A nonlinear differential operator series that commutes with any function

@inproceedings{Olver1992AND,
  title={A nonlinear differential operator series that commutes with any function},
  author={Peter J. Olver},
  year={1992}
}
  • Peter J. Olver
  • Published 1992
  • Mathematics
  • A natural differential operator series is one that commutes with every function. The only linear examples are the formal series operators $e^{\alpha zD} $ representing translations. This paper discusses a surprising natural nonlinear “normally ordered” differential operator series, arising from the Lagrange inversion formula. The operator provides a wide range of new higher-order derivative identities and identities among Bell polynomials. These identities specialize to a large variety of… CONTINUE READING

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    Now, to complete the proof of Theorem 4, it suffices to notice that '(u), as defined by (A1), being a particular case of (A2), satisfies the three conditions of Lemma A3

    • But then the series @('(u)) also satisfies them since, for example, r((u)) '((u))y((u)) o

    Then , are uniquely determined by the function Vo(Uo)= @(Uo)

    • Then , are uniquely determined by the function Vo(Uo)= @(Uo)