In mathematics, a Szegő polynomial is one of a family of orthogonal polynomials for the Hermitian inner product
⟨
f
|
g
⟩
=
∫
−
π
π
f
(
e
i
θ
)
g
(
e
i
θ
)
¯
d
μ
{\displaystyle \langle f|g\rangle =\int _{-\pi }^{\pi }f(e^{i\theta }){\overline {g(e^{i\theta })}}\,d\mu }
where dμ is a given positive measure on [−π, π]. Writing
ϕ
n
(
z
)
{\displaystyle \phi _{n}(z)}
for the polynomials, they obey a recurrence relation
ϕ
n
+
1
(
z
)
=
z
ϕ
n
(
z
)
+
ρ
n
+
1
ϕ
n
∗
(
z
)
{\displaystyle \phi _{n+1}(z)=z\phi _{n}(z)+\rho _{n+1}\phi _{n}^{*}(z)}
where
ρ
n
+
1
{\displaystyle \rho _{n+1}}
is a parameter, called the reflection coefficient or the Szegő parameter.