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Taylor-like expansion in terms of a rational function obtained by means of fractional derivatives. (English) Zbl 1270.26007

Using the Cauchy integral formula for fractional derivatives, T. Osler [SIAM J. Math. Anal. 2, 37–48 (1971; Zbl 0215.12101)], generalized Taylor’s series for fractional derivatives. Motivated by this work, R. Tremblay and B. J. Fugère [Appl. Math. Comput. 187, No. 1, 507–529 (2007; Zbl 1116.33001)] obtained the power series of an analytic function \(f (z)\) in terms of quadratic functions with some applications. They also deduced the region of validity of the derived formula. In this paper the authors obtain the power series of an analytic function \(f (z)\) in terms of the rational expression \((z-z_1)/ (z-z_2)\) where \(z_1\) and \(z_2\) are two arbitrary points inside the region of convergence \(R\) of \(f (z)\). The authors provide certain definitions of the fractional derivative using Pochhammer’s contour. Some examples of expansions in terms of a rational function are given.

MSC:

26A33 Fractional derivatives and integrals
33C20 Generalized hypergeometric series, \({}_pF_q\)
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