Related papers: Multiple binomial sums
The binomial convolution of two sequences $\{a_n\}$ and $\{b_n\}$ is the sequence whose $n$th term is $\sum_{k=0}^{n} \binom{n}{k} a_k b_{n-k}$. If $\{a_n\}$ and $\{b_n\}$ have rational generating functions then so does their binomial…
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…
Symbolic summation as an active research topic of symbolic computation provides efficient algorithmic tools for evaluating and simplifying different types of sums arising from mathematics, computer science, physics and other areas. Most of…
The Fibonacci number is the residue of a rational function, from which follows that Fibonacci number summation identities can be derived with the integral representation method, a method also used to derive combinatorial identities. A…
A generating function for reciprocal binomial coefficients is written down, integral representations of this function are obtained, generating functions for sums of reciprocal binomial coefficients are derived, new identities are obtained,…
Multiple harmonic-like numbers are studied using the generating function approach. A closed form is stated for binomial sums involving these numbers and two additional parameters. Several corollaries and examples are presented which are…
Working with generating functions, the combinatorics of a recurrence relation can be expressed in a way that allows for more efficient calculation of the quantity. This is true of the Catalan numbers for an ordered binary tree…
We study generalized sums of linear orders. These are binary operations that, given linear orders $A$ and $B$, return an order $A \oplus B$ that can be decomposed as an isomorphic copy of $A$ interleaved with a copy of $B$. We show that…
By combining the telescoping method with an algebraic relation, four classes of binomial moments are examined. Several explicit summation formulae are established.
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 a recent article a generalization of the binomial distribution associated with a sequence of positive numbers was examined. The analysis of the nonnegativeness of the formal expressions was a key-point to allow to give them a statistical…
We consider binomial and inverse binomial sums at infinity and rewrite them in terms of a small set of constants, such as powers of $\pi$ or $\log(2)$. In order to perform these simplifications, we view the series as specializations of…
One can find lists of whole numbers having equal sum and product. We call such a creature a bioperational multiset. No one seems to have seriously studied them in areas outside whole numbers such as the rationals, Gaussian integers, or…
Binomial coefficients have been used for centuries in a variety of fields and have accumulated numerous definitions. In this paper, we introduce a new way of defining binomial coefficients as repeated sums of ones. A multitude of binomial…
Let A be a class of objects, equipped with an integer size such that for all n the number a(n) of objects of size n is finite. We are interested in the case where the generating fucntion sum_n a(n) t^n is rational, or more generally…
A summation is a shift-invariant ${\rm R}$-module homomorphism from a submodule of ${\rm R}[[\sigma]]$ to ${\rm R}$ or another ring. [11] formalized a method for extending a summation to a larger domain by telescoping. In this paper, we…
Combining the derivative operator with a binomial sum from the telescoping method, we establish a family of summation formulas involving generalized harmonic numbers.
This is a compendium of generating functions involving single, double sums and definite integrals. These generating functions also involve special functions in both the summand function and closed form solution.
We define integer multimodal sequences, which are generalizations of unimodal sequences having multiple local peaks of equal size. The generating functions for multimodal sequences represent novel types of $q$-series that combine generating…
In this paper some generalizations of the sum of powers of natural numbers is considered. In particular, the class of sums whose generating function is the power of the generating function for the classical sums of powers is studying. The…