Fuchsian equivalence and Fuchsian classification
A (formal, genuine) Fuchsian operator of order is a (formal, resp., converging) series of the form with the coefficients from the ring of polynomials in the variable with .
The polynomial is called the Euler part of .
Two Fuchsian operators are -equivalent (formally or analytically), if there exist two Fuchsian operators such that and the Euler parts of are mutually prime in .
Unlike the Weyl algebra , the collection of Fuchsian operators is not a subalgebra, although it is “multiplicatively” (compositionally) closed.
The Fuchsian equivalence is indeed reflexive, transitive and symmetric. The first two properties are obvious, to prove the last one an additional effort is required. Indeed, for two Fuchsian operators of order with mutually prime Euler parts, one can construct two operators with holomorphic coefficients so that the identity holds, but in the leading terms of may well degenerate, thus violating the Fuchsian condition. However, one can always find such pair of operators of order greater by 1, which will still be Fuchsian. The rest is easy.
The following results can be proved by more or less direct computation in the algebra :
- A Fuchsian operator with a nonresonant Euler part is -equivalent to its Euler part.
- Any Fuchsian operator is -equivalent to a polynomial operator from .
- Any Fuchsian operator is -equivalent to a polynomial operator of the form with being polynomials without free terms, , which is Liouville integrable.
- A Fuchsian operator has trivial (identical) monodromy if and only if it is -equivalent to an Euler operator with pairwise different integer roots. The corresponding equation has an apparent singularity (all solutions are analytic) if and only if all these roots are pairwise different nonnegative integers.