Extended Delta and Nabla Operators and its Sums

This paper aims to the generaized α(q)- difference operator and its sums. By equating the closed and summation forms of inverse operators, we derive novel summation formulae. These advancements contribute to a deeper understanding of the calculus framework and its applications across various mathematical and scientific fields. This results can be applied to improve dicrete dynamic systems, enchance computational techniques and provide new insights into the behaviour of discrete functions.