The main purpose of this paper is to present new families of Jacobi type matrix valued orthogonal polynomials obtained from the underlying group $SU\left(n\right)$ and its representations. These polynomials are eigenfunctions of some symmetric second order hypergeometric differential operator with matrix coefficients. The final result holds for arbitrary values of the parameters $\alpha ,\beta \>-1$, but it is derived only for those values that come from the group theoretical setup.

Let p,q be positive integers. The groups $U_p\left(ℂ\right)$ and $U_p\left(ℂ\right)\times U_q\left(ℂ\right)$ act on the Heisenberg group $H_{p,q}:=M_{p,q}\left(ℂ\right)\times ℝ$ canonically as groups of automorphisms, where $M_{p,q}\left(ℂ\right)$ is the vector space of all complex p × q matrices. The associated orbit spaces may be identified with ${\Pi }_q\times ℝ$ and ${\Xi }_q\times ℝ$ respectively, ${\Pi }_q$ being the cone of positive semidefinite matrices and ${\Xi }_q$ the Weyl chamber $x\in {ℝ}^q:x₁\ge \cdots \ge x_q\ge 0$. In this paper we compute the associated convolutions on ${\Pi }_q\times ℝ$ and ${\Xi }_q\times ℝ$ explicitly, depending on p. Moreover, we extend these convolutions by analytic continuation to series of convolution...

We present a new criterion for the weighted $L^p-L^q$ boundedness of multiplier operators for Laguerre and Hermite expansions that arise from a Laplace-Stieltjes transform. As a special case, we recover known results on weighted estimates for Laguerre and Hermite fractional integrals with a unified and simpler approach.