Related papers: An integral representation for Herglotz-Nevanlinna…
Integral representations are considered of solutions of the inhomogeneous Airy differential equation $w''-z w=\pm1/\pi$. The solutions of these equations are also known as Scorer functions. Certain functional relations for these functions…
We extend the study of the Pick class, the set of complex analytic functions taking the upper half plane into itself, to the noncommutative setting. R. Nevanlinna showed that elements of the Pick class have certain integral representations…
We give a functional equation for the refined Herglotz-Zagier function. It is analogous to a result in the theory of modular forms.
In the paper, the authors find some integral representations and discover some properties of Lah numbers.
In the present article a new method of deriving integral representations of combinations and partitions in terms of harmonic products has been established. This method may be relevant to statistical mechanics and to number theory.
The paper is devoted to study the $H$-function defined by the Mellin-Barnes integral $$H^{m,n}_{\thinspace p,q}(z)={\frac1{2\pi i}}\int_{\Lss} \HHs^{m,n}_{\thinspace p,q}(s)z^{-s}ds,$$ where the function $\HH^{m,n}_{\thinspace p,q}(s)$ is a…
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 establish new analytic and numerical results on a general class of rational operator Nevanlinna functions that arise e.g. in modelling photonic crystals. The capability of these dielectric nano-structured materials to control the flow of…
Generalizations of classical theta functions are proposed that include any even number of analytic parameters for which conditions of quasi-periodicity are fulfilled and that are representations of extended Heisenberg group. Differential…
This survey shows how, for the Nevanlinna class N of the unit disc, one can define and often characterize the analogues of well-known objects and properties related to the algebra of bounded analytic functions $ H^\infty$: interpolating…
We establish the meromorphic continuation of certain multiple zeta functions of generalized Hurwitz type. From this meromorphic continuation, we obtain explicit formulas for their (derivative) values at nonpositive integers along a given…
We introduce different classical characteristics used to regularize a subharmonic function and compare them. As an application we give a complete proof of a useful characterization of the modulus of continuity of such functions in terms of…
Based on a Problem and its solution published on the pages of SIAM Review, we give an interesting integral representation for the Lambert $W$ function in this short note. In particular, our result yields a new integral representation for…
Let $f$ be a $r\times m$-matrix of holomorphic functions that is generically surjective. We provide explicit integral representation of holomorphic $\psi$ such that $\phi=f\psi$, provided that $\phi$ is holomorphic and annihilates a certain…
We give an asymptotic expansion (the higher Stirling formula) and an infinite product representation (the Weierstrass product representation) of the Vign\'{e}ras multiple gamma functions by considering the classical limit of the multiple…
We investigate functionals defined on manifolds through parameterizations. If they are to be meaningful, from a geometrical viewpoint, they ought to be invariant under reparameterizations. Standard, local, integral functionals with this…
The Nevanlinna transform K(z), of a measure and a real constant, plays an important role in the complex analysis and more recently in the free probability theory (boolean convolution). It is shown that its restriction k(it) (the restricted…
We present a conceptual and uniform interpretation of the methods of integral representations of L-functions (period integrals, Rankin-Selberg integrals). This leads to: (i) a way to classify of such integrals, based on the classification…
The hypergeometric function method naturally provides the analytic expressions of scalar integrals from concerned Feynman diagrams in some connected regions of independent kinematic variables, also presents the systems of homogeneous linear…
In the last decade there has been a growing interest in superoscillations in various fields of mathematics, physics and engineering. However, while in applications as optics the local oscillatory behaviour is the important property, some…