Dictionary of Difference Equations with Polynomial Coefficients

2016 ◽  
pp. 113-123
Author(s):  
Leonard C. Maximon
Keyword(s):  
Difference Equations ◽  
Polynomial Coefficients

scholarly journals Meromorphic Solutions of Linear Difference Equations with Polynomial Coefficients

2017 ◽  
Vol 59 (1) ◽  
pp. 159-168
Author(s):  
Y. Zhang ◽  
Z. Gao ◽  
H. Zhang
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Necessary Conditions ◽  
Meromorphic Solution ◽  
Linear Difference Equation ◽  
Meromorphic Solutions ◽  
Linear Difference Equations ◽  
Polynomial Coefficients ◽  
Difference Analogue ◽  
Borel Exceptional Values

AbstractWe study the growth of the transcendental meromorphic solution f(z) of the linear difference equation:where q(z), p0(z), ..., pn-(z) (n ≥ 1) are polynomials such that p0(z)pn(z) ≢ 0, and obtain some necessary conditions guaranteeing that the order of f(z) satisfies σ(f) ≥ 1 using a difference analogue of the Wiman-Valiron theory. Moreover, we give the form of f(z) with two Borel exceptional values when two of p0(z), ..., pn(z) have the maximal degrees.

ARITHMETICAL RESULTS ON CERTAIN q-SERIES, II

2009 ◽  
Vol 05 (07) ◽  
pp. 1231-1245 ◽  
Author(s):  
PETER BUNDSCHUH ◽  
KEIJO VÄÄNÄNEN
Keyword(s):  
Vector Space ◽  
Lower Bounds ◽  
Difference Equations ◽  
Quadratic Field ◽  
Imaginary Quadratic Field ◽  
Order Linear ◽  
Polynomial Coefficients ◽  
New Feature ◽  
Extra Parameter

As in Part I, entire transcendental solutions of certain mth order linear q-difference equations are investigated arithmetically, where now the polynomial coefficients are much more general. The purpose of this paper is to produce again lower bounds for the dimension of the K-vector space generated by 1 and the values of these solutions at m successive powers of q, where K is the rational or an imaginary quadratic field. A new feature in the proof is to use simultaneously positive and negative powers of q as interpolation points leading to an extra parameter in the main result extending its applicability.

On the number of linearly independent admissible solutions to linear differential and linear difference equations

2020 ◽  
pp. 1-36
Author(s):  
Janne Heittokangas ◽  
Hui Yu ◽  
Mohamed Amine Zemirni
Keyword(s):  
Difference Equations ◽  
Infinite Order ◽  
Entire Functions ◽  
Linear Difference Equations ◽  
Deficient Value ◽  
Polynomial Coefficients ◽  
Linearly Independent ◽  
Admissible Solutions ◽  
Linear Differential ◽  
Growth Of Solutions

Abstract A classical theorem of Frei states that if $A_p$ is the last transcendental function in the sequence $A_0,\ldots ,A_{n-1}$ of entire functions, then each solution base of the differential equation $f^{(n)}+A_{n-1}f^{(n-1)}+\cdots +A_{1}f'+A_{0}f=0$ contains at least $n-p$ entire functions of infinite order. Here, the transcendental coefficient $A_p$ dominates the growth of the polynomial coefficients $A_{p+1},\ldots ,A_{n-1}$ . By expressing the dominance of $A_p$ in different ways and allowing the coefficients $A_{p+1},\ldots ,A_{n-1}$ to be transcendental, we show that the conclusion of Frei’s theorem still holds along with an additional estimation on the asymptotic lower bound for the growth of solutions. At times, these new refined results give a larger number of linearly independent solutions of infinite order than the original theorem of Frei. For such solutions, we show that $0$ is the only possible finite deficient value. Previously, this property has been known to hold for so-called admissible solutions and is commonly cited as Wittich’s theorem. Analogous results are discussed for linear differential equations in the unit disc, as well as for complex difference and complex q-difference equations.

The approximate solution of difference equations with polynomial coefficients

1976 ◽  
Vol 27 (6) ◽  
pp. 678-682
Author(s):  
B. M. Mikhailets ◽  
L. I. Savchenko
Keyword(s):  
Approximate Solution ◽  
Difference Equations ◽  
Polynomial Coefficients

ARITHMETICAL RESULTS ON CERTAIN q-SERIES, I

2008 ◽  
Vol 04 (01) ◽  
pp. 25-43 ◽  
Author(s):  
PETER BUNDSCHUH
Keyword(s):  
Vector Space ◽  
Number Field ◽  
Lower Bounds ◽  
Difference Equations ◽  
Quadratic Number ◽  
Quadratic Number Field ◽  
Dimension Estimate ◽  
Polynomial Coefficients ◽  
Asymptotic Evaluation ◽  
Imaginary Quadratic Number

Entire transcendental solutions of certain mth order linear q-difference equations with polynomial coefficients are considered. The aim of this paper is to give, under appropriate arithmetical conditions, lower bounds for the dimension of the K-vector space generated by 1 and the values of these solutions at m successive powers of q, where K is the rational or an imaginary quadratic number field. The main ingredients of the proofs are, first, Nesterenko's dimension estimate and its various generalizations, and secondly, Popov's method (in Töpfer's version) for the asymptotic evaluation of certain complex integrals.

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