Related papers: A Note on Gamma Function and Convolution
Motivated mainly by certain interesting recent extensions of the Gamma, Beta and hypergeometric functions, we introduce here new extensions of the Beta function, hypergeometric and confluent hypergeometric functions. We systematically…
We consider several possible approaches to evaluating an integral involving the digamma function and a related logarithmic series.
In this expository article, we provide a self-contained overview of the notion of convolution embedded in different theories: from the classical Fourier theory to the theory of algebraic signal processing. We discuss their relations and…
We define a new class of generating function transformations related to polylogarithm functions, Dirichlet series, and Euler sums. These transformations are given by an infinite sum over the $j^{th}$ derivatives of a sequence generating…
This note is a short survey of two topics: Archimedean zeta functions and Archimedean oscillatory integrals. We have tried to portray some of the history of the subject and some of its connections with similar devices in mathematics. We…
In this paper we deal with the generalized Gamma processes and their compositions. For the compositions of two or more than two generalized Gamma processes we give, when possible, the explicit law whereas, in the other cases the…
``Quasi-elliptic'' functions can be given a ring structure in two different ways, using either ordinary multiplication, or convolution. The map between the corresponding standard bases is calculated. A related structure has appeared…
We define the notion of an invariant function on a cluster ensemble with respect to an action of the cluster modular group on its associated function fields. We realize many examples of previously studied functions as elements of this type…
We study several variants of Euler sums by using the methods of contour integration and residue theorem. These variants exhibit nice properties such as closed forms, reduction, etc., like classical Euler sums. In addition, we also define a…
We define the generalized-Euler-constant function $\gamma(z)=\sum_{n=1}^{\infty} z^{n-1} (\frac{1}{n}-\log \frac{n+1}{n})$ when $|z|\leq 1$. Its values include both Euler's constant $\gamma=\gamma(1)$ and the "alternating Euler constant"…
One familiar with the Euler zeta function, which established the remarkable relationship between the prime and composite numbers, might naturally ponder the results of the application of this special function in cases where there is no…
In the paper, the authors establish an inequality involving exponential functions and sums, introduce a ratio of many gamma functions, discuss properties, including monotonicity, logarithmic convexity, (logarithmically) complete…
This paper continues a series of investigations on converging representations for the Riemann Zeta function. We generalize some identities which involve Riemann's zeta function, and moreover we give new series and integrals for the zeta…
The main object of this paper is to present generalizations of gamma, beta and hypergeometric functions. Some recurrence relations, transformation formulas, operation formulas and integral representations are obtained for these new…
We show that an apparently overlooked result of Euler from \cite{E421} is essentially equivalent to the general multiplication formula for the $\Gamma$-function that was proven by Gauss in \cite{Ga28}.
Recently the construction of various integral transforms for slice monogenic functions has gained a lot of attention. In line with these developments, the article at hand introduces the slice Fourier transform. In the first part, the kernel…
This paper deals with the convolution powers of the characteristic function of $[0,1], \chi_{[0,1]}$ and its function-derivatives. The importance that such convolution products have can be seen, for an instance, at \cite{DahmenLatour} where…
In this paper we deal with Mellin convolution of generalized Gamma densities which leads to integrals of modified Bessel functions of the second kind. Such convolutions allow us to explicitly write the solutions of the time-fractional…
In this note we present a method for obtaining a wide class of combinatorial identities. We give several examples, in particular, based on the Gamma and Beta functions. Some of them have already been considered by previously, and other are…
We give closed-form expressions for the Dirichlet beta function at even positive integers and for the Dirichlet lambda function at odd positive integers, based on the function J(s) defined via convergent integral. We also show fundamental…