Higher Schwarzian Operators and Combinatorics of Schwarzian Derivative

摘要

A family of Mobius invariant nonlinear differential operators (higher Schwarzianoperators) is constructed in the context of the best Mobius approximations of analytic functions in one complex variable. Viewing the Schwarzian equation purely algebraically, we obtain a sequence of polynomials (Schwarzian polynomials) with positive integral coefficients expressing the higher Schwarzian operators in terms of derivatives of the classical Schwarzian derivative, and vice versa. These Schwarzian polynomials can be expressed in terms of the partial Bell polynomials. Integrality, positivity, and negativity properties of the Taylor coefficients of solutions to the Schwarzian equation are shown. The higher Schwarzian operators allow us to prove that any Mobius invariant differential operators can be derived from the classical Schwarzian derivative. In the Appendix, we give examples of Schwarzian polynomials.