Discrete version of fundamental theorems of fractional order integration for nabla operator

The goal of this paper is to develop and present a precise theory for integer and fractional order & ell;-nabla integration and its fundamental theorems. In our research work, we take two forms of higher order difference equation such as closed form and summation form. But most of the authors are focused only on the summation part only. Instead of finding the solution for the summation part, finding the solution for the closed gives the exact solution. To find the closed form solution for the integer order using the & ell;-nabla operator, we used the factorial-coefficient method. For developing the theory of fractional order & ell;-nabla operator and its integration, we introduce a function called N-nu-type function. If the summation series is huge, this approach can help us to find the solution quickly. Suitable examples are provided for verification. Finally, we provide the application for detecting viral transmission using the nabla operator.