A sum up method for solving summations of the form $\sum_{k=n_0}^{n} A_{n,k}$ and rising and falling factorial transforms

Abstract

In this paper, we discuss a method that utilizes the recurrence of $A_{n,k}$ to solve summations of the form $\sum_{k=n_0}^{n} A_{n,k}$. It is observed that by repeating the procedure, the upper bound of summation is reduced and tilts toward the lower bound. This method of summation is mostly suitable for combinatorial sequences such as binomial coefficients, Stirling numbers of both kinds, etc. After the main method is displayed, some examples are illustrated. Some useful identities about Stirling and r-Stirling numbers are obtained. Finally, two transforms called rising and falling factorial transforms which turn the basis of power polynomials into factorial basis are derived. These transforms verify and simplify the results obtained in the examples section. Also, these transforms describe the relationship between fractional derivatives (or fractional integrals) and falling factorial (or rising factorial) by its series expansion.