The Fuchsian theory of linear differential equations, which is named after Lazarus Immanuel Fuchs, provides a characterization of various types of singularities and the relations among them.
At any ordinary point of a homogeneous linear differential equation of order
linearly independent power series solutions.
A non-ordinary point is called a singularity.
At a singularity the maximal number of linearly independent power series solutions may be less than the order of the differential equation.
denotes the falling factorial notation.
be a linear differential operator of order
be expandable as Laurent series with finite principle part at
is an ordinary point, the resulting indicial equation is given by
This implies that the degree of the indicial polynomial relative to
is equal to the order of the differential equation,
Certainly, at least one coefficient of the lower derivatives pushes the exponent of
Inevitably, the coefficient of a lower derivative is of smallest exponent.
The degree of the indicial polynomial relative to
We have given a homogeneous linear differential equation
with coefficients that are expandable as Laurent series with finite principle part.
The goal is to obtain a fundamental set of formal Frobenius series solutions relative to any point
This can be done by the Frobenius series method, which says: The starting exponents are given by the solutions of the indicial equation and the coefficients describe a polynomial recursion.
is an ordinary point, a fundamental system is formed by the
linearly independent formal Frobenius series solutions
is a regular singularity, one has to pay attention to roots of the indicial polynomial that differ by integers.
In this case the recursive calculation of the Frobenius series' coefficients stops for some roots and the Frobenius series method does not give an
The following can be shown independent of the distance between roots of the indicial polynomial: Let
-fold root of the indicial polynomial relative to
linearly independent formal solutions where
linearly independent formal solutions, because the indicial polynomial relative to a regular singularity is of degree
[4] One can show that a linear differential equation of order
linearly independent solutions of the form where
Hence, a differential equation is of Fuchsian type if and only if for all
there exists a fundamental system of Frobenius series solutions with