arXiv subjects
Publications and source records attributed to Tibor K. Pogány.
We prove that the distribution of the product of two correlated normal random variables with arbitrary means and arbitrary variances is infinitely divisible. We also obtain exact formulas for the probability density function of the sum of independent copies of such random variables.
A celebrated result in probability theory is that a simple symmetric random walk on the $d$-dimensional lattice $\mathbb{Z}^d$ is recurrent for $d=1,2$ and transient for $d\geq 3$. In this note, we derive a closed-form expression, in terms of the Lauricella function $F_C$, for the return probability for all $d\geq3$. Previously, a closed-form formula had only been available for $d=3$.
The cumulative distribution function of the non-central chi-square distribution $χ_ν'^2(λ),\, ν\in\mathbb{R}^+$ possesses an integral representation in terms of a generalized Marcum $Q$-function. Regarding some already known results, here we derive a simpler form of the cumulative distribution function for $ν= 2n \in\mathbb{N}$ degrees of freedom. Also, we express these representations in terms of an incomplete Fox-Wright function ${}_pΨ_q^{(γ)}$ and the generalized incomplete hypergeometric functions concerning the important special cases as ${}_1Γ_1,\, {}_2Γ_1$ and ${}_2γ_1$. New identities are established between ${}_1Γ_1$ and ${}_2Γ_1$ as well.
We prove that for $ν>n-1$ all zeros of the $n$th derivative of Bessel function of the first kind $J_ν$ are real and simple. Moreover, we show that the positive zeros of the $n$th and $(n+1)$th derivative of Bessel function of the first kind $J_ν$ are interlacing when $ν\geq n,$ and $n$ is a natural number or zero. Our methods include the Weierstrassian representation of the $n$th derivative, properties of the Laguerre-Pólya class of entire functions, and the Laguerre inequalities. Some similar results for the zeros of the first and second derivative of the Struve function of the first kind $\mathbf{H}_ν$ are also proved. The main results obtained in this paper generalize and complement some classical results on the zeros of Bessel functions of the first kind. Some open problems related to Hurwitz theorem on the zeros of Bessel functions are also proposed, which may be of interest for further research.
In this brief note an integral expression is presented for the COM-Poisson renormalization constant $Z(λ, ν)$ on the real axis.
Motivated by certain current results by Parmar and Pogány [9] in which the authors introduced the so-called $p$-extended Mathieu series the main aim of this paper is to present a connection between such series and a various types of Schlömilch series.
Motivated by several generalizations of the well--known Mathieu series, the main object of this paper is to introduce new extension of generalized Mathieu series and to derive various integral representations of such series. Finally master bounding inequality is established using the newly derived integral expression.
Considering the recently studied Gamma exponentiated exponential Weibull ${\rm GEEW}(θ)$ probability distribution \cite{PoganySaboor} surprising infinite summations are obtained for series which building blocks are special functions like lower and upper incomplete Gamma, Fox--Wright Psi, Meijer $G$ or Whittaker $W$ functions.
Introducing the discrete probability distribution by means of the Prabhakar (or the three--parameter Mittag--Leffler) function, we establish explicit expressions for raw and factorial moments and also general fractional order moments. Applying an elementary moment inequality we obtain functional upper bounds for the Turánian difference for Prabhakar function. Finally, a Laguerre inequality is proved and functional upper bound has been given for Laguerreian difference for Prabhakar function.
The main purpose of this paper is to present closed integral form expressions for the Mathieu-type $\boldsymbol a$-series and its associated alternating version whose terms contain a $(p, q)$-extended Gauss' hypergeometric function. Certain upper bounds for the two series are also given.
This paper is concerned with new results for the circular Eisenstein series $\varepsilon_r(z)$ as well as with a novel approach to Hilbert-Eisenstein series $\mathfrak h_r(z)$, introduced by Michael Hauss in 1995. The latter turn out to be the product of the hyperbolic sinh--function with an explicit closed form linear combination of digamma functions. The results, which include differentiability properties and integral representations, are established by independent and different argumentations. Highlights are new results on the Butzer--Flocke--Hauss Omega function, one basis for the study of Hilbert-Eisenstein series, which have been the subject of several recent papers.
In this short research note we obtain double definite integral expressions for the Kapteyn type series built by Kummer's $M$ (or confluent hypergeometric ${}_1F_1$) functions. These kind of series unify in natural way the similar fashion results for Neumann-, Schlömilch- and Kapteyn-Bessel series recently established by Pogány, Süli, Baricz and Jankov Maširević.
Closed form expressions are proposed for the Feynman integral $$ I_{D, m}(p,q) = \int\frac{d^my}{(2π)^m}\int\frac{d^Dx}{(2π)^D} \frac1{(x-p/2)^2+(y-q/2)^4} \frac1{(x+p/2)^2+(y+q/2)^4} $$ over $d=D+m$ dimensional space with $(x,y),\,(p,q)\in \mathbb R^D \oplus \mathbb R^m$, in the special case $D=1$. We show that $I_{1,m}(p,q)$ can be expressed in different forms involving real and imaginary parts of the complex variable Gauss hypergeometric function $_2F_1$, as well as generalized hypergeometric $_2F_2$ and $_3F_2$, Horn $H_4$ and Appell $F_2$ functions. Several interesting relations are derived between the real and imaginary parts of $_2F_1$ and the function $H_4$.
In this paper a necessary and sufficient condition is deduced for the close-to-convexity of a cross product of Bessel and modified Bessel functions of the first kind and their derivatives by using a result of Shah and Trimble about transcendental entire functions with univalent derivatives, the newly discovered power series and infinite product representation of this cross-product, as well as a slightly modified version of a result of Lorch on the monotonicity of the zeros of the cross product with respect to the order.
In this note we offer some log-concavity properties of certain functions related to Bessel functions of the first kind and modified Bessel functions of the first and second kind, by solving partially a recent conjecture on the log-convexity/log-concavity properties for modified Bessel functions of the first kind and their derivatives. Moreover, we give an application of the mentioned results by extending two inequalities of van der Corput to Bessel and modified Bessel functions of the first kind. Similar inequalities are proved also for modified Bessel functions of the second kind, as well as for log-concave probability density functions.
In this paper our aim is to present an elementary proof of an identity of Calogero concerning the zeros of Bessel functions of the first kind. Moreover, by using our elementary approach we present a new identity for the zeros of Bessel functions of the first kind, which in particular reduces to some other new identities. We also show that our method can be applied for the zeros of other special functions, like Struve functions of the first kind, and modified Bessel functions of the second kind.
In this note our aim is to deduce some new monotonicity properties for a special combination of Bessel functions of the first kind by using a recently developed Mittag-Leffler expansion for the derivative of a normalized Bessel function of the first kind. These monotonicity properties are used to obtain some new inequalities for Bessel functions of the first kind.
Various miscellaneous functional inequalities are deduced for the so-called generalized inverse trigonometric and hyperbolic functions. For instance, functional inequalities for sums, difference and quotient of generalized inverse trigonometric and hyperbolic functions are given, as well as some Grünbaum inequalities with the aid of the classical Bernoulli inequality. Moreover, by means of certain already derived bounds, bilateral bounding inequalities are obtained for the generalized hypergeometric ${}_3F_2$ Clausen function.