MSC 2010: 33E12, 30A10, 30D15, 30E15We consider some families of 3-index generalizations of the classical Mittag-Le²er functions and study the behaviour of these functions in domains of the complex plane. First, some inequalities in the complex plane and on its compact subsets are obtained. We also prove an asymptotic formula for the case of "large" values of the indices of these functions. Similar results have also been obtained by the author for the classical Bessel functions and their Wright's...

We first characterize the increasing eigenfunctions associated to the following family of integro-differential operators, for any α, x>0, γ≥0 and fa smooth function on ${\Re }^{+}$, $$\begin{array}{c}\hfill {𝐋}^{\left(\gamma \right)}f\left(x\right)=x^{-\alpha }(\frac{\sigma }{2}{x}^2{f}^{\text{'}\text{'}}\left(x\right)+(\sigma \gamma +b)x{f}^{\text{'}}\left(x\right)\\ \hfill +{\int }_0^{\infty }\left(f\left({\mathrm{e}}^{-r}x\right)-f\left(x\right)\right){\mathrm{e}}^{-r\gamma }+x{f}^{\text{'}}\left(x\right)r{𝕀}_{\{r\le 1\}}\nu \left(\mathrm{d}r\right)),(0.1)\end{array}$$ where the coefficients $b\in \Re$,σ≥0 and the measure ν, which satisfies the integrability condition ∫0∞(1∧r2)ν(dr)<+∞, are uniquely determined by the distribution of a spectrally negative, infinitely divisible random variable, with characteristic exponent ψ. L(γ) is known to be the infinitesimal generator of a positive...

Mathematics Subject Classification: 33C05, 33C10, 33C20, 33C60, 33E12, 33E20, 40A30The main purpose of this paper is to present a number of potentially useful integral representations for the generalized Mathieu series as well as for its alternating versions via Mittag-Leffler type functions.

Mathematical Subject Classification 2010: 35R11, 42A38, 26A33, 33E12.The method of integral transforms based on using a fractional generalization of the Fourier transform and the classical Laplace transform is applied for solving Cauchy-type problem for the time-space fractional diffusion equation expressed in terms of the Caputo time-fractional derivative and a generalized Riemann-Liouville space-fractional derivative.

In this work we consider the Dunkl operator on the complex plane, defined by $${𝒟}_{k}f\left(z\right)=\frac{d}{dz}f\left(z\right)+k\frac{f\left(z\right)-f(-z)}{z},k\ge 0.$$ We define a convolution product associated with ${𝒟}_k$ denoted ${*}_k$ and we study the integro-differential-difference equations of the type $\mu {*}_{k}f={\sum }_{n=0}^{\infty }{a}_{n,k}{𝒟}_k^{n}f$, where $\left(a_{n,k}\right)$ is a sequence of complex numbers and $\mu$ is a measure over the real line. We show that many of these equations provide representations for particular classes of entire functions of exponential type.

A procedure is proposed in order to expand $w={\prod }_{j=1}^N{P}_{i_j}\left(x\right)={\sum }_{k=0}^M{L}_k{P}_k\left(x\right)$ where $P_i\left(x\right)$ belongs to aclassical orthogonal polynomial sequence (Jacobi, Bessel, Laguerre and Hermite) ($M={\sum }_{j=1}^N{i}_j$). We first derive a linear differential equation of order $2^N$ satisfied by w, fromwhich we deduce a recurrence relation in k for the linearizationcoefficients $L_k$. We develop in detail the two cases ${\left[P_i\left(x\right)\right]}^N$, $P_i\left(x\right)P_j\left(x\right)P_k\left(x\right)$ and give the recurrencerelation in some cases (N=3,4), when the polynomials $P_i\left(x\right)$are monic Hermite orthogonal polynomials.

Let $P_k$ be any sequence of classical orthogonal polynomials of a discrete variable. We give explicitly a recurrence relation (in k) for the coefficients in $P_i{P}_j={\sum }_{k}c(i,j,k)P_k$, in terms of the coefficients σ and τ of the Pearson equation satisfied by the weight function ϱ, and the coefficients of the three-term recurrence relation and of two structure relations obeyed by $P_k$.

The purpose of these notes is to give a short survey of an interesting connection between partition functions of supersymmetric gauge theories and hypergeometric functions and to present the recent progress in this direction.

MSC 2010: 15A15, 15A52, 33C60, 33E12, 44A20, 62E15 Dedicated to Professor R. Gorenflo on the occasion of his 80th birthdayA connection between fractional calculus and statistical distribution theory has been established by the authors recently. Some extensions of the results to matrix-variate functions were also considered. In the present article, more results on matrix-variate statistical densities and their connections to fractional calculus will be established. When considering solutions of fractional...