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A new class of 2-orthogonal polynomials satisfying orthogonality conditionswith respect to a pair of linear functionals $(u_0,u_1)$ was presented in DouakK & Maroni P [On a new class of 2-orthogonal polynomials, I: the recurrencerelations and some properties. Integral Transforms Spec Funct.2021;32(2):134-153]. Six interesting special cases were pointed out there. Foreach case, we precisely deal with the integral representation problem for thefunctionals associated to these polynomials. The focus is on the matrixdifferential equation $\big({\bf\Phi U}\big)'+{\bf\Psi U}=0$, with ${\bfU}={^t}(u_0 , u_1)$ and ${\bf\Phi}$, ${\bf\Psi}$ are $2\times2$ polynomialmatrices, from which we establish the differential equations satisfied by thetwo 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 orpositive real line and defined in terms of various special functions. In orderfor certain integral representations to exist, addition of Dirac mass isnecessary.

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