Related papers: On generalized Lambert function
Exact formulas are derived for the probability density functions of the sum and difference of two independent non-central gamma distributed random variables, with both series and integral representations of the density presented. These…
In this paper, we introduce and investigate a new extension of the beta function by means of an integral operator involving a product of Bessel-Struve kernel functions. We also define a new extension of the well-known beta distribution, the…
In 2000, Hafner and Stopple proved a conjecture of Zagier which states that the constant term of the automorphic function $|\Delta(x+iy)|^2$ i.e., the Lambert series $\sum_{n=1}^\infty \tau(n)^2 e^{-4 \pi n y}$ can be expressed in terms of…
Consider the mutually catalytic branching process with finite branching rate $\gamma$. We show that as $\gamma\to\infty$, this process converges in finite-dimensional distributions (in time) to a certain discontinuous process. We give…
We generalize the Mittag-Leffler function by attaching an exponent to its Taylor coefficients. The main result is an asymptotic formula valid in sectors of the complex plane, which extends work by Le Roy [Bull. des sciences math. 24, 1900]…
We develop new representations for the Levy measures of the beta and gamma processes. These representations are manifested in terms of an infinite sum of well-behaved (proper) beta and gamma distributions. Further, we demonstrate how these…
The classical beta function B(x; y) is one of the most fundamental special functions, due to its important role in various fields in the mathematical, physical, engineering and statistical sciences. Useful extensions of the classical Beta…
A general class of probability density functions \[u(x,t)=Ct^{-\alpha d}\left (1-\left (\frac{\|x\|}{ct^{\alpha}}\right )^{\beta}\right )_+^{\gamma},\quad x\in \mathbb{R}^d,t>0,\] is considered, containing as particular case the Barenblatt…
This paper generalizes beta divergence beyond its classical form associated with power variance functions of Tweedie models. Generalized form is represented by a compact definite integral as a function of variance function of the…
An interesting line of research is the investigation of the laws of random variables known as Dirichlet means. However, there is not much information on interrelationships between different Dirichlet means. Here, we introduce two…
We construct the generalized entropy optimized by a given arbitrary statistical distribution with a finite linear expectation value of a random quantity of interest. This offers, via the maximum entropy principle, a unified basis for a…
A dynamical model based on a continuous addition of colored shot noises is presented. The resulting process is colored and non-Gaussian. A general expression for the characteristic function of the process is obtained, which, after a scaling…
We consider a generalisation of a definite integral involving the Bessel function of the first kind. It is shown that this integral can be expressed in terms of the Fox-Wright function ${}_p\Psi_q(z)$ of one variable. Some consequences of…
We address the question and related controversy of the formulation of the $q$-entropy, and its relative entropy counterpart, for models described by continuous (non-discrete) sets of variables. We notice that an $L_p$ normalized functional…
We calculate some infinite sums containing the digamma function in closed-form. These sums are related either to the incomplete beta function or to the Bessel functions. The calculations yield interesting new results as by-products, such as…
Let the summatory function of the M\"{o}bius function be denoted $M(x)$. We deduce in this article conditional results concerning $M(x)$ assuming the Riemann Hypothesis and a conjecture of Gonek and Hejhal on the negative moments of the…
Problems formulated in terms of logarithmic or exponential equations often use the Lambert $W$ function in their solutions. Expansions, approximations and bounds on $W$ have been derived in an effort to gain a better understanding of the…
We establish sharp inequalities involving the incomplete Beta and Gamma functions. These inequalities arise in the approximation of generalized Bernstein functions by higher order Thorin-Bernstein functions. Furthermore, new properties of a…
Let ${\mathcal A}$ be the ${\mathcal L}^q-$functional of a stable L\'evy process starting from one and killed when crossing zero. We observe that ${\mathcal A}$ can be represented as the independent quotient of two infinite products of…
We discuss in detail the asymptotic distribution of sample expectiles. First, we show uniform consistency under the assumption of a finite mean. In case of a finite second moment, we show that for expectiles other then the mean, only the…