 
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 nonstandard, 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
