Related papers: Periodic Weighted Sums of Binomial Coefficients

Balancing numbers possess, as Fibonacci numbers, a Binet formula. Using this, partial sums of arbitrary powers of balancing numbers can be summed explicitly. For this, as a first step, a power $B_n^l$ is expressed as a linear combination of…

We find and prove relationships between Riemann zeta values and central binomial sums. We also investigate alternating binomial sums (also called Ap\'ery sums). The study of non-alternating sums leads to an investigation of different types…

These lecture notes provide an informal introduction to the theory of nonnegative polynomials and sums of squares. We highlight the history and some recent developments, especially the new connections with classical (complex) algebraic…

In this work, we introduce a symmetric algorithm obtained by the recurrence relation a_{n}^{k}=a_{n-1}^{k}+a_{n}^{k-1}. We point out that this algorithm can be apply to hyperharmonic-, ordinary and incomplete Fibonacci- and Lucas numbers.…

In this article we study a generalization of Fibonomials, replacing the Fibonacci sequences by bivariate s-Fibonacci polynomial sequences. We call the obtained objects "Bivariate s-Fibopolynomials".

We study the distribution of the extended binomial coefficients by deriving a complete asymptotic expansion with uniform error terms. We obtain the expansion from a local central limit theorem and we state all coefficients explicitly as…

In this note we present examples of cumulative connection constants included new fibonomial ones. All examples posses combinatorial interpretation.

In this paper, we consider several special polynomials related to associated sequences of polynomials. Finally, we give some new and interesting identities of those polynomials arising from transfer formula for the associated sequences.

A recent paper of A. Sofo proves some results about sums of products of quadratic alternating harmonic numbers and reciprocal binomial coefficients. In this paper, we extend his result to cubic alternating harmonic number sums and develop…

We describe an effective method for calculating certain infinite sums, generalizations of the classical Bernoulli polynomials. As shown by Edward Witten in his papers on two-dimensional gauge theories, the correlation functions of…

This is a survey of old and new problems and results in additive number theory.

We give explicit upper bounds for coefficients of polynomials appearing in Gauss-Kra\"{i}tchik formula for cyclotomic polynomials. We use a certain relation between elementary symmetric polynomials and power sums polynomials.

The alternating and non-alternating harmonic sums and other algebraic objects of the same equivalence class are connected by algebraic relations which are induced by the product of these quantities and which depend on their index class…

Ismail et al. (Constr. Approx. {\bf 15} (1999) 69--81) proved the positivity of some trigonometric polynomials with single binomial coefficients. In this paper, we prove some similar results by replacing the binomial coefficients with…

Using the WZ-method we find some of the easiest Ramanujan's formulae and also some new interesting Ramanujan-like sums.

Power series in which the summand satisfies a linear recurrence relation with polynomial coefficients are shown to be the solution of a linear differential or algebraic equation. Solving the associated differential or algebraic equation…

We use elementary methods to establish three key recurrence relations: one for derangement numbers, a second for harmonic numbers, and a third for degenerate harmonic numbers. Our results not only contribute to the understanding of the…

Recent developments in the theory and application of the Hardy-Littlewood method are discussed, concentrating on aspects associated with diagonal diophantine problems. Recent efficient differencing methods for estimating mean values of…

In this note we show a simple formula for the coefficients of the polynomial associated with the sums of powers of the terms of an arbitrary arithmetic progression. This formula consists of a double sum involving only ordinary binomial…

We propose higher-order generalizations of Jacobsthal's $p$-adic approximation for binomial coefficients. Our results imply explicit formulae for linear combinations of binomial coefficients $\binom{ip}{p}$ ($i=1,2,\dots$) that are…