Related papers: Identities for the shifted harmonic numbers and bi…
This paper presents new formulae for the harmonic numbers of order $k$, $H_{k}(n)$, and for the partial sums of two Fourier series associated with them, denoted here by $C^m_{k}(n)$ and $S^m_{k}(n)$. I believe this new formula for…
For any $m,n\in\mathbb{N}$ we first give new proofs for the following well known combinatorial identities \begin{equation*} S_n(m)=\sum\limits_{k=1}^n\binom{n}{k}\frac{(-1)^{k-1}}{k^m}=\sum\limits_{n\geq r_1\geq r_2\geq...\geq r_m\geq…
In this sequel to arXiv:0905.3327, we continue to study the congruence properties of the alternating version of multiple harmonic sums. As contrast to the study of multiple harmonic sums where Bernoulli numbers and Bernoulli polynomials…
We present some new linear, quadratic, cubic and quartic binomial Fibonacci, Lucas and Fibonacci--Lucas summation identities.
A new deformed canonical commutation relation, generalizing various known deformations, is defined together with its structure function of deformation. Then, the related irreducible representations are characterized and classified. Finally,…
We define a class of rational numbers including, as a particular case, the classical harmonic numbers. For one particular instance we apply it to the expansion into powers series of a special function, and also detail its relashionship with…
Nested sums containing binomial coefficients occur in the computation of massive operator matrix elements. Their associated iterated integrals lead to alphabets including radicals, for which we determined a suitable basis. We discuss…
In this paper, we establish some expressions of Mneimneh-type binomial sums involving multiple harmonic-type sums in terms of finite sums of Stirling numbers, Bell numbers and some related variables. In particular, we present some new…
We study sums of the form $\sum_{k=m}^n a_{nk} b_{km}$, where $a_{nk}$ and $b_{km}$ are binomial coefficients or unsigned Stirling numbers. In a few cases they can be written in closed form. Failing that, the sums still share many common…
In this paper, we continue our investigation of double sums where the inner sum is binomial but incomplete. We prove many new results for these types of double sums associated with binomial transform pairs. As applications we deduce new…
We present a different proof of the following identity due to Munarini, which generalizes a curious binomial identity of Simons. \begin{align*} \sum_{k=0}^{n}\binom{\alpha}{n-k}\binom{\beta+k}{k}x^k…
We develop a systematic and fully explicit approach to the evaluation of binomial sums involving reciprocals of binomial coefficients based on Beta integral techniques. Starting from a simple integral representation, we provide a derivation…
This paper gives new explicit formulas for sums of powers of integers and their reciprocals.
In the present article a new method of deriving integral representations of combinations and partitions in terms of harmonic products has been established. This method may be relevant to statistical mechanics and to number theory.
Given two infinite sequences with known binomial transforms, we compute the binomial transform of the product sequence. Various identities are obtained and numerous examples are given involving sequences of special numbers: Harmonic…
In this paper, we explore a variety of series involving the central binomial coefficients, highlighting their structural properties and connections to other mathematical objects. Specifically, we derive new closed-form representations and…
In this paper we investigate congruence relationships of particular finite generalized harmonic numbers sums. We suggest more transparent and simpler method to analyse these sums and present several additional results for certain special…
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…
We present several types of ordinary generating functions involving central binomial coefficients, harmonic numbers, and odd harmonic numbers. Our results complement those of Boyadzhiev from 2012 and Chen from 2016. Based on these…
We introduce the associated Lah numbers. Some recurrence relations and convolution identities are established. An extension of the associated Stirling and Lah numbers to the r-Stirling and r-Lah numbers are also given. For all these…