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

Abstract

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

Functional Power Series

“…Boas [20] worked with function expansions and developed some useful theorems. Campos [17] has also made a good effort in this direction by using diffintegration concepts. The results are generalizations of the known series expansions.…”

Further properties of Osler's generalized fractional integrals and derivatives with respect to another function

“…We mention the papers [22,23,26], where a similar concept of fractional derivative was studied, and a Leibniz and chain formulas were proven. In [6], a Taylor's formula is extended to the same type of derivative operator, and in [11], summation formulas are obtained by using the generalized chain rule. In [17,34], that concept of fractional derivative is generalized by means of a representation based on the Pochhammer's contour of integration.…”