الفهرس 
Introduction and PreliminariesOnly 14 pages are availabe for public view
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Abstract
Quantum calculus, sometimes called calculus without limits, is equivalent to traditional infinitesimal calculus without the notion of limits. It defines ”qcalculus”; q stands for quantum. In q-calculus we are looking for q-analogues of mathematical objects that have the original object as limits when q tends to 1. There are two types of qaddition, the Nalli-Ward-Al-Salam q-addition (NWA) and the Jackson-Hahn-Cigler q-addition (JHC). The first one is commutative and associative, while the second one is neither. This is one of the reasons why sometimes more than one q-analogue exists. The two types above form the basis of the method which unities hypergeometric series and q-hypergeometric series and gives many formulas of q-calculus. Since the eighties, an intensive and interest in the subject appeared in many areas of mathematics and applications including new difference calculus and orthogonal polynomials, q-combinatorics, q-arithmetics, q-integrable systems and variational q-calculus. The q-analogue of a mathematical concept is a polynomial expression in a realvalued variable which reduces to a simple, classical object in the limit 1. The most basic of these is the q-integer, which takes the form of the partial sum of a geometric series in and produces a normal integer in the limit case. For n belong to N,We can easily see that if1, this sum is simply the sum of ones, and is therefore definitively equal to The q-difference equations has many applications in different fields, such as statistical physics , fractal geometry, number theory, quantum mechanics, orthogonal polynomials and other fields of sciences including quantum theory, mechanics and theory of relativity.