scholarly journals Polynomial solutions of algebraic difference equations and homogeneous symmetric polynomials

2021 ◽  
Vol 103 ◽  
pp. 22-45 ◽  
Author(s):  
Olha Shkaravska ◽  
Marko van Eekelen
Keyword(s):  
Difference Equations ◽  
Symmetric Polynomials ◽  
Polynomial Solutions ◽  
Algebraic Difference

scholarly journals Asymptotic iteration method for solving Hahn difference equations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Lucas MacQuarrie ◽  
Nasser Saad ◽  
Md. Shafiqul Islam
Keyword(s):  
Difference Equations ◽  
Iteration Method ◽  
Difference Operator ◽  
Sufficient Conditions ◽  
Asymptotic Iteration Method ◽  
Order Linear ◽  
Polynomial Solutions ◽  
Hahn Difference Equations ◽  
Iteration Methods ◽  
Necessary And Sufficient

AbstractHahn’s difference operator $D_{q;w}f(x) =({f(qx+w)-f(x)})/({(q-1)x+w})$ D q ; w f ( x ) = ( f ( q x + w ) − f ( x ) ) / ( ( q − 1 ) x + w ) , $q\in (0,1)$ q ∈ ( 0 , 1 ) , $w>0$ w > 0 , $x\neq w/(1-q)$ x ≠ w / ( 1 − q ) is used to unify the recently established difference and q-asymptotic iteration methods (DAIM, qAIM). The technique is applied to solve the second-order linear Hahn difference equations. The necessary and sufficient conditions for polynomial solutions are derived and examined for the $(q;w)$ ( q ; w ) -hypergeometric equation.

scholarly journals Finiteness Properties of Affine Difference Algebraic Groups

2020 ◽  
Author(s):  
Michael Wibmer
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Difference Equations ◽  
General Linear Group ◽  
Algebraic Groups ◽  
The Matrix ◽  
Algebraic Difference ◽  
The Difference ◽  
Linear Differential ◽  
Finiteness Properties

Abstract We establish several finiteness properties of groups defined by algebraic difference equations. One of our main results is that a subgroup of the general linear group defined by possibly infinitely many algebraic difference equations in the matrix entries can indeed be defined by finitely many such equations. As an application, we show that the difference ideal of all difference algebraic relations among the solutions of a linear differential equation is finitely generated.

Algebraic Methods for the Construction of Algebraic-Difference Equations With Desired Behavior

2018 ◽  
Vol 34 ◽  
pp. 1-17 ◽  
Author(s):  
Lazaros Moysis ◽  
Nicholas Karampetakis
Keyword(s):  
Difference Equations ◽  
Polynomial Matrix ◽  
Elementary Divisor ◽  
System Of Equations ◽  
Polynomial Matrices ◽  
Linear System Of Equations ◽  
System A ◽  
Auto Regressive ◽  
Algebraic Difference ◽  
Forward Operator

For a given system of algebraic and difference equations, written as an Auto-Regressive (AR) representation $A(\sigma)\beta(k)=0$, where $\sigma $ denotes the shift forward operator and $A\left( \sigma \right) $ a regular polynomial matrix, the forward-backward behavior of this system can be constructed by using the finite and infinite elementary divisor structure of $A\left( \sigma \right) $. This work studies the inverse problem: Given a specific forward-backward behavior, find a family of regular or non-regular polynomial matrices $A\left( \sigma \right) $, such that the constructed system $A\left( \sigma \right) \beta \left( k\right) =0$ has exactly the prescribed behavior. It is proved that this problem can be reduced either to a linear system of equations problem or to an interpolation problem and an algorithm is proposed for constructing a system satisfying a given forward and/or backward behavior.

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