Session S32 - Special functions and orthogonal polynomials

Monday, July 12, 11:00 ~ 12:00 UTC-3

Linear spectral transforms, matrix factorizations and orthogonal polynomials

Francisco Marcellán

Universidad Carlos III de Madrid, Spain   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

Let $u$ be a quasi-definite linear functional defined on the linear space of polynomials $\mathbb{P}.$ For such a functional we can define a sequence of monic orthogonal polynomials (SMOP in short) $(P_n)_{n\geq 0},$ which satisfies a three term recurrence relation. Shifting one unity the recurrence coefficient indices we get the sequence of associated polynomials of the first kind$(P_n^{(1)})_{n\geq 0}$ which are orthogonal with respect to a linear functional denoted by $u^{(1)}$.

In the literature two special spectral transformations of the functional $u$ are studied: the canonical Christoffel transformation $\widetilde{u}=(x-c) u$ and the canonical Geronimus transformation $\widehat {u}= (x-c)^{-1} {u}+M\delta_c$ , where $c$ is a fixed complex number, $M $ is a free parameter and $\delta_c$ is the linear functional defined on $\mathbb{P}$ as $<\delta_{c},p(x) >=p(c).$ They constitue a generating system of the so called linear spectral transformation set analyzed in [2]. For the Christoffel transformation with SMOP $(\widetilde P_n)_{n\geq 0}$, we are interested in analyzing the relation between the linear functionals $ u^{(1)}$ and $\widetilde{u}^{(1)}.$ There, the super index denotes the linear functionals associated with the orthogonal polynomial sequences of the first kind $(P_n^{(1)})_{n\geq 0}$ and $(\widetilde P_n^{(1)})_{n\geq 0},$ respectively. This problem is also studied for Geronimus transformations. Here we give close relations between their corresponding monic Jacobi matrices by using the LU and UL factorizations.

For more information, see [1].

References

[1] J. C. García Ardila, F. Marcellán, P. H. Villamil-Hernández, \textit{Associated orthogonal polynomials of the first kind and Darboux transformations}, arXiv:2103.0232, [math.CA] 23 March 2021

[2] A. Zhedanov, \textit{Rational spectral transformations and orthogonal polynomials}, J. Comput. Appl. Math. \textbf{85} (1997), 67-86.

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Joint work with Juan Carlos Garcia Ardila (Universidad Politécnica de Madrid, España) and Paul H. Villamil-Hernández (Universidad Carlos III de Madrid, España).

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