Gamma function, Beta function and combinatorial identities

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 function to the standard Beta function have been provided. We have also established inequalities, which allow to approximate this new function.

In this paper we introduce the Two Parameter Gamma Function, Beta Function and Pochhammer Symbol. We named them, as p - k Gamma Function, p - k Beta Function and p - k Pochhammer Symbol and denoted as $_{p}\Gamma_{k}(x), $ $_{p}B_{k}(x,y) $ and $_{p}(x)_{n,k} $ respectively. We prove the several identities for $_{p}\Gamma_{k}(x), $ $_{p}B_{k}(x,y) $ and $_{p}(x)_{n,k} $ those satisfied by the classical Gamma, Beta and Pochhammer Symbol. Also we provide the integral representa...

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 d\'eterminer est la meilleure g\'en\'eralisation possible des factorielles et des coefficients du bin\^oome. On s'interesse \`a plusieurs exemples, \`a leurs propri\'et\'es combinatoires, et aux differentes relations qu'ils mettent en jeu.

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 classical harmonic numbers. For the binomial coefficients the definition with gamma functions is used, thus also allowing non-integer arguments in the identities. The generalized harmonic numbers in this case are harmonic numbers with a complex ...

In this short note we present a set of interesting and useful properties of a one-parameter family of sequences including factorial and subfactorial, and their relations to the Gamma function and the incomplete Gamma function.

In this paper we formulate combinatorial identities that give representation of positive integers as linear combination of even powers of 2 with binomial coefficients. We present side by side combinatorial as well as computer generated proofs using the Wilf-Zeilberger(WZ) method.

The problem of counting derangements was initiated by Pierre Remonde de Motmort in 1708. A derangement is a permutation that has no fixed points and the derangement number Dn is the number of fixed point free permutations on an n element set. Furthermore, the derangement polynomials are natural extensions of the derangement numbers. In this paper, we study the derangement polynomials and numbers, their connections with cosine-derangement polynomials and sine-derangement polyn...

We give two new proofs of the Chaundy-Bullard formula $$ (1-x)^{n+1} \sum_{k=0}^m {n+k\choose k} x^k +x^{m+1}\sum_{k=0}^n {m+k\choose k} (1-x)^k=1 $$ and we prove the "twin formula" $$ \frac{ (1-x)^{(n+1)}}{(n+1)!} \sum_{k=0}^m \frac{n+1}{n+k+1} \frac{ x^{(k)}}{k!} + \frac{ x^{(m+1)}}{(m+1)!} \sum_{k=0}^n \frac{m+1}{m+k+1} \frac{ (1-x)^{(k)}}{k!}=1, $$ where $z^{(n)}$ denotes the rising factorial. Moreover, we present identities involving the incomplete beta function and a ce...

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 immediate consequences of the main result. Finally, combinatorial identities involving harmonic-like numbers and other prominent sequences like hyperharmonic numbers and odd harmonic numbers are offered.

Modifying an idea of E. Brietzke we give simple proofs for the recurrence relations of some sequences of binomial sums which have previously been obtained by other more complicated methods.