Parham Zarghami | Searcharxiv

arXiv2023

On the properties of falling and rising factorial transforms

In this paper, we will extend the falling and rising factorial transforms \cite{ref. 1} which in this case every arbitrary function can be applied. Then, the properties of these transforms will be investigated and some corollaries will be shown. These transforms have interesting properties that led to new series expansions, representation of functions in terms of operators, connection between different series and operators, etc. By converting the power series into the Newton series, useful identities can be derived.

math.CA↗

arXiv2023

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

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.

math.NT↗