Related papers: The Binomial Coefficient for Negative Arguments
An identity for binomial symbols modulo an odd positive integer $n$ relating to the least prime factor of $n$ is proved. The identity is discussed within the context of Pell conics.
In the paper, we establish an inequality involving the gamma and digamma functions and use it to prove the negativity and monotonicity of a function involving the gamma and digamma functions.
We consider Tuenter polynomials as linear combinations of descending factorials and show that coefficients of these linear combinations are expressed via a Catalan triangle of numbers. We also describe a triangle of coefficients in terms of…
We introduce a generalization of Pascal triangle based on binomial coefficients of finite words. These coefficients count the number of times a word appears as a subsequence of another finite word. Similarly to the Sierpi\'nski gasket that…
The paper describes a new algorithm of construction of the nonlinear arithmetic triangle on the basis of numerical simulation and the binary system. It demonstrates that the numbers that fill the nonlinear arithmetic triangle may be…
We consider here a particular quadratic equation linking two elements of a C-Algebra. By analysing powers of the unknowns, it appears a double sequence of polynomials related to classical Bernoulli polynomials. We get the generating…
The binomial notation (w u) represents the number of occurrences of the word u as a (scattered) subword in w. We first introduce and study possible uses of a geometrical interpretation of (w ab) and (w ba) when a and b are distinct letters.…
In this article, we define a special function called the Bigamma function. It provides a generalization of Euler's gamma function. Several algebraic properties of this new function are studied. In particular, results linking this new…
We formulate several polynomial identities. One side of these identities has a nice simple form. Whereas the other has a form of a polynomial whose coefficients contain binomial coefficients double factorials or (and) rising factorials. The…
The binomial coefficients and Catalan triangle numbers appear as weight multiplicities of the finite-dimensional simple Lie algebras and affine Kac--Moody algebras. We prove that any binomial coefficient can be written as weighted sums…
In an earlier article [3], we presented an algorithm that can be used to rigorously check whether a specific cosine or sine polynomial is nonnegative in a given interval or not. The algorithm proves to be an indispensable tool in…
This paper is about counting the number of distinct (scattered) subwords occurring in a given word. More precisely, we consider the generalization of the Pascal triangle to binomial coefficients of words and the sequence $(S(n))_{n\ge 0}$…
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.
We pose the question of what is the best generalization of the factorial and the binomial coefficient. We give several examples, derive their combinatorial properties, and demonstrate their interrelationships. On cherche ici \`a…
We obtain formulas for the coefficients of positive and negative powers of a partial theta function.
The summatory function of the number of binomial coefficients not divisible by a prime is known to exhibit regular periodic oscillations, yet identifying the less regularly behaved minimum of the underlying periodic functions has been open…
A product difference equation is proved and used for derivation by elementary methods of four combinatorial identities, eight combinatorial identities involving generalized harmonic numbers and eight combinatorial identities involving…
We compute an analogue of Pascal's triangle enriched in bilinear forms over a finite field. This gives an arithmetically meaningful count of the ways to choose $j$ ring homomorphisms into an algebraic closure from an \'etale extension of…
For a prime p and nonnegative integers n,k, consider the set A_{n,k}^{(p)}={x is in [0,1,...,n]: p^k||binom {n} {x}}. Let the expansion of n+1 in base p be: n+1=alpha_{0} p^{\nu}+alpha_{1}p^{nu-1}+...+alpha_{nu}, where 0<=alpha_{i}<=…
In this paper, the formal derivative operator defined with respect to context-free grammars is used to prove some properties about binomial coefficients and multifactorial numbers. In addition, we extend the formal derivative operator to…