Related papers: Note on extensions of the beta function
In this paper we define a symmetric zeta function. We show that it can be analytically continued to a meromorphic function on $\mathbb{C}^3$ with only simple poles at some special hyperplanes. We also calculate the value of a multiple…
This paper studies the log-convexity of the extended beta functions. As a consequence, Tur\'an-type inequalities are established.The monotonicity, log-convexity, log-concavity of extended hypergeometric functions are deduced by using the…
The secondary zeta function is defined as a generalized zeta series over the imaginary parts of non-trivial zeros assuming (RH). This function admits Laurent series expansion at the double pole at $s=1$. In this article, we derive a new…
We compute the dependence on the classical action "gauge" parameters of the beta functions of the standard topological sigma model in flat space. We thus show that their value is a "gauge" artifact indeed. We also show that previously…
We expand the classical balayage of measures and subharmonic functions on a system of rays $S$ with a common origin on the complex plane $\mathbb C$. This allows for an arbitrary subharmonic function $v$ of finite order on $\mathbb C$ build…
The computation and inversion of the binomial and negative binomial cumulative distribution functions play a key role in many applications. In this paper, we explain how methods used for the central beta distribution function (described in…
We consider integration of functions with values in a partially ordered vector space, and two notions of extension of the space of integrable functions. Applying both extensions to the space of real valued simple functions on a measure…
Special functions are often defined as a Fourier or Laplace transform of a positive measure, and the positivity of the measure manifests as positive definiteness of certain matrices. The purpose of this expository note is to give a sample…
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…
An analogue of Brylinski's knot beta function is defined for a compactly supported (Schwartz) distribution $T$ on $d$-dimensional Euclidean space. This is a holomorphic function on a right half-plane. If $T$ is a (uniform) double-layer on a…
A multidimensional generalization of the Bernstein class of functions and the properties of functions of the introduced class are examined. In particular, a new proof of the integral representation of Bernstein functions of many variables…
We study the concept of extended derivations of algebras which expands diverse definitions of generalized derivations given in the literature. We concentrate on the family of the anti-commutative algebras and classify such spaces of…
In the present paper we introduce and studied two subclasses of multivalent functions denoted by $\mathcal{M}^{\lambda}_{p,n}(\gamma;\beta)$ and $\mathcal{N}^{\lambda}_{p,n}(\mu,\eta;\delta)$. Further, by giving specific values of the…
A generalization of a well-known relation between the Riemann zeta function $\zeta(s)$ and Bernoulli numbers $B_n$ is obtained. The formula is a new representation of the Riemann zeta function in terms of a nested series of Bernoulli…
Fourier transforms are ubiquitous mathematical tools in basic and applied sciences. We here report classical and quantum optical realizations of the discrete fractional Fourier transform, a generalization of the Fourier transform. In the…
We construct a family of continuous biorthogonal functions related to an elliptic analogue of the Gauss hypergeometric function. The key tools used for that are the elliptic beta integral and the integral Bailey chain introduced earlier by…
In the paper, after reviewing the history, background, origin, and applications of the functions $\frac{b^{t}-a^{t}}{t}$ and $\frac{e^{-\alpha t}-e^{-\beta t}}{1-e^{-t}}$, we establish sufficient and necessary conditions such that the…
In this paper we will focus on the study of relationships that can exist between odd numbers and different traditional functions like the gamma function, Riemann zeta function or function of von Mangoldt. Number theory applies to this…
We deduce several curious q-series expansions by applying inverse relations to certain identities for basic hypergeometric series. After rewriting some of these expansions in terms of q-integrals, we obtain, in the limit q -> 1, some…
The classical perspective of a function is a construction which transforms a convex function into one that is jointly convex with respect to an auxiliary scaling variable. Motivated by applications in several areas of applied analysis, we…