Solving Differential Equations in Terms of Bessel Functions

40 CHAPTER 2. TRANSFORMATIONS
where λ1,λ2 ∈ Z and S1,S2 ∈ k((t))[ln(t)]. Thus,
ry + sy ′
e + λ3
= exp dt S3,
t
where λ3 ∈ Z depends on λ1 and λ2 and is such that the Puiseux series S3 ∈
k((t))[ln(t)] starts with a non-zero constant term. Hence, ē is a generalized exponents
of L and ē = e + λ3 = e mod Z.
The exponent difference ∆ has the following property.
Corollary 2.15 Two operators L1,L2 ∈ K[∂] with L1 −→EG L2 satisfy ∆(L1, p) =
∆(L2, p) mod Z for each point p, i.e. ∆(L1, p) mod Z is invariant under −→EG.
This result will be used in the following theorem.
Theorem 2.16 Let L ∈ K[∂] be a differential operator and let p be a fixed point.
If there exists an operator M ∈ K[∂] where p is regular such that M −→EG L, then
the solutions of L are not logarithmic and ∆(L, p) ∈ Z.
Proof. Let M and p be as required. Then there exist rational functions r0,r1,r2 ∈ K
and ˜M ∈ K[∂] such that
M r0 r1,r2
−→E ˜M −→G L.
Furthermore, let p be a regular point of M. The generalized exponents at p are
0 and 1. Hence, ∆(M, p) ∈ Z and from the previous corollary it follows that
∆(L, p) ∈ Z.
Let
y1 =
∞
∑
i=0
ait i p,a0 = 0 and y2 =
∞
∑
i=1
bit i p,b1 = 0
be linear independent local solutions of M at the point p, i.e.
V (M) = {c1y1 + c2y2 | c1,c2 ∈ K}.
After the exp-product the solution space is
V ( ˜M)
= exp (c1y1 + c2y2) c1,c2 ∈ K
and the gauge transformation changes this into
r0
V (L) = r1z + r2z ′ | z ∈ V ( ˜M) .