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A large family of symmetric Laguerre–Hahn polynomials of class two

A regular form (linear functional) u is said to be Laguerre–Hahn, if the Stieltjes function S(u) satisfies a Riccati differential equation Φ (z) S′(u)(z)=B(z) S 2(u)(z)+C(z) S(v)z)+D(z), where Φ≠0, B, C and D are polynomials. Recently, all the symmetric Laguerre–Hahn forms of class s≤1 have been determined. In this paper, by means of the quadratic decomposition, we carry out the complete description of the symmetric Laguerre–Hahn forms of class s=2, when Φ and B vanish at zero. Essentially, three canonical cases appear. The Laguerre–Hahn forms found generalize those of class s=1.