Studies on Generalization of Special Equations at the q-Integers

Abstract

q-integers are knots which divide intervals between consecutive knots, in the form of a geometric sequence. The property of q-integers is, the difference between knots progress decreasingly for 0 is less than q is less than 1 and progress increasingly for q is less than 1. In recent studies about the usage of q-integers the most impressing point is, to be used in the area of numerical analysis and in the approximation theory. Because by the help of numeric integration and the various well qualified polynomials, it gives the best approximation. In this study the polynomials such Splines, B-Splines, Bernstein, Newton divided differences and forward differences interpolation are generalized on q-integers. Moreover; the studies are concerned with application of numerical integration rules, difference identities, Cauchy identity, Binomial identity, Factorial equation, Leibniz Formula and expression of the difference operators have been represented in chronological order between 1994 and 2010.