Taylor-like expansion in terms of a rational function obtained by means of fractional derivatives

In 2007, Tremblay and Fugère [The use of fractional derivatives to expand analytical functions in terms of quadratic functions with applications to special functions, Appl. Math. Comput. 187 (2007), pp. 507–529] motivated by a generalization of Taylor's series of f(z) obtained in 1971 by Osler [Taylor's series generalized for fractional derivatives and applications, SIAM J. Math. Anal. 2 (1971), pp. 37–48] presented 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. They also deduced the region of validity of this formula. In this paper, we present the power series expansion of an analytic function f(z) in ℛ in the case where t is equal to the rational function ((z−z 1)/(z−z 2)), q(t)=1, z 1≠z 2 and z 1 and z 2 are two arbitrary points in ℛ.

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