By S. Elaydi, J. Cushing, R. Lasser, V. Papageorgiou, A. Ruffing
This quantity includes talks given at a joint assembly of 3 groups operating within the fields of distinction equations, detailed features and purposes (ISDE, OPSFA, and SIDE). The articles mirror the variety of the themes within the assembly yet have distinction equations as universal thread. Articles hide themes in distinction equations, discrete dynamical structures, specified capabilities, orthogonal polynomials, symmetries, and integrable distinction equations.
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Additional resources for Difference Equations, Special Functions and Orthogonal Polynomials: Proceedings of the International Conference: Munich, Germany 25-30, July 2005
Example text
Conclusions We have reviewed some recent works on the Pascal matrix. It has been shown t h a t it is strictly related to other important matrices in applications as well as to classical polynomials. It arises also as a solution of differential and partial difference equations. In all cases, its use considerably simplifies both the relations among them and, consequently, the algebraic manipulations involved. References 1. L. Aceto and D. Trigiante, On the A-stable methods in the GBDF class, Nonlinear Anal.
In the sequel the following set will play an important role o 7 = { u > 0 : f ( u , u )= u } . From now on, let f E C ( ( 0 ,oo) x (0, oo),(0, oo)) be a 3-cycle for (1). Now we are stating some properties of fiber maps associate to f . The first one is concerning the monotonicity of such fiber maps. Using (2) and (3) it is easy to prove that both cp, and t+bu are injective maps. 1. I f f generates a 3-cycle, then: (I) For any u,9, and $J, are strictly monotone maps. (2) If w,w' are different positive real numbers, cpw(x) # cpw'(x) f o r all x > 0.
Let a, p E C((0,co),(0, co)) and consider f ( z ,y) = a(z) . p ( y ) . Suppose that zn+2 = f(x,+l, z), is a 3-cycle. Then for some constant K > 0 we have Proof. Suppose that the fiber maps (pu,$ju verify (pu = $ju for all u > 0. ) . ( a for some constant C > 0, and for all u > 0. On the other hand, since (pu = G,, u > 0, by (7) we find . 4u) . 4 ( P u ( x ) ) c . a(u). a ( a ( u ) p ( z ) )= c . (a . u(Ca(u)a(z)), z, u > 0. * > 0, for instance x = 1. Then Eq. (a u > 0. 1 allows us to introduce the change of variable s = Ca(u), s E (0,oo).