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A CHARACTERIZATION OF ULTRASPHERICAL POLYNOMIALS MICHAEL ANSHELEVICH
 

Summary: A CHARACTERIZATION OF ULTRASPHERICAL POLYNOMIALS
MICHAEL ANSHELEVICH
ABSTRACT. We show that the only orthogonal polynomials with a generating function of the form
F xz - z2
are the ultraspherical, Hermite, and Chebyshev polynomials of the first kind. The
generating function for the Chebyshev case is non-standard, although it is easily derived from the
usual one.
1. THE QUESTION
Hermite polynomials Hn(x) are one of the most important families of orthogonal polynomials in
mathematics, appearing in probability theory, mathematical physics, differential equations, combi-
natorics, etc. One of the simplest ways to construct them is through their generating function,

n=0
1
n!
Hn(x)zn
= exp xz - z2
/2 .
On the other hand, Chebyshev polynomials of the second kind Un(x) are another important fam-
ily of orthogonal polynomials (appearing in numerical analysis, for example), with a generating

  

Source: Anshelevich, Michael - Department of Mathematics, Texas A&M University

 

Collections: Mathematics