## 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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### References:

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