Abstract
Abstract In this paper, we introduce generalizations of rising factorials and falling factorials, respectively, and study their relations with the well-known Stirling numbers, Lah numbers, and so on. The first stage is to define poly-falling factorial sequences in terms of the polyexponential functions, reducing them to falling factorials if k = 1 k=1 , necessitating a demonstration of the relations: between poly-falling factorial sequences and the Stirling numbers of the first and second kind, respectively; between poly-falling factorial sequences and the poly-Bell polynomials; between poly-falling factorial sequences and the poly-Bernoulli numbers; between poly-falling factorial sequences and poly-Genocchi numbers; and recurrence formula of these sequences. The later part of the paper deals with poly-rising factorial sequences in terms of the polyexponential functions, reducing them to rising factorial if k = 1 k=1 . We study some relations: between poly-falling factorial sequences and poly-rising factorial sequences; between poly-rising factorial sequences and the Stirling numbers of the first kind and the power of x x ; and between poly-rising factorial sequences and Lah numbers and the poly-falling factorial sequences. We also derive recurrence formula of these sequences and reciprocal formula of the poly-falling factorial sequences.
Highlights
We demonstrate the relations: between poly-falling factorial sequences and the Stirling numbers of the first and second kind, respectively; between poly-falling factorial sequences and the poly-Bell polynomials; between poly-falling factorial sequences and the poly-Bernoulli numbers; between poly-falling factorial sequences and poly-Genocchi numbers; and recurrence formula of these sequences
We study recurrence formula of these sequences and a reciprocal formula of the poly-falling factorial sequences
We demonstrated the following: the nth poly-falling factorial sequences of x were expressed in terms of the Stirling numbers of the first kind and the power of x; the nth power of x were expressed in terms of the Stirling numbers of the second kind and polyfalling factorial sequences of x; the poly-Bell polynomials were represented in terms of the poly-falling factorial sequences; the poly-falling poly-falling factorial sequences were represented in terms of the polyBell polynomials; recurrence formula; and the poly-falling factorial sequences were expressed in terms of the poly-Bernoulli numbers and poly-Genocchi numbers, respectively
Summary
In the study of falling factorials and rising factorials, the following properties produce important characteristics for special numbers such as Stirling numbers and Lah numbers: the nth falling factorial of x is expressed in terms of the Stirling numbers of the first kind and the power of x; the nth power of x is expressed in terms of the Stirling numbers of the second kind S2(n, l); the falling factorials of x; the nth rising factorial of x is expressed in terms of Lah numbers and falling factorials; and the nth falling factorial of x can be expressed in terms of Lah numbers and rising factorials [1,2,3,4,5,6,7,8,9,10]. We intend to study the above-mentioned properties by generalizing falling factorials and rising factorials, respectively, using the polyexponential functions. We demonstrate the relations: between poly-falling factorial sequences and the Stirling numbers of the first and second kind, respectively; between poly-falling factorial sequences and the poly-Bell polynomials; between poly-falling factorial sequences and the poly-Bernoulli numbers; between poly-falling factorial sequences and poly-Genocchi numbers; and recurrence formula of these sequences. As the multivariate version of the Stirling numbers S(n, k) of the second kind, the generation function of the incomplete Bell polynomials Bn,k(x1, x2, ...,xn−k+1) is given by.
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