scholarly journals TRANSCENDENCE OF SPECIAL VALUES OF POCHHAMMER FUNCTIONS

2009 ◽  
Vol 05 (04) ◽  
pp. 667-677
Author(s):  
MARVIN D. TRETKOFF ◽  
PAULA TRETKOFF
Keyword(s):  
Hypergeometric Functions ◽  
Higher Order ◽  
Algebraic Numbers ◽  
Special Values

In this paper, we examine the set of algebraic numbers at which higher order hypergeometric functions take algebraic values. In particular, we deduce criteria for this set to be finite and for it to be infinite.

scholarly journals Explicit logarithmic formulas of special values of hypergeometric functions 3F2

2019 ◽  
Vol 22 (05) ◽  
pp. 1950040
Author(s):  
Masanori Asakura ◽  
Toshifumi Yabu
Keyword(s):  
Linear Combination ◽  
Hypergeometric Function ◽  
Hypergeometric Functions ◽  
Rational Numbers ◽  
Explicit Description ◽  
Generalized Hypergeometric Function ◽  
Algebraic Numbers ◽  
Nagoya Math ◽  
Special Values ◽  
Geometric Study

In [M. Asakura, N. Otsubo and T. Terasoma, An algebro-geometric study of special values of hypergeometric functions [Formula: see text], to appear in Nagoya Math. J.; https://doi.org/10.1017/nmj.2018.36 ], we proved that the value of [Formula: see text] of the generalized hypergeometric function is a [Formula: see text]-linear combination of log of algebraic numbers if rational numbers [Formula: see text] satisfy a certain condition. In this paper, we present a method to obtain an explicit description of it.

AN ALGEBRO-GEOMETRIC STUDY OF SPECIAL VALUES OF HYPERGEOMETRIC FUNCTIONS

2018 ◽  
Vol 236 ◽  
pp. 47-62
Author(s):  
MASANORI ASAKURA ◽  
NORIYUKI OTSUBO ◽  
TOMOHIDE TERASOMA
Keyword(s):  
Hypergeometric Functions ◽  
Sufficient Condition ◽  
Algebraic Numbers ◽  
Special Values ◽  
Special Value ◽  
Geometric Study

For a certain class of hypergeometric functions $_{3}F_{2}$ with rational parameters, we give a sufficient condition for the special value at $1$ to be expressed in terms of logarithms of algebraic numbers. We give two proofs, both of which are algebro-geometric and related to higher regulators.

scholarly journals Moving breathers and breather-to-soliton conversions for the Hirota equation

2015 ◽  
Vol 471 (2180) ◽  
pp. 20150130 ◽  
Author(s):  
A. Chowdury ◽  
A. Ankiewicz ◽  
N. Akhmediev
Keyword(s):  
Schrodinger Equation ◽  
Higher Order ◽  
Nonlinear Schrodinger Equation ◽  
Second Order ◽  
Periodic Oscillations ◽  
Hirota Equation ◽  
First Order ◽  
Special Values ◽  
Analytic Expression ◽  
Free Parameters

We find that the Hirota equation admits breather-to-soliton conversion at special values of the solution eigenvalues. This occurs for the first-order, as well as higher orders, of breather solutions. An analytic expression for the condition of the transformation is given and several examples of transformations are presented. The values of these special eigenvalues depend on two free parameters that are present in the Hirota equation. We also find that higher order breathers generally have complicated quasi-periodic oscillations along the direction of propagation. Various breather solutions are considered, including the particular case of second-order breathers of the nonlinear Schrödinger equation.

scholarly journals On Integrability of Christou’s Sixth Order Solitary Wave Equations

2019 ◽  
Vol 60 (5) ◽  
pp. 1172-1179
Author(s):  
M. Allami
Keyword(s):  
Solitary Wave ◽  
Wave Equations ◽  
Higher Order ◽  
Sixth Order ◽  
Special Values

We examine the integrability in terms of Painlevè analysis for several models of higher order nonlinear solitary wave equations which were recently derived by Christou. Our results point out that these equations do not possess Painlevè property and fail the Painlevè test for some special values of the coefficients; and that indicates a non-integrability criteria of the equations by means of the Painlevè integrability.

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