Related papers: On the binomial sums of Horadam sequence

In this paper, we propose new generalizations of amicable numbers. We also give examples and prove properties of these new concepts.

We define a special function related to the digamma function and use it to evaluate in closed form various series involving binomial coefficients and harmonic numbers.

In this paper we give a short introduction to our results on the holonomy of gerbe-connections and explain our motivation coming from state-sum models.

In this article a new method of generating sums of like powers is presented.

The aim of this paper is to construct general forms of ordinary generating functions for special numbers and polynomials involving Fibonacci type numbers and polynomials, Lucas numbers and polynomials, Chebyshev polynomials, Sextet…

We study properties of functions of binomial coefficients mod 2 and derive a set of recurrence relations for sums of products of binomial coefficients mod 2. We show that they result in sequences that are the run length transforms of well…

We give an expression of polynomials for higher sums of powers of integers via the higher order Bernoulli numbers.

In this article, we give a formula for the generalization of the binomial coefficient to the complex numbers as a linear combination of $\sinc$ functions. We then give a general formula to compute the integral on the real line of the…

In this note, we explore two families of sequences associated to a suitable integer sequence: the gap-sum sequence and the gap-product sequence. These are the sums and the products of consecutive numbers not in the original sequence. We…

We improve a result of Bennett concerning certain sequences involving sums of powers of positive integers.

We perform certain alternating binomial summations with parameters that occur in the analysis of algorithms. A combination of integral and special function and special number representations is used. The results are sufficiently general to…

In a recent paper (Appl. Math. Comput. 215, 1622--1645, 2009), the authors proposed a method of summation of some slowly convergent series. The purpose of this note is to give more theoretical analysis for this transformation, including the…

This note highlights an interesting connection between Euler sums of even weight and prime numbers.

In this article, we derive a congruence property of particular sum rules involving prime numbers. The resulting expression involves Bernoulli numbers and polynomials, for which we obtain, as a consequence, a general congruence relation as…

We present several sequences involving harmonic numbers and the central binomial coefficients. The calculational technique is consists of a special summation method that allows, based on proper two-valued integer functions, to calculate…

The main purpose of this note is to establish the continuity of seminorms on finite-dimensional vector spaces over the real or complex numbers.

In [16], we obtained some congruences for Lucas quotients of two infinite families of Lucas sequences by studying the combinatorial sum $$\sum_{k\equiv r(\mbox{mod}m)}{n\choose k}a^k.$$ In this paper, we show that the sum can be expressed…

We derive weighted sums, including binomial and double binomial sums, for the generalized Fibonacci sequence $\{G_m\}$ where for $m\ge 2$, $G_m=G_{m-1}+G_{m-2}$ with initial values $G_0$ and $G_1$.

We develop new closed form representations of sums of (n + {\alpha})th shifted harmonic numbers and reciprocal binomial coefficients in terms of {\alpha}th shifted harmonic numbers. Some interesting new consequences and illustrative…

In two recent articles we have examined a generalization of the binomial distribution associated with a sequence of positive numbers, involving asymmetric expressions of probabilities that break the symmetry {\it win-loss}. We present in…