
A Generalization of the Weyl Algebra 337
Remark 8.145 In the proof of Proposition 8.144 we used the structure of higher derivatives
of f
/f. In [607] (see also [38]) combinatorial aspects of higher derivatives of inverses 1/f
were considered. More recently, Jakimczuk [590] considered higher derivatives of arbitrary
fractions h/f .
Combining (8.81) and Proposition 8.144, we conclude that Be
s;h
satisfies an algebraic
differential equation, provided Se
s;h
satisfies an algebraic differential equation.
Lemma 8.146 Let h ∈ C \{0}.ThenSe
s;h
satisfies for all s ∈ R the algebraic differential
equation
Se
s;h
Se
s;h
− (s +1)(Se
s;h
)
2
=0. (8.134)
In the particular case ...