Related papers: Two Parameter Gamma Function and its Properties
In the present paper, unification of Bessel, modified Bessel, spherical Bessel and Bessel-Clifford functions via the generalized Pochhammer symbol [ Srivastava HM, Cetinkaya A, K{\i}ymaz O. A certain generalized Pochhammer symbol and its…
The celebrated Zeilberger algorithm which finds holonomic recurrence equations for definite sums of hypergeometric terms $F(n,k)$ is extended to certain nonhypergeometric terms. An expression $F(n,k)$ is called a hypergeometric term if both…
New expansions for some functions related to the Zeta function in terms of the Pochhammer's polynomials are given (coefficients b(k), d(k), d_(k) and d__(k). In some formal limit our expansion b(k) obtained via the alternating series gives…
In this paper the incomplete gamma function $\gamma(\alpha,x)$ and its derivative is considered for negative values of $\alpha $ and the incomplete gamma type function $\gamma_*(\alpha,x_-)$ is introduced. Further the polygamma functions…
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…
In this expository article we show explicitly how to compute Gamma(p/q) in terms of Beta function values which in turn are Kontsevich-Zagier Periods.
In this paper, the incomplete Pochhammer ratios are defined in terms of the incomplete beta function $B_{y}(x,z)$. With the help of these incomplete Pochhammer ratios, we introduce new incomplete Gauss, confluent hypergeometric and Appell's…
We present several identities with a form of polynomials or rational functions that involve Pochhammer and q-Pochhammer symbols and q-binomials (i.e. Gauss polynomials). All these identities were obtained by some analytical methods based on…
Quasisymmetric functions have recently been used in time series analysis as polynomial features that are invariant under, so-called, dynamic time warping. We extend this notion to data indexed by two parameters and thus provide warping…
We consider nested sums involving the Pochhammer symbol at infinity and rewrite them in terms of a small set of constants, such as powers of $\pi,$ $\log(2)$ or zeta values. In order to perform these simplifications, we view the series as…
We show the modular properties of the multiple 'elliptic' gamma functions, which are an extension of those of the theta function and the elliptic gamma function. The modular property of the theta function is known as Jacobi's…
Using a probabilistic approach, we derive several interesting identities involving beta functions. Our results generalize certain well-known combinatorial identities involving binomial coefficients and gamma functions.
We aim to achieve the following three goals. First of all, we collect all known definitions, transformation properties and functional identities of Barnes double gamma function $G(z;\tau)$. Second, we derive an algorithm for numerically…
We derive several identities for the Hurwitz and Riemann zeta functions, the Gamma function, and Dirichlet $L$-functions. They involve a sequence of polynomials $\alpha_k(s)$ whose study was initiated in an earlier paper. The expansions…
Two representations of the extended gamma functions $\Gamma^{2,0}_{0,2}[(b,x)]$ are proved. These representations are exploited to find a transformation relation between two Fox's $H$-functions. These results are used to solve Fox's…
In this paper we will give a proof of a certain summation formula for Gamma functions utilizing Gegenbauer polynomials.
Motivated by the integral representation of the Euler Beta function, we introduce its Cauchy siblings and investigate some of their properties. Two of these newly introduced functions happen to coincide with some classical means, such as…
In this paper, the power series and hypergeometric series representations of the beta and Ramanujan functions \begin{equation*} \mathcal{B}\left( x\right) =\frac{\Gamma \left( x\right)^{2}}{\Gamma \left( 2x\right) }\text{ and…
In this paper, we establish the following two identities involving the Gamma function and Bernoulli polynomials, namely $$ \sum_{k\leq x}\frac{1}{k^s} \sum_{j=1}^{k^s}\log\Gamma\left(\frac{j}{k^s}\right) \sum_{\substack{d|k \\…
We study q-integral representations of the q-gamma and the q-beta functions. This study leads to a very interesting q-constant. As an application of these integral representations, we obtain a simple conceptual proof of a family of…