scholarly journals Parametric Stokes phenomena of the Gauss hypergeometric differential equation with a large parameter

2016 ◽  
Vol 68 (3) ◽  
pp. 1099-1132 ◽  
Author(s):  
Takashi AOKI ◽  
Mika TANDA
Keyword(s):  
Differential Equation ◽  
Large Parameter ◽  
Hypergeometric Differential Equation ◽  
Stokes Phenomena

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

The Smoluchowski–Kramers approximation for the stochastic Liénard equation by mean-field

1991 ◽  
Vol 23 (2) ◽  
pp. 303-316 ◽  
Author(s):  
Kiyomasa Narita
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Diffusion Process ◽  
Langevin Equation ◽  
Mean Field ◽  
Two Dimensional ◽  
Large Parameter ◽  
Rigorous Evaluation ◽  
One Dimensional ◽  
Limit Diffusion

The oscillator of the Liénard type with mean-field containing a large parameter α < 0 is considered. The solution of the two-dimensional stochastic differential equation with mean-field of the McKean type is taken as the response of the oscillator. By a rigorous evaluation of the upper bound of the displacement process depending on the parameter α, a one-dimensional limit diffusion process as α → ∞is derived and identified. Then our result extends the Smoluchowski–Kramers approximation for the Langevin equation without mean-field to the McKean equation with mean-field.

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

scholarly journals The Intersection Probability of Brownian Motion and SLEκ

2015 ◽  
Vol 2015 ◽  
pp. 1-5
Author(s):  
Shizhong Zhou ◽  
Shiyi Lan
Keyword(s):  
Differential Equation ◽  
Brownian Motion ◽  
Poisson Kernel ◽  
Order Differential Equation ◽  
Kernel Method ◽  
Second Order ◽  
Second Order Differential Equation ◽  
Planar Brownian Motion ◽  
Hypergeometric Differential Equation ◽  
Upper Half Plane

By using excursion measure Poisson kernel method, we obtain a second-order differential equation of the intersection probability of Brownian motion andSLEκ. Moreover, we find a transformation such that the second-order differential equation transforms into a hypergeometric differential equation. Then, by solving the hypergeometric differential equation, we obtain the explicit formula of the intersection probability for the trace of the chordalSLEκand planar Brownian motion started from distinct points in an upper half-planeH-.

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