Related papers: Finite sums associated with some polynomial identi…
The main result of this paper is to show that all binomial identities are orderable. This is a natural statement in the combinatorial theory of finite sets, which can also be applied in distributed computing to derive new strong bounds on…
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 present various constructions of sequences of polynomials satisfying the Binomial Theorem in finite characteristic based on the theory of additive polynomials. Various actions on these constructions are also presented. It is an open…
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…
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…
The harmonic numbers and generalized harmonic numbers appear frequently in many diverse areas such as combinatorial problems, many expressions involving special functions in analytic number theory and analysis of algorithms. The aim of this…
We give some theoretical and computational results on "random" harmonic sums with prime numbers, and more generally, for integers with a fixed number of prime factors.
Combining the derivative operator with a binomial sum from the telescoping method, we establish a family of summation formulas involving generalized harmonic numbers.
In a recent paper, Yu. A. Brychkov derived a series of identities for multiples sums of special functions, using generating functions. Among these identities, a particularly interesting one involves multiples sums of Bessel $I_{\nu}$…
The notion of block divisibility naturally leads one to introduce unitary cyclotomic polynomials. We formulate some basic properties of unitary cyclotomic polynomials and study how they are connected with cyclotomic, inclusion-exclusion and…
In a recent article, Apagodu and Zeilberger (http://arxiv.org/abs/1606.03351)discuss some applications of an algorithm for finding and proving congruence identities (modulo primes) of indefinite sums of many combinatorial sequence. At the…
Combinatorial $t$-designs have been an interesting topic in combinatorics for decades. It was recently reported that the image sets of a fixed size of certain special polynomials may constitute a $t$-design. Till now only a small amount of…
Our paper deals about identities involving Bell polynomials. Some identities on Bell polynomials derived using generating function and successive derivatives of binomial type sequences. We give some relations between Bell polynomials and…
We investigate some classes of infinite series involving central binomial coefficients, particularly focusing on those arising from ratios such as $\binom{2n}{n}\binom{4n}{2n}^{-1}$,$\binom{4n}{2n}\binom{2n}{n}^{-1}$ and related…
In this paper, we derive eight basic identities of symmetry in three variables related to generalized Bernoulli polynomials and generalized power sums. All of these are new, since there have been results only about identities of symmetry in…
We propose investigating a summation analog of the paradigm for parallel integration. We make some first steps towards an indefinite summation method applicable to summands that rationally depend on the summation index and a P-recursive…
We give several families of polynomials which are related by Eulerian summation operators. They satisfy interesting combinatorial properties like being integer-valued at integral points. This involves nearby-symmetries and a recursion for…
Let $(G_n(x))_{n=0}^\infty$ be a $d$-th order linear recurrence sequence having polynomial characteristic roots, one of which has degree strictly greater than the others. Moreover, let $m\geq 2$ be a given integer. We ask for…
We give a simplified presentation of some results about recurrences of certain sequences of binomial sums in terms of (generalized) Fibonacci and Lucas polynomials.
We study the combinatorial properties of final types, which are certain non-decreasing sequences of integers, together with the partitions naturally associated with them. As a consequence, we obtain an identity expressing the $n$-nacci…