Result: The use of fractional derivatives to expand analytical functions in terms of quadratic functions with applications to special functions

In 1971, T.J. Osler propose a generalization of Taylor's series of f(z) in which the general term is [Dan+γ z0-f(z0)](z-z0)an+γ/Γ(an + γ+ 1), where 0 < a ≤ 1, b ≠ z0 and y is an arbitrary complex number and Dαz is the fractional derivative of order α. In this paper, we present a new expansion of an analytic function f(z) in 9? in terms of a power series 0(t) = tq(t), where q(t) is any regular function and t is equal to the quadratic function [(z - z1)(z- z2)], z1 ≠z2, where z1 and z2 are two points in R and the region of validity of this formula is also deduced. To illustrate the concept, if q(t)= 1, the coefficient of (z - zi)n(z- z2)n in the power series of the function (z - z1)α(z- z2)βf(z) is D -α+n z1-z2 [f(z1)(z1-z2 β-n-1(z1-z2 +z- w)] |w=z1/Γ(1-α+n) where α and β are arbitrary complex numbers. Many special forms are examined and some new identities involving special functions and integrals are obtained.