Each conservation law of a given partial differential equation is determined (up to equivalence) by a function known as the characteristic. This function is used to find conservation laws, to prove equivalence between conservation laws, and to prove the converse of Noether’s Theorem. Transferring these results to difference equations is nontrivial, largely because difference operators are not d...

We prove a result on the existence of periodic solutions of a class of systems of differentialdifference equations partially solved with respect to the first-order derivative by constructing it in a form of a series.

Our aim in this paper is to obtain sufficient conditions for the oscillation of every solution of first order difference equations xn+1 − xn + m ∑ i=1 (pixn−ki − qixn−li) = 0, n = 0, 1, 2, ... and xn+1 − xn + m ∑ i=1 (pixn+ki − qixn+li) = 0, n = 0, 1, 2, ... where pi, qi ∈ R and ki, li ∈ N for i = 1, 2, . . . ,m.

The asymptotic contracted measure of zeros of a large class of orthogonal polynomials is explicitly given in the form of a Lauricella function. The polynomials are defined by means of a three-term recurrence relation whose coefficients may be unbounded but vary regularly and have a different behaviour for even and odd indices. Subclasses of systems of orthogonal polynomials having their contrac...

*Correspondence: [email protected] 1Modern Business and Management Department, Guangdong Construction Polytechnic, Guangzhou, 510440, China Full list of author information is available at the end of the article Abstract In this paper, we investigate the solutions of boundary value problems for second-order p-Laplacian difference equations. By using the critical point theory, the existence and mul...

Starting from the D!-Riccati Diierence equation satissed by the Stieltjes function of a linear functional, we work out an algorithm which enables us to write the general fourth-order diierence equation satissed by the associated of any integer order of orthogonal polynomials of the-Laguerre-Hahn class. Moreover, in classical situations (Meixner, Charlier, Krawtchouk and Hahn), we give these dii...

We isolate conditions on the relative asymptotic size of sets of natural numbers A,B that guarantee a nonempty intersection of the corresponding sets of distances. Such conditions apply to a large class of zero density sets. We also show that a variant of Khintchine’s Recurrence Theorem holds for all infinite sets A = {a1 < a2 < . . .} where ann n3/2.

In this note we nd the exact solution for the minimal recurrence S n = min bn=2c k=1 faS n?k + bS k g; where a and b are positive integers and a b. We prove that the solution is the same as that of the recurrence relation S n = aS dn=2e + bS bn=2c. In other words, S n = S 1 +(a+b?1)S 1 P n?1 i=1 a z(i) b blg nc?z(i) , where z(i) is the number of zeros in the binary representation of i. The proo...