On a new class of 2-orthogonal polynomials, II_The integral representations

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 relations and some properties. Integral Transforms Spec Funct. 2021;32(2):134-153]. Six interesting special cases were pointed out there. For each case, we precisely deal with the integral representation problem for the functionals associated to these polynomials. The focus is on the matrix differential equation $\big({\bf\Phi U}\big)'+{\bf\Psi U}=0$, with ${\bf U}={^t}(u_0 , u_1)$ and ${\bf\Phi}$, ${\bf\Psi}$ are $2\times2$ polynomial matrices, from which we establish the differential equations satisfied by the two functionals. Based on this, depending on the case, we show that $u_0$ and $u_1$ are represented via weight functions supported on the real line or positive real line and defined in terms of various special functions. In order for certain integral representations to exist, addition of Dirac mass is necessary.