Abstract
In this paper, we develop theorems on finite and infinite summation formulas by utilizing the q and (q,h) anti-difference operators, and also we extend these core theorems to q(α) and (q,h)α difference operators. Several integer order theorems based on q and q(α) difference operator have been published, which gave us the idea to derive the fractional order anti-difference equations for q and q(α) difference operators. In order to develop the fractional order anti-difference equations for q and q(α) difference operators, we construct a function known as the quantum geometric and alpha-quantum geometric function, which behaves as the class of geometric series. We can use this function to convert an infinite summation to a limited summation. Using this concept and by the gamma function, we derive the fractional order anti-difference equations for q and q(α) difference operators for polynomials, polynomial factorials, and logarithmic functions that provide solutions for symmetric difference operator. We provide appropriate examples to support our results. In addition, we extend these concepts to the (q,h) and (q,h)α difference operators, and we derive several integer and fractional order theorems that give solutions for the mixed symmetric difference operator. Finally, we plot the diagrams to analyze the (q,h) and (q,h)α difference operators for verification.