In this paper, we perform a further investigation for finite trigonometric sums. We establish some connections between the higher-order trigonometric functions and Zeta functions. As applications, ...various known finite trigonometric sums are explicitly expressed as linear combinations of the Bernoulli and Euler polynomials and numbers.
A weighted extension of Fibonacci numbers Bhatnagar, Gaurav; Kumari, Archna; Schlosser, Michael J.
Journal of difference equations and applications,
07/03/2023, Volume:
29, Issue:
7
Journal Article
Peer reviewed
Open access
We extend Fibonacci numbers with arbitrary weights and generalize a dozen Fibonacci identities. As a special case, we propose an elliptic extension which extends the q-Fibonacci polynomials appearing ...in Schur's work. The proofs of most of the identities are combinatorial, extending the proofs given by Benjamin and Quinn, and in the q-case, by Garrett. Some identities are proved by telescoping.
We study set partitions with r distinguished elements and block sizes found in an arbitrary index set S. The enumeration of these (S, r)-partitions leads to the introduction of (S, r)-Stirling ...numbers, an extremely wide-ranging generalization of the classical Stirling numbers and the r-Stirling numbers. We also introduce the associated (S, r)-Bell and (S, r)-factorial numbers. We study fundamental aspects of these numbers, including recurrence relations and determinantal expressions. For S with some extra structure, we show that the inverse of the (S, r)-Stirling matrix encodes the Möbius functions of two families of posets. Through several examples, we demonstrate that for some S the matrices and their inverses involve the enumeration sequences of several combinatorial objects. Further, we highlight how the (S, r)-Stirling numbers naturally arise in the enumeration of cliques and acyclic orientations of special graphs, underlining their ubiquity and importance. Finally, we introduce related (S, r) generalizations of the poly-Bernoulli and poly-Cauchy numbers, uniting many past works on generalized combinatorial sequences.
By using the theory of Riordan arrays, we establish four pairs of general r-Stirling number identities, which reduce to various identities on harmonic numbers, hyperharmonic numbers, the Stirling ...numbers of the first and second kind, the r-Stirling numbers of the first and second kind, and the r-Lah numbers. We further discuss briefly the connections between the r-Stirling numbers and the Cauchy numbers, the generalized hyperharmonic numbers, and the poly-Bernoulli polynomials. Many known identities are shown to be special cases of our results, and the combinatorial interpretations of several particular identities are also presented as supplements.
The goal of the paper is twofold. First, we present an analytic method leading to a class of combinatorial identities with Bernoulli, Euler and Catalan numbers based on considering specific multiple ...zeta-like series and infinite products. The developed method allows us to naturally extend Hoffman’s combinatorial identity that led to the famous evaluation of the multiple zeta value
ζ
(
{
2
}
k
)
in 1992. Second, we present new evaluations of two multiple zeta-like series with their consequences to combinatorial identities, and, as a by-product of our technical considerations, we establish two combinatorial identities with a trinomial coefficient and Stirling numbers respectively.
Using the software package Sigma developed by Schneider, we automatically discover and prove some combinatorial identities involving harmonic numbers, from which we deduce some supercongruences on ...partial sums of hypergeometric series. These results confirm some conjectural generalizations of van Hamme's supercongruences in some special cases, which were recently proposed by Guo (2017).
The trace method for cotangent sums Ejsmont, Wiktor; Lehner, Franz
Journal of combinatorial theory. Series A,
January 2021, 2021-01-00, Volume:
177
Journal Article
Peer reviewed
Open access
This paper presents a combinatorial study of sums of integer powers of the cotangent. Our main tool is the realization of the cotangent values as eigenvalues of a simple self-adjoint matrix with ...complex integer entries. We use the trace method to draw conclusions about integer values of the sums and series expansions of the generating function to provide explicit evaluations; it is remarkable that throughout the calculations the combinatorics are governed by the higher tangent and arctangent numbers exclusively. Finally, we indicate a new approximation of the values of the Riemann zeta function at even integer arguments.
NEW FAMILY OF JACOBI-STIRLING NUMBERS Cakić, Nenad P.; El-Desouky, Beih. S.; Gomaa, Rabab. S.
Applicable analysis and discrete mathematics,
10/2023, Volume:
17, Issue:
2
Journal Article
Peer reviewed
Open access
The Jacobi-Stirling numbers of the first and second kind were introduced in 2007 by Everitt et al. In this article we find new explicit formulas for Jacobi Stirling numbers. Furthermore, we derive ...and study new class of the Jacobi Stirling numbers so-called generalized Jacobi-Stirling numbers. Some special cases such as Legendre-Stirling numbers are given. Some interesting combinatorial identities are obtained.
We give an elementary proof of an interesting combinatorial identity which is of particular interest in graph theory and its applications. Two applications to enumeration of forests with closed-form ...expressions are given.
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