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Higher order coherent pairs. (English) Zbl 1262.42009

The present work concerns a study of \((1,1)\)-coherent pairs of order \(r\).
Let \(\{P_n\}_{n\geq0}\) and \(\{R_n\}_{n\geq0}\) be two sequences of monic orthogonal polynomials with respect to the quasi-definite linear functionals \(\mathcal{U}\), \(\mathcal{V}\), respectively, satisfy the relation \[ P_n^{[r]}(x)+a_{n-1,r}P_{n-1}^{[r]}(x)=R_{n-r}(x)+b_{n-1,r}R_{n-r-1}(x),\quad a_{n-1,r}\neq0,\;n\geq r+1, \] where \(f^{[r]}(x)=\frac{(n-r)!}{n!}f^{(r)}(x)\), \(f^{(r)}(x)\) denoting the \(r\)-th derivative of \(f(x)\) for a monic polynomial \(f\) of degree \(n\) and \(r\in \mathbb{N}\).
In this case, the pair \((\mathcal{U},\mathcal{V})\) is called \((1,1)\)-coherent pair of order \(r\); in the particular case when \(b_{n,r}=0,\,n\geq n+1\), \((\mathcal{U},\mathcal{V})\) is said to be a \((1,0)\)-coherent pair of order \(r\).
When \((\mathcal{U}\), \(\mathcal{V})\) are positive definite linear functionals, there exist two positive Borel measures \(\mu_0\) and \(\mu_1\) supported on the real line such that \[ \langle \mathcal{U},f\rangle=\int_{\mathbb{R}}f(x)d\mu_0, \langle \mathcal{V},f\rangle=\int_{\mathbb{R}}f(x)d\mu_1, \] for all polynomials \(f\).
In this case, the authors give necessary and sufficient conditions ensuring that the pair \((\mathcal{U},\mathcal{V})\) is a \((1,1)\)-coherent pair of order \(r\). Moreover, they compute the Sobolev orthogonal polynomials with respect to the inner product \[ \langle p(x),q(x)\rangle_{\lambda,r}=\int\limits_{-\infty}^{+\infty}p(x)q(x)d\mu_0+\lambda \int\limits_{-\infty}^{+\infty}p^{(r)}(x)q^{(r)}(x)d\mu_0,\,\lambda>0, \] where \(p,q\) are two polynomials.
For \((1,0)\) (or \((1,1)\))-coherent pairs of order \(r\) \((\mathcal{U}, \mathcal{V})\), the authors give some algebraic equations connecting the forms \(\mathcal{U}\), \(\mathcal{V}\) and \(D^k\mathcal{V}\), where \( D^k\mathcal{V}\) denotes the derivatives of the linear functional \(\mathcal{V}\), for \(k=1,2,3\) and \(r\geq k\).
In the case where \(\mathcal{U}\) is classical, they show that \(\mathcal{V}\) is semiclassical of class at most two.
Finally, some relations for the formal Stieltjes series associated with the linear functionals in a \((1,1)\)-coherent pair of order \(r\) are given.

MSC:

42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
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