(p,q)-Hahn Difference Operator

Abstract

ABSTRACT: One of the main starting point for the theory of calculus is the differentiation operation, which is defined as follows. Firstly, divide the difference of two function values by the difference of the corresponding two arguments, and then take the limit as the two arguments converge to each other. The result of this limit is called the derivative of the original function. Many variants of this basic operation have been proposed, giving rise to different theories and types of calculus. In this thesis, I will study some particular variants in which the limiting process is omitted but the two arguments in the quotient expression are linear functions of each other. The most basic one is the q-calculus (or quantum calculus), which is a particular case of both the (q,ω)-calculus (or Hahn calculus) and the (p,q)-calculus, which are the both special cases of the new type called (p,q)-Hahn calculus. These approaches give more discrete theories than the original calculus, more applicable to quantum physics. But a lot of the structure remains the same: in all cases there are derivatives, integrals, product and chain rules, exponential and Appell functions. In this thesis, I will study important properties and special functions associated with each of these three known types of calculus, and finally, I introduce the new (p,q)-Hahn Calculus. Keywords: q-calculus or quantum calculus, q,ω-calculus or Hahn Calculus, (p,q)-calculus, (p,q)-Hahn Calculus, Exponential Functions, Appell Polynomials.