneous k-hypergeometric differential equations.

scholarly journals THE GENERALIZED MATRIX VALUED HYPERGEOMETRIC EQUATION

2010 ◽  
Vol 21 (02) ◽  
pp. 145-155 ◽  
Author(s):  
P. ROMÁN ◽  
S. SIMONDI
Keyword(s):  
Differential Equation ◽  
Orthogonal Polynomials ◽  
Unit Circle ◽  
Hypergeometric Functions ◽  
Spherical Functions ◽  
Hypergeometric Equation ◽  
Necessary Condition ◽  
Hypergeometric Differential Equation ◽  
The Matrix ◽  
Generalized Matrix

The matrix valued analog of the Euler's hypergeometric differential equation was introduced by Tirao in [4]. This equation arises in the study of matrix valued spherical functions and in the theory of matrix valued orthogonal polynomials. The goal of this paper is to extend naturally the number of parameters of Tirao's equation in order to get a generalized matrix valued hypergeometric equation. We take advantage of the tools and strategies developed in [4] to identify the corresponding matrix hypergeometric functions nFm. We prove that, if n = m + 1, these functions are analytic for |z| < 1 and we give a necessary condition for the convergence on the unit circle |z| = 1.

Hypergeometric Differential Equation

1991 ◽  
pp. 25-116
Author(s):  
Katsunori Iwasaki ◽  
Hironobu Kimura ◽  
Shun Shimomura ◽  
Masaaki Yoshida
Keyword(s):  
Differential Equation ◽  
Hypergeometric Differential Equation

scholarly journals Affine Schwarz Map for the Hypergeometric Differential Equation

2008 ◽  
Vol 51 (2) ◽  
pp. 281-305 ◽  
Author(s):  
Ryoichi Kobayashi ◽  
Tatsuya Nishizaka ◽  
Shoji Shinzato ◽  
Masaaki Yoshida
Keyword(s):  
Differential Equation ◽  
Hypergeometric Differential Equation

Study on unbiased interval grey number prediction model with new information priority

2019 ◽  
Vol 10 (1) ◽  
pp. 1-11 ◽  
Author(s):  
Ye Li ◽  
Juan Li
Keyword(s):  
Differential Equation ◽  
Difference Equation ◽  
Prediction Model ◽  
Response Function ◽  
Iteration Method ◽  
Time Response ◽  
Initial Value ◽  
Content Type ◽  
New Information ◽  
Grey Number

Purpose The purpose of this paper is to construct an unbiased interval grey number prediction model with new information priority for dealing with the jumping errors from difference equation to the differential equation in the prediction model of interval grey number. Design/methodology/approach First, this study obtains a set of linear equations about the model parameters by taking the minimum error sum of squares between the accumulative sequence and its simulation values as criterion, and solves them on the basis of the Crammer rule. Then, according to the new information priority principle, it selects the last number of the accumulated generation sequence as the initial value and gives the expression of the time response function by the recursive iteration method to establish the interval grey number prediction model. Findings This paper provides an unbiased interval grey number prediction model with new information priority, and the example analysis shows that the method proposed in this paper has higher prediction precision and practicality. Research limitations/implications If there is a better method to whiten the interval grey number, so as to fully tap the grey information contained in it, the accuracy of the model will be higher. Practical implications The model proposed in this paper can avoid the error caused by jumping from difference equation to differential equation and make full use of new information. It can be better used in a problem where new information has a great influence on prediction results. Originality/value This paper selects the last number of the accumulated generation sequence as the initial value and gives the expression of the time response function by the recursive iteration method. Then, it constructs an unbiased interval grey number prediction model with new information priority.

scholarly journals Spherical curves and quadratic relationships for special functions

1985 ◽  
Vol 27 (1) ◽  
pp. 111-124
Author(s):  
Even Mehlum ◽  
Jet Wimp
Keyword(s):  
Differential Equation ◽  
Hypergeometric Functions ◽  
Bessel Functions ◽  
Special Functions ◽  
Position Vector ◽  
Third Order ◽  
Generalized Hypergeometric Functions ◽  
Transcendental Functions ◽  
Horn Functions ◽  
Vector Differential Equation

AbstractWe show that the position vector of any 3-space curve lying on a sphere satisfies a third-order linear (vector) differential equation whose coefficients involve a single arbitrary function A(s). By making various identifications of A(s), we are led to nonlinear identities for a number of higher transcendental functions: Bessel functions, Horn functions, generalized hypergeometric functions, etc. These can be considered natural geometrical generalizations of sin2t + cos2t = 1. We conclude with some applications to the theory of splines.

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