The principal objective here is to look at some algebraic properties of the orthogonal polynomials
Ψ
n
(
b
,
s
,
t
)
\Psi _n^{(b,s,t)}
with respect to the Sobolev inner product on the unit circle
\[
⟨
f
,
g
⟩
S
(
b
,
s
,
t
)
=
(
1
−
t
)
⟨
f
,
g
⟩
μ
(
b
)
+
t
f
(
1
)
¯
g
(
1
)
+
s
⟨
f
′
,
g
′
⟩
μ
(
b
+
1
)
,
\langle f,g\rangle _{S^{(b,s,t)}} = (1-t)\, \langle f,g\rangle _{\mu ^{(b)}} + t\, \overline {f(1)}\,g(1) + s\, \langle f^{\prime },g^{\prime }\rangle _{\mu ^{(b+1)}},
\]
where
⟨
f
,
g
⟩
μ
(
b
)
=
τ
(
b
)
2
π
∫
0
2
π
f
(
e
i
θ
)
¯
g
(
e
i
θ
)
(
e
π
−
θ
)
I
m
(
b
)
(
sin
2
(
θ
/
2
)
)
R
e
(
b
)
d
θ
.
\langle f,g\rangle _{\mu ^{(b)}} = \frac {\tau (b)}{2\pi } \int _{0}^{2\pi }\overline {f(e^{i\theta })} \,g(e^{i\theta })\,(e^{\pi -\theta })^{\mathcal {I}m(b)} (\sin ^{2}(\theta /2))^{\mathcal {R}e(b)} d\theta .
Here,
R
e
(
b
)
>
−
1
/
2
\mathcal {R}e(b) > -1/2
,
0
≤
t
>
1
0 \leq t > 1
,
s
>
0
s > 0
and
τ
(
b
)
\tau (b)
is taken to be such that
⟨
1
,
1
⟩
μ
(
b
)
=
1
\langle 1,1\rangle _{\mu ^{(b)}} = 1
. We show that, for example, the monic Sobolev orthogonal polynomials
Ψ
n
(
b
,
s
,
t
)
\Psi _n^{(b,s,t)}
satisfy the recurrence
Ψ
n
(
b
,
s
,
t
)
(
z
)
−
β
n
(
b
,
s
,
t
)
Ψ
n
−
1
(
b
,
s
,
t
)
(
z
)
=
Φ
n
(
b
,
t
)
(
z
)
,
\Psi _n^{(b,s,t)}(z) - \beta _n^{(b,s,t)} \Psi _{n-1}^{(b,s,t)}(z) = \Phi _n^{(b,t)}(z),
n
≥
1
n \geq 1
, where
Φ
n
(
b
,
t
)
\Phi _n^{(b,t)}
are the monic orthogonal polynomials with respect to the inner product
⟨
f
,
g
⟩
μ
(
b
,
t
)
=
(
1
−
t
)
⟨
f
,
g
⟩
μ
(
b
)
+
t
f
(
1
)
¯
g
(
1
)
\langle f,g\rangle _{\mu ^{(b,t)}} = (1-t)\, \langle f,g\rangle _{\mu ^{(b)}} + t\, \overline {f(1)}\,g(1)
. Some related bounds and asymptotic properties are also given.