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UID:f4bd2d6e702f0519322f71f0464f81fe
CATEGORIES:Lie Group Quantum Mathematics Seminar
CREATED:20240520T214739
SUMMARY:Hypergeometric orthogonal polynomial families
LOCATION:Hill 705
DESCRIPTION:Motivated by the theory of hypergeometric orthogonal polynomials, we consid
er quasi-orthogonal polynomial families - those that are orthogonal with re
spect to a non-degenerate bilinear form defined by a linear functional - in
which the ratio of successive coefficients is given by a rational function
f(u,s) which is polynomial in u. We call this a family of Jacobi type. Our
main result is that, up to rescaling and renormalization, there are only f
ive families of Jacobi type.\nThese are the classical families of Jacobi, L
aguerre and Bessel polynomials, and two more one parameter families $E^c, F
^c$. Each family arises as a specialization of some hypergeometric series.
The last two families can also be expressed through Lommel polynomials, and
they are orthogonal with respect to a positive measure on the real line fo
r c>0 and c>-1 respectively.\nWe also consider the more general ratio
nal families, i.e. quasi-orthogonal families in which the ratio f(u,s) of s
uccessive coefficients is allowed to be rational in u as well. I will formu
late the two main theorems, one on Jacobi families and one on rational fami
lies, as well as the main ideas of the proofs. This is joint work with Jose
ph Bernstein and Siddhartha Sahi.\n
X-ALT-DESC;FMTTYPE=text/html:Motivated by the theory of hypergeometric orthogonal polynomials, we con
sider quasi-orthogonal polynomial families - those that are orthogonal with
respect to a non-degenerate bilinear form defined by a linear functional -
in which the ratio of successive coefficients is given by a rational funct
ion f(u,s) which is polynomial in u. We call this a family of Jacobi type.
Our main result is that, up to rescaling and renormalization, there are onl
y five families of Jacobi type.

These are the classical families of J
acobi, Laguerre and Bessel polynomials, and two more one parameter families
$E^c, F^c$. Each family arises as a specialization of some hypergeometric
series. The last two families can also be expressed through Lommel polynomi
als, and they are orthogonal with respect to a positive measure on the real
line for c>0 and c>-1 respectively.

We also consider the more
general rational families, i.e. quasi-orthogonal families in which the rati
o f(u,s) of successive coefficients is allowed to be rational in u as well.
I will formulate the two main theorems, one on Jacobi families and one on
rational families, as well as the main ideas of the proofs. This is joint w
ork with Joseph Bernstein and Siddhartha Sahi.

CONTACT:Dmitry Gourevitch, Weizmann Institute
DTSTAMP:20240714T111048
DTSTART;TZID=America/New_York:20240611T130000
DTEND;TZID=America/New_York:20240611T140000
SEQUENCE:0
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