## View abstract

### 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. document.getElementById('cloak005571c81ce82d038ce4b64f38f3828b').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy005571c81ce82d038ce4b64f38f3828b = 'p&#97;c&#111;m&#97;rc' + '&#64;'; addy005571c81ce82d038ce4b64f38f3828b = addy005571c81ce82d038ce4b64f38f3828b + '&#105;ng' + '&#46;' + '&#117;c3m' + '&#46;' + '&#101;s'; var addy_text005571c81ce82d038ce4b64f38f3828b = 'p&#97;c&#111;m&#97;rc' + '&#64;' + '&#105;ng' + '&#46;' + '&#117;c3m' + '&#46;' + '&#101;s';document.getElementById('cloak005571c81ce82d038ce4b64f38f3828b').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy005571c81ce82d038ce4b64f38f3828b + '\'>'+addy_text005571c81ce82d038ce4b64f38f3828b+'<\/a>';

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.