On generalizations of the series of Taylor, Lagrange, Laurent and Teixeira

The classical theorems of Taylor, Lagrange, Laurent and Teixeira, are extended from the representation of a complex function F ( z ) , to its derivative F ( ν ) ( z ) of complex order ν , understood as either a Liouville (1832) or a Rieman (1847) differintegration (Campos 1984, 1985); these results are distinct from, and alternative to, other extensions of Taylor's series using differintegrations (Osler 1972, Lavoie & Osler & Tremblay 1976). We consider a complex function F ( z ) , which is analytic (has an isolated singularity) at ζ , and expand its derivative of complex order F ( ν ) ( z ) , in an ascending (ascending-descending) series of powers of an auxiliary function f ( z ) , yielding the generalized Teixeira (Lagrange) series, which includes, for f ( z ) = z − ζ , the generalized Taylor (Laurent) series. The generalized series involve non-integral powers and/or coefficients evaluated by fractional derivatives or integrals, except in the case ν = 0 , when the classical theorems of Taylor (1715), Lagrange (1770), Laurent (1843) and Teixeira (1900) are regained. As an application, these generalized series can be used to generate special functions with complex parameters (Campos 1986), e.g., the Hermite and Bessel types.