In the paper the basic analytical properties of the MacDonald function (the modified Bessel function of the second kind) are summarized and the properties of some subclasses of distribution functions based on MacDonald function, especially of the types $x^n{K}_n\left(x\right),x\ge 0,{\left|x\right|}^n{K}_n\left(x\left|x\right|\right),x\in 𝐑$ and $x^{n+1}{K}_n\left(x\right),x\ge 0$ are discussed. The distribution functions mentioned are useful for analytical modelling of composed (mixed) distributions, especially for products of random variables having distributions of the exponential type. Extensive and useful applications...

For any odd prime p we obtain q-analogues of van Hamme’s and Rodriguez-Villegas’ supercongruences involving products of three binomial coefficients such as ${\sum }_{k=0}^{(p-1)/2}{\left[\genfrac{}{}{0pt}{}{2k}{k}\right]}_{q²}^3\left(q^{2k}\right)/\left((-q²;q²){²}_k(-q;q){²}_{2k}²\right)\equiv 0\left(mod\left[p\right]²\right)$ for p≡ 3 (mod 4), ${\sum }_{k=0}^{(p-1)/2}{\left[\genfrac{}{}{0pt}{}{2k}{k}\right]}_{q³}\left({(q;q³)}_k{(q²;q³)}_k{q}^{3k}\right)\left({(q⁶;q⁶)}_k²\right)\equiv 0\left(mod\left[p\right]²\right)$ for p≡ 2 (mod 3), where $\left[p\right]=1+q+\cdots +q^{p-1}$ and $(a;q)ₙ=(1-a)(1-aq)\cdots (1-a{q}^{n-1})$. We also prove q-analogues of the Sun brothers’ generalizations of the above supercongruences. Our proofs are elementary in nature and use the theory of basic hypergeometric series and combinatorial q-binomial identities including a new q-Clausen type summation formula.

The Humbert matrix polynomials were first studied by Khammash and Shehata (2012). Our goal is to derive some of their basic relations involving the Humbert matrix polynomials and then study several generating matrix functions, hypergeometric matrix representations, matrix differential equation and expansions in series of some relatively more familiar matrix polynomials of Legendre, Gegenbauer, Hermite, Laguerre and modified Laguerre. Finally, some definitions of generalized Humbert matrix polynomials...

The classical Hermite-Hermite matrix polynomials for commutative matrices were first studied by Metwally et al. (2008). Our goal is to derive their basic properties including the orthogonality properties and Rodrigues formula. Furthermore, we define a new polynomial associated with the Hermite-Hermite matrix polynomials and establish the matrix differential equation associated with these polynomials. We give the addition theorems, multiplication theorems and summation formula for the Hermite-Hermite...