A description of second degree semiclassical forms of class two arising via cubic decomposition. (English) Zbl 1504.33008
Summary: In this work, we consider orthogonal polynomials via cubic decomposition in the framework of the second degree semiclassical class. We give a complete description, using the formal Stieltjes function and the moments, of semiclassical linear forms of class two which are of second degree such that their corresponding orthogonal polynomials sequences \(\{W_n\}_{n\ge 0}\) are obtained via the cubic decomposition \(W_{3n}(x)=P_n(x^3)\), \(n\ge 0\). We focus our attention, not only on the link between all these forms and the second degree classical forms \(\mathcal{T}_{p, q}=\mathcal{J}(p-1/2, q-1/2)\), \(p, q\in\mathbb{Z}\), \(p+q\ge 0\), but also on their connection with the Tchebychev form of the first kind \(\mathcal{T}=\mathcal{J}(-1/2, -1/2)\).
MSC:
| 33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |
| 42C05 | Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis |
Keywords:
orthogonal polynomials; classical and semiclassical forms; Stieltjes function; cubic decomposition; second degree formsReferences:
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