Asymptotically polynomial solutions of difference equations

Polynomial solutions of differential-difference equations

1 We investigate the zeros of polynomial solutions to the differential-difference equation P n+1 (x) = A n (x)P ′ n (x) + B n (x)P n (x), n = 0, 1,. .. where A n and B n are polynomials of degree at most 2 and 1 respectively. We address the question of when the zeros are real and simple and whether the zeros of polynomials of adjacent degree are interlac-ing. Our result holds for general classe...

Univariate polynomial solutions of algebraic difference equations

Contrary to linear difference equations, there is no general theory of difference equations of the form G(P (x − τ1), . . . , P (x − τs)) + G0(x)=0, with τi ∈ K, G(x1, . . . , xs) ∈ K[x1, . . . , xs] of total degree D ≥ 2 and G0(x) ∈ K[x], where K is a field of characteristic zero. This article is concerned with the following problem: given τi, G and G0, find an upper bound on the degree d of a...

Univariate Polynomial Solutions of Nonlinear Polynomial Difference Equations

We study real-polynomial solutions P (x) of difference equations of the formG(P (x−τ1), . . . , P (x− τs)) +G0(x)=0, where τi are real numbers, G(x1, . . . , xs) is a real polynomial of a total degree D ≥ 2, and G0(x) is a polynomial in x. We consider the following problem: given τi, G and G0, find an upper bound on the degree d of a real-polynomial solution P (x), if exists. We reduce this pro...

Polynomial Solutions to Difference Equations Connected to Painleve’ Ii-vi

On meromorphic solutions of certain type of difference equations

‎We mainly discuss the existence of meromorphic (entire) solutions of‎ ‎certain type of non-linear difference equation of the form‎: ‎$f(z)^m+P(z)f(z+c)^n=Q(z)$‎, ‎which is a supplement of previous‎ ‎results in [K‎. ‎Liu‎, ‎L. Z‎. ‎Yang and X‎. ‎L‎. ‎Liu‎, ‎Existence of entire solutions of nonlinear difference‎ ‎equations‎, ‎Czechoslovak Math. J. 61 (2011)‎, no. 2, ‎565--576‎, and X‎. ‎G‎. ‎Qi‎...