turkmath.org

Although the calculus and the theory of difference/differential equations have been deeply analyzed since the discovery of time scales, the study on a general time scale may have deficiencies and inapplicabilities even in some elementary subjects such as polynomials, exponential functions, Taylor series. To overcome these deficiencies, in this talk, we present two approaches which unify and extend discrete time scales. First approach is based on the study of a special time scale, namely $(q,h)$-time scale. We briefly introduce the calculus on delta and nabla $(q,h)$-time scales. As an application, we focus on $(q,h)$-analogue of Bessel equation and Bessel function which reduce $h$-, $q$- and ordinary Bessel equations and functions under proper limits. Furthermore, we present a second approach which is based on the construction of a new time scale, namely $\alpha$-time scale. For this purpose, we offer a weighted jump operator $\alpha$, which generates the $\alpha$-time scale, $\alpha$-derivative and $\alpha$-polynomials.