Related papers: Recurrence relations for harmonic and derangement …
Many authors have recently studied the degenerate harmonic numbers. This paper makes two main contributions. First, we derive several explicit expressions for these numbers, which are a degenerate version of the ordinary harmonic numbers.…
The aim of this paper is to investigate some properties, recurrence relations and identities involving degenerate hyperharmonic numbers, hyperharmonic numbers and degenerate harmonic numbers. In particular, we derive an explicit expression…
Harmonic numbers have been studied since antiquity, while hyperharmonic numbers were intoduced by Conway and Guy in 1996. The degenerate harmonic numbers and degenerate hyperharmonic numbers are their respective degenerate versions. The aim…
We evaluate binomial series with harmonic number coefficients, providing recursion relations, integral representations, and several examples. The results are of interest to analytic number theory, the analysis of algorithms, and…
We define recursive harmonic numbers as a generalization of harmonic numbers. The table of recursive harmonic numbers, which is like Pascal's triangle, is constructed. A formula for recursive harmonic numbers containing binomial…
Recently, the degenerate Stirling numbers of the first kind were introduced. In this paper, we give some formulas for the degenerate Stirling numbers of the first kind in the terms of the complete Bell polynomials with higher-order harmonic…
Recently, Kim-Kim investigated the degenerate harmonic numbers and the degenerate hyperharmonic numbers as degenerate versions of the harmonic numbers and the hyperharmonic numbers, respectively. The aim of this paper is to study the…
Multiple harmonic-like numbers are studied using the generating function approach. A closed form is stated for binomial sums involving these numbers and two additional parameters. Several corollaries and examples are presented which are…
We obtain new recurrence relations, an explicit formula, and convolution identities for higher order geometric polynomials. These relations generalize known results for geometric polynomials, and lead to congruences for higher order…
The classical derangement numbers count fixed point-free permutations. In this paper we study the enumeration problem of generalized derangements, when some of the elements are restricted to be in distinct cycles in the cycle decomposition.…
In combinatorics, a derangement is a permutation of the elements of a set, such that no element appears in its original position. The number of derangement of an n-element set is called the nth derangement number. Recently, the degenerate…
We apply the theory of disconjugate linear recurrence relations to the study of irrational quantities in number theory. In particular, for an irrational number associated with solutions of three-term linear recurrence relations we show that…
We study three classes of combinatorial sums involving central binomial coefficients and harmonic numbers, odd harmonic numbers, and even indexed harmonic numbers, respectively. In each case we use summation by parts to derive recursive…
Recently, the degenerate harmonic and the degenerate hyperharmonic numbers are introduced respectively as degenerate versions of the harmonic and the hyperharmonic numbers. The aim of this paper is to introduce the degenerate…
In the paper, the author derives several "diagonal" recurrence relations, constructs some inequalities, finds monotonicity, and poses a conjecture related to Stirling numbers of the second kind.
A product difference equation is proved and used for derivation by elementary methods of four combinatorial identities, eight combinatorial identities involving generalized harmonic numbers and eight combinatorial identities involving…
The aim of this paper is to investigate some properties, recurrence relations and identities involving degenerate Stirling numbers of both kinds associated with degenerate hyperharmonic numbers and also with degenerate Bernolli, degenerate…
In the paper, the author presents diagonal recurrence relations for the Stirling numbers of the first kind. As by-products, the author also recovers three explicit formulas for special values of the Bell polynomials of the second kind.
Given any two sequences of complex numbers, we establish simple relations between their binomial convolution and the binomial convolution of their individual binomial transforms. We employ these relations to derive new identities involving…
We introduce new refinements of the Bell, factorial, and unsigned Stirling numbers of the first and second kind that unite the derangement, involution, associated factorial, associated Bell, incomplete Stirling, restricted factorial,…