Recent zbMATH articles in MSC 44A20https://www.zbmath.org/atom/cc/44A202021-02-27T13:50:00+00:00WerkzeugIntegrals involving the aleph-function of several variables.https://www.zbmath.org/1453.330092021-02-27T13:50:00+00:00"Ayant, Frederic"https://www.zbmath.org/authors/?q=ai:ayant.frederic-ySummary: In this paper, we evaluate four integrals involving the product of elementary special functions and the multivariable Aleph-function. The integrals are quite general in nature and from them a large number of new results can be obtained simply by specializing the parameters of the multivariable Aleph-function.Some inverse Laplace transforms that contain the Marcum \(Q\) function and an expanded property of the Marcum \(Q\) function.https://www.zbmath.org/1453.440022021-02-27T13:50:00+00:00"Veestraeten, Dirk"https://www.zbmath.org/authors/?q=ai:veestraeten.dirkIt appears that the current paper is an extension and generalization of [\textit{D.~Veestraeten}, Integral Transforms Spec. Funct. 26, No.~10, 859--871 (2015; Zbl 1331.33005)], where the inverse Laplace transform for the product of two parabolic cylinder functions is obtained using the Ornstein-Uhlenbeck process.
In Section~2, basic definitions of transition distributions and the transition density are presented.
Employing a Feller process, the Laplace transform of the transition density and of the transition distribution are given.
Using a new substitution in the transition distribution (2.5) and specifying the transition density (3.1), the transition distribution is derived in terms of the Marcum $Q$-function in Section~3.
In Section~4, the inverse Laplace transform is established from the transition density, which contains products of Kummer and Tricomi confluent hypergeometric functions (as Theorem~4.1) under certain initial conditions.
Using the transition distribution under different conditions six new inverse Laplace transforms are derived in Theorems 5.1--5.4, four of which contain the Marcum $Q$-function. Section~6 is divided into two subsections, one of which discusses the inverse Laplace transform (Theorem~6.1), which involves the relation between two Marcum $Q$-functions with reversed arguments, and in the other the Marcum $Q$-function is represented (Theorem~6.2) in terms of the generalized hypergeometric function involving integer and fractional order values.
Reviewer: Deshna Loonker (Jodhpur)On certain polynomials associated with the multivariable aleph-function.https://www.zbmath.org/1453.330082021-02-27T13:50:00+00:00"Ayant, Frederic"https://www.zbmath.org/authors/?q=ai:ayant.frederic-ySummary: In an attempt to unify various bilateral generating functions obtained earlier by
\textit{H. M. Srivastava} and \textit{R. Panda}, J. Reine Angew. Math. 283/284, 265--274 (1976; Zbl 0315.33003),
\textit{R. K. Raina}, Proc. Natl. Acad. Sci. India, Sect. A 46, 300--304 (1976; Zbl 0438.33004)
and \textit{H. M. Srivastava} and \textit{R. K. Raina}, Hokkaido Math. J. 10, 34--45 (1981; Zbl 0438.33005),
we study here a few new sets of polynomials associated with the multivariable Aleph-function and give certain theorems concerning the generating functions of these polynomials. In the sequel we also show that these theorems can be applied to yield several bilateral generating function for some polynomial sets.