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

Abstract In 1971, T.J. Osler propose a generalization of Taylor’s series of f ( z ) in which the general term is [ D z 0 - b an + γ f ( z 0 ) ] ( z - z 0 ) an + γ / Γ ( an + γ + 1 ) , where 0  a  ⩽ 1, b  ≠  z 0 and γ is an arbitrary complex number and D z α is the fractional derivative of order α . In this paper, we present a new expansion of an analytic function f ( z ) in R in terms of a power series θ ( t ) =  tq ( t ), where q ( t ) is any regular function and t is equal to the quadratic function [( z  −  z 1 )( z  −  z 2 )] , z 1  ≠  z 2 , where z 1 and z 2 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  −  z 1 ) n ( z  −  z 2 ) n in the power series of the function ( z  −  z 1 ) α ( z  −  z 2 ) β f ( z ) is D z 1 - z 2 - α + n [ f ( z 1 ) ( z 1 - z 2 ) β - n - 1 ( z 1 - z 2 + z - w ) ] | w = z 1 / 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.

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