On differential equations for Sobolev-type Laguerre polynomials
Authors: Koekoek J. , Koekoek R. , Bavinck H. 
Published: July 2002

The Sobolev-type Laguerre polynomials

{

L
n

α
,
M
,
N

(
x
)

}

n
=
0

∞

\{L_n^{\alpha ,M,N}(x)\}_{n=0}^{\infty }

are orthogonal with respect to the inner product
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mathematical left-angle f comma g mathematical right-angle equals StartFraction 1 Over normal upper Gamma left-parenthesis alpha plus 1 right-parenthesis EndFraction integral Subscript 0 Superscript normal infinity Baseline x Superscript alpha Baseline e Superscript negative x Baseline f left-parenthesis x right-parenthesis g left-parenthesis x right-parenthesis d x plus upper M f left-parenthesis 0 right-parenthesis g left-parenthesis 0 right-parenthesis plus upper N f prime left-parenthesis 0 right-parenthesis g prime left-parenthesis 0 right-parenthesis comma">

⟨
f
,
g
⟩

=

1

Γ
(
α
+
1
)

∫
0

∞

x

α

e

−
x

f
(
x
)
g
(
x
)
d
x
+
M
f
(
0
)
g
(
0
)
+
N

f
′

(
0
)

g
′

(
0
)
,

\langle f,g\rangle \;=\frac {1}{\Gamma (\alpha +1)}\int _0^{\infty }x^{\alpha }e^{-x}f(x)g(x)dx+Mf(0)g(0)+ Nf’(0)g’(0),

\]
where

α
>
−
1

\alpha >-1

,

M
≥
0

M\ge 0

and

N
≥
0

N\ge 0

. In 1990 the first and second author showed that in the case

M
>
0

M>0

and

N
=
0

N=0

the polynomials are eigenfunctions of a unique differential operator of the form
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M sigma-summation Underscript i equals 1 Overscript normal infinity Endscripts a Subscript i Baseline left-parenthesis x right-parenthesis upper D Superscript i Baseline plus x upper D squared plus left-parenthesis alpha plus 1 minus x right-parenthesis upper D comma">

M

∑

i
=
1

∞

a
i

(
x
)

D
i

+
x

D
2

+
(
α
+
1
−
x
)
D
,

M\sum _{i=1}^{\infty }a_i(x)D^i+xD^2+(\alpha +1-x)D,

\]
where

{

a
i

(
x
)
}

i
=
1

∞

\left \{a_i(x)\right \}_{i=1}^{\infty }

are independent of

n
n

. This differential operator is of order

2
α
+
4

2\alpha +4

if

α
\alpha

is a nonnegative integer, and of infinite order otherwise. In this paper we construct all differential equations of the form
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row 1st Column Blank 2nd Column a m p semicolon upper M sigma-summation Underscript i equals 0 Overscript normal infinity Endscripts a Subscript i Baseline left-parenthesis x right-parenthesis y Superscript left-parenthesis i right-parenthesis Baseline left-parenthesis x right-parenthesis plus upper N sigma-summation Underscript i equals 0 Overscript normal infinity Endscripts b Subscript i Baseline left-parenthesis x right-parenthesis y Superscript left-parenthesis i right-parenthesis Baseline left-parenthesis x right-parenthesis 2nd Row 1st Column Blank 2nd Column a m p semicolon plus upper M upper N sigma-summation Underscript i equals 0 Overscript normal infinity Endscripts c Subscript i Baseline left-parenthesis x right-parenthesis y Superscript left-parenthesis i right-parenthesis Baseline left-parenthesis x right-parenthesis plus x y left-parenthesis x right-parenthesis plus left-parenthesis alpha plus 1 minus x right-parenthesis y prime left-parenthesis x right-parenthesis plus n y left-parenthesis x right-parenthesis equals 0 comma EndLayout">

a
m
p
;
M

∑

i
=
0

∞

a
i

(
x
)

y

(
i
)

(
x
)
+
N

∑

i
=
0

∞

b
i

(
x
)

y

(
i
)

(
x
)

a
m
p
;

+
M
N

∑

i
=
0

∞

c
i

(
x
)

y

(
i
)

(
x
)
+
x
y
(
x
)
+
(
α
+
1
−
x
)

y
′

(
x
)
+
n
y
(
x
)
=
0
,

\begin{align} &M\sum _{i=0}^{\infty }a_i(x)y^{(i)}(x)+ N\sum _{i=0}^{\infty }b_i(x)y^{(i)}(x)\\ &\hspace {1cm}{}+MN\sum _{i=0}^{\infty }c_i(x)y^{(i)}(x)+ xy(x)+(\alpha +1-x)y’(x)+ny(x)=0, \end{align}

where the coefficients

{

a
i

(
x
)
}

i
=
1

∞

\left \{a_i(x)\right \}_{i=1}^{\infty }

,

{

b
i

(
x
)
}

i
=
1

∞

\left \{b_i(x)\right \}_{i=1}^{\infty }

and

{

c
i

(
x
)
}

i
=
1

∞

\left \{c_i(x)\right \}_{i=1}^{\infty }

are independent of

n
n

and the coefficients

a
0

(
x
)

a_0(x)

,

b
0

(
x
)

b_0(x)

and

c
0

(
x
)

c_0(x)

are independent of

x
x

, satisfied by the Sobolev-type Laguerre polynomials

{

L
n

α
,
M
,
N

(
x
)

}

n
=
0

∞

\{L_n^{\alpha ,M,N}(x)\}_{n=0}^{\infty }

. Further, we show that in the case

M
=
0

M=0

and

N
>
0

N>0

the polynomials are eigenfunctions of a linear differential operator, which is of order

2
α
+
8

2\alpha +8

if

α
\alpha

is a nonnegative integer and of infinite order otherwise. Finally, we show that in the case

M
>
0

M>0

and

N
>
0

N>0

the polynomials are eigenfunctions of a linear differential operator, which is of order

4
α
+
10

4\alpha +10

if

α
\alpha

is a nonnegative integer and of infinite order otherwise.