scholarly journals DETERMINING EQUATIONS FOR HIGHER-ORDER DECOMPOSITIONS OF EXPONENTIAL OPERATORS

1995 ◽  
Vol 09 (25) ◽  
pp. 3241-3268 ◽  
Author(s):  
ZENGO TSUBOI ◽  
MASUO SUZUKI
Keyword(s):  
General Scheme ◽  
Higher Order ◽  
Explicit Formulas ◽  
Decomposition Theory ◽  
Lyndon Words

The general decomposition theory of exponential operators is briefly reviewed. A general scheme to construct independent determining equations for the relevant decomposition parameters is proposed using Lyndon words. Explicit formulas of the coefficients are derived.

A Higher-Order Period Function and Its Application

2015 ◽  
Vol 25 (10) ◽  
pp. 1550140 ◽  
Author(s):  
Linping Peng ◽  
Lianghaolong Lu ◽  
Zhaosheng Feng
Keyword(s):  
Periodic Orbits ◽  
Upper Bound ◽  
Critical Period ◽  
Higher Order ◽  
Quadratic System ◽  
Critical Periods ◽  
Explicit Formulas ◽  
Period Function ◽  
Isochronous System ◽  
Isochronous Centers

This paper derives explicit formulas of the q th period bifurcation function for any perturbed isochronous system with a center, which improve and generalize the corresponding results in the literature. Based on these formulas to the perturbed quadratic and quintic rigidly isochronous centers, we prove that under any small homogeneous perturbations, for ε in any order, at most one critical period bifurcates from the periodic orbits of the unperturbed quadratic system. For ε in order of 1, 2, 3, 4 and 5, at most three critical periods bifurcate from the periodic orbits of the unperturbed quintic system. Moreover, in each case, the upper bound is sharp. Finally, a family of perturbed quintic rigidly isochronous centers is shown, which has three, for ε in any order, as the exact upper bound of the number of critical periods.

scholarly journals A three-term recurrence formula for the generalized Bernoulli polynomials

2021 ◽  
Vol 39 (6) ◽  
pp. 139-145
Author(s):  
Mohamed Amine Boutiche ◽  
Ghania Guettai ◽  
Mourad Rahmani ◽  
Madjid Sebaoui
Keyword(s):  
Recurrence Formula ◽  
Bernoulli Polynomials ◽  
Higher Order ◽  
Explicit Formulas ◽  
Whitney Numbers ◽  
Generalized Bernoulli Polynomials

In the present paper, we propose some new explicit formulas of the higher order Daehee polynomials in terms of the generalized r-Stirling and r-Whitney numbers of the second kind. As a consequence, we derive a three-term recurrence formula for the calculation of the generalized Bernoulli polynomials of order k.

scholarly journals Some Properties of the Hermite Polynomials and Their Squares and Generating Functions

2016 ◽  
Author(s):  
Feng Qi ◽  
Bai-Ni Guo
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Generating Functions ◽  
Recurrence Relations ◽  
Hermite Polynomials ◽  
Higher Order ◽  
Explicit Formulas ◽  
Derivatives Of

In the paper, the authors consider the generating functions of the Hermite polynomials and their squares, present explicit formulas for higher order derivatives of the generating functions of the Hermite polynomials and their squares, which can be viewed as ordinary differential equations or derivative polynomials, find differential equations that the generating functions of the Hermite polynomials and their squares satisfy, and derive explicit formulas and recurrence relations for the Hermite polynomials and their squares.

Some Explicit Formulas for the Frobenius-Euler Polynomials of Higher Order

2017 ◽  
Vol 11 (2) ◽  
pp. 621-626 ◽  
Author(s):  
H. M. Srivastava ◽  
Mohamed Amine Boutiche ◽  
Mourad Rahmani
Keyword(s):  
Higher Order ◽  
Euler Polynomials ◽  
Explicit Formulas

Lepage forms in Finsler geometry: Second-order generalization

2019 ◽  
Vol 16 (supp02) ◽  
pp. 1941005
Author(s):  
Demeter Krupka
Keyword(s):  
Finsler Geometry ◽  
Higher Order ◽  
Second Order ◽  
Basic Requirement ◽  
Jet Spaces ◽  
Explicit Formulas ◽  
Geometric Concepts

Projectability of Lepage forms, defined on higher-order jet spaces, onto the corresponding Grassmann fibrations, is a basic requirement for the extension of the theory of Lepage forms to integral variational functionals for submanifolds. In this paper, projectability of second-order Lepage forms is considered for variational functionals of the Finsler type. Underlying geometric concepts such as regular 2-velocities, contact elements and second-order Grassmann fibrations of rank 1 are discussed. It is shown that a Lepage form is projectable if and only if its Hamiltonian vanishes identically. In this case, explicit formulas for the Lagrange functions and the projected Lepage forms are given in terms of the adapted coordinates.

BIFURCATIONS OF SOLITARY WAVES AND DOMAIN WALL WAVES FOR KdV-LIKE EQUATION WITH HIGHER ORDER NONLINEARITY

2002 ◽  
Vol 12 (02) ◽  
pp. 397-407 ◽  
Author(s):  
ZHENGRONG LIU ◽  
JIBIN LI
Keyword(s):  
Domain Wall ◽  
Solitary Wave ◽  
Solitary Waves ◽  
Bifurcation Theory ◽  
Higher Order ◽  
Bifurcation Parameter ◽  
Solitary Wave Solutions ◽  
Explicit Formulas ◽  
Wave Solutions ◽  
Planar Dynamical Systems

Bifurcations of solitary waves and domain wall waves for a KdV-like equation with higher order nonlinearity are studied, by using bifurcation theory of planar dynamical systems. Bifurcation parameter sets are shown. Numbers of solitary waves and domain wall waves are given. Under some parameter conditions, a lot of explicit formulas of solitary wave solutions and domain wall solutions are obtained.

scholarly journals Some Identities for a Sequence of Unnamed Polynomials Connected with the Bell Polynomials

2017 ◽  
Author(s):  
Feng Qi ◽  
Da-Wei Niu ◽  
Bai-Ni Guo
Keyword(s):  
Stirling Numbers ◽  
Higher Order ◽  
Recursive Relation ◽  
Order Derivative ◽  
Bell Polynomials ◽  
Explicit Formulas ◽  
Derivative Formula ◽  
Higher Order Derivative ◽  
Determinantal Expression

In the paper, using two inversion theorems for the Stirling numbers and binomial coecients, employing properties of the Bell polynomials of the second kind, and utilizing a higher order derivative formula for the ratio of two dierentiable functions, the authors present two explicit formulas, a determinantal expression, and a recursive relation for a sequence of unnamed polynomials, derive two identities connecting the sequence of unnamed polynomials with the Bell polynomials, and recover a known identity connecting the sequence of unnamed polynomials with the Bell polynomials.

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