Related papers: A Matrix Integral Solution to [P,Q]=P and Matrix L…

Integral representations of two $q$-difference operators are provided in terms of special functions arising in the theory of asymptotic solutions to $q$-difference equations in the complex domain. Both representations are unified through…

Series of finite dimensional representations of the superalgebras spl(p,q) can be formulated in terms of linear differential operators acting on a suitable space of polynomials. We sketch the general ingredients necessary to construct these…

The recurrence matrix relations, differentiation formulas, and analytical and fractional integral properties of incomplete gamma matrix functions $\gamma(Q, x)$ and $\Gamma(Q, x)$ are all covered in this article. The generalized incomplete…

An operator theoretic approach to invariant integration theory on non-compact quantum spaces is introduced on the example of the quantum (n,1)-matrix ball O_q(Mat_{n,1}). In order to prove the existence of an invariant integral, operator…

We define a class of pseudo-differential operators in a completely new way, which is called the abstract operators and expounded systematically the theory of abstract operators. By combining abstract operators with the Laplace transform, we…

We first strictly expressed the basic notions and research methods of abstract operators, which systematically expounded the main results of abstract operator theory. By combining abstract operators with the Laplace transform, we can easily…

Two $(p,q)$-Laplace transforms are introduced and their relative properties are stated and proved. Applications are made to solve some $(p,q)$-linear difference equations.

The aim of this paper is to study the q-Laplace operator and q-harmonic polynomials on the quantum complex vector space generated by z_i,w_i, i=1,2,...,n, on which the quantum group GL_q(n) (or U_q(n)) acts. The q-harmonic polynomials are…

A new class of 2-orthogonal polynomials satisfying orthogonality conditions with respect to a pair of linear functionals $(u_0,u_1)$ was presented in Douak K & Maroni P [On a new class of 2-orthogonal polynomials, I: the recurrence…

We construct a new class of operators that act on symmetric functions with two deformation parameters $q$ and $t$. Our combinatorial construction associates each operator with a specific lattice path, whose steps alternate between moving up…

The partition function for unitary two matrix models is known to be a double KP tau-function, as well as providing solutions to the two dimensional Toda hierarchy. It is shown how it may also be viewed as a Borel sum regularization of…

We generalize the classic Fourier transform operator $\mathcal{F}_{p}$ by using the Henstock-Kurzweil integral theory. It is shown that the operator equals the $HK$-Fourier transform on a dense subspace of $\mathcal{ L}^p$, $1<p\leq 2$. In…

We study existence, uniqueness and regularity of solutions for linear equations in infinitely many derivatives. We develop a natural framework based on Laplace transform as a correspondence between appropriate $L^p$ and Hardy spaces: this…

Two square matrices of (arbitrary) order N are introduced. They are defined in terms of N arbitrary numbers z_{n}, and of an arbitrary additional parameter (a respectively q), and provide finite-dimensional representations of the two…

For the linear partial differential equation $P(\partial_x,\partial_t)u=f(x,t)$, where $x\in\mathbb{R}^n,\;t\in\mathbb{R}^1$, with $P(\partial_x,\partial_t)$ is $\prod^m_{i=1}(\frac{\partial}{\partial{t}}-a_iP(\partial_x))$ or…

Let $p,q,n$ be natural numbers such that $p+q=n$. Let $\FF$ be either $\CC$, the complex numbers field, or $\HH$, the quaternionic division algebra. We consider the Heisenberg group $N(p,q,\FF)$ defined as $N(p,q,\FF)=\FF^{n}\times…

Generalized convolution symmetries of integrable hierarchies of KP and 2KP-Toda type multiply the Fourier coefficients of the elements of the Hilbert space $\HH= L^2(S^1)$ by a specified sequence of constants. This induces a corresponding…

Any homogeneous polynomial $P(x, y, z)$ of degree $d$, being restricted to a unit sphere $S^2$, admits essentially a unique representation of the form $\lambda_0 + \sum_{k = 1}^d \lambda_k [\prod_{j = 1}^k L_{kj}]$, where $L_{kj}$'s are…

In Part 1 we study the spherical functions on compact symmetric pairs of arbitrary rank under a suitable multiplicity freeness assumption and additional conditions on the branching rules. The spherical functions are taking values in the…

We consider the matrix spherical function related to the compact symmetric pair $(G,K)=(\mathrm{SU}(n+m),\mathrm{S}(\mathrm{U}(n)\times\mathrm{U}(m)))$. The irreducible $K$ representations $(\pi,V)$ in the ${\rm U}(n)$ part are considered…