Modified Bessel Functions
We shall develop the theory of modified Bessel functions from their gen
erating function. Explicitly, define
f
(
t, u
) = exp
{
1
2
t
(
u
+
u

1
)
}
for
t
and
u
6
= 0 real (though we may also allow complex when appropriate).
Note that if in the (absolutely convergent) expansion
f
(
t, u
) =
X
a
>
0
1
a
!
(
t
2
)
a
u
a
X
b
>
0
1
b
!
(
t
2
)
b
u

b
we collect together terms according to powers of
u
then we obtain
f
(
t, u
) =
X
n
∈
Z
I
n
(
t
)
u
n
where (letting
k
=
b
and
a
=
b
+
n
) if
n
>
0 then
I

n
=
I
n
and
I
n
(
t
) =
X
k
>
0
(
t
2
)
2
k
+
n
k
!(
k
+
n
)!
= (
t
2
)
n
X
k
>
0
(
t
2
)
2
k
k
!(
k
+
n
)!
is the
modified Bessel function
of order
n
.
Differentiation of
f
(
t, u
) by
t
and
u
yields relations between consecutive
modified Bessel functions and their derivatives. By
t
:
∂f
∂t
=
1
2
(
u
+
u

1
)
f
whence
X
n
∈
Z
I
0
n
(
t
)
u
n
=
1
2
X
n
∈
Z
(
I
n
(
t
)
u
n
+1
+
I
n
(
t
)
u
n

1
) =
X
n
∈
Z
(
I
n

1
(
t
) +
I
n
+1
(
t
))
u
n
1
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and therefore
2
I
0
n
(
t
) =
I
n

1
(
t
) +
I
n
+1
(
t
)
.
By
u
:
∂f
∂u
=
1
2
t
(1

u

2
)
f
whence
X
n
∈
Z
I
n
(
t
)
nu
n

1
=
1
2
t
X
n
∈
Z
(
I
n
(
t
)
u
n

I
n
(
t
)
u
n

2
) =
1
2
t
X
n
∈
Z
(
I
n

1
(
t
)

I
n
+1
(
t
))
u
n

1
and therefore
2
nI
n
(
t
) =
t
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 Summer '10
 Staff
 Algebra, Recurrence relation, 2k, Bessel function, Helmholtz equation

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