Related papers: Combinatorial sums derived from properties of Lege…
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…
Finite trigonometric sums occur in various branches of physics, mathematics, and their applications. These sums may contain various powers of one or more trigonometric functions. Sums with one trigonometric function are known, however sums…
In this note we augment the poly-Bernoulli family with two new combinatorial objects. We derive formulas for the relatives of the poly-Bernoulli numbers using the appropriate variations of combinatorial interpretations. Our goal is to show…
Classical binomial identities are established by giving probabilistic interpretations to the summands. The examples include Vandermonde identity and some generalizations.
This paper introduces a symbolic calculus-based approach for deriving closed-form expressions for the sums of arithmetic sequences. The method extends beyond constant-difference sequences to those with polynomially increasing steps,…
Using a probabilistic approach, we derive some interesting combinatorial identities involving gamma and beta functions. These results generalize certain well-known combinatorial identities involving binomial coefficients and special…
We present a relationship between the generalized hyperharmonic numbers and the poly-Bernoulli polynomials, motivated from the connections between harmonic and Bernoulli numbers. This relationship yields numerous identities for the…
Spivey presented a new approach to evaluate combinatorial sums by using finite differences. We present some closed forms for sums involving the binomial coefficients, Fibonacci and Lucas numbers in terms of the falling factorial.
When two boundary-parabolic representations of knot groups are given, we introduce the connected sum of these representations and show several natural properties including the unique factorization property. Furthermore, the complex volume…
We survey some results that provide different versions of classical results through different summability methods. Specifically, in order to adapt such classical results, we analyze which properties should satisfy the summability methods.…
We present a definition for the sum of a sequence of combinatorial games. This sum coincides with the classical sum in the case of a converging sequence of real numbers and with the infinitary natural sum in the case of a sequence of…
In this paper, we study non-linear differential equations associated with Legendre polynomials and their applications. From our study of non- linear differential equations, we derive some new and explicit identities for Legendre…
We reduce the calculation of the simplest Hodge integrals to some sums over decorated trees. Since Hodge integrals are already calculated, this gives a proof of a rather interesting combinatorial theorem and a new representation of…
We give an expression of polynomials for higher sums of powers of integers via the higher order Bernoulli numbers.
We prove a closed formula for the derivative, of any order, of a implicit function, in terms of some binomial building blocks, and explain the combinatorics behind the coefficients appearing in the formula.
The present note considers a certain family of sums indexed by the set of fixed length compositions of a given number. The sums in question cannot be realized as weighted compositions. However they can be be related to the hypergeometric…
Let $p$ be an odd prime. In the paper, by using the properties of Legendre polynomials we prove some congruences for $\sum_{k=0}^{\frac{p-1}2}\binom{2k}k^2m^{-k}\mod {p^2}$. In particular, we confirm several conjectures of Z.W. Sun. We also…
In this article, we introduce combinatorial models for poly-Bernoulli polynomials and poly-Euler numbers of both kinds. As their applications, we provide combinatorial proofs of some identities involving poly-Bernoulli polynomials.
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…
Starting from a small number of well-motivated axioms, we derive a unique definition of sums with a noninteger number of addends. These "fractional sums" have properties that generalize well-known classical sum identities in a natural way.…