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.


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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