The use of fractional derivatives to expand analytical functions in terms of quadratic functions with applications to special functions
type: Item
instance of: scholarly article
title: The use of fractional derivatives to expand analytical functions in terms of quadratic functions with applications to special functions
zbMATH Open document ID: 1116.33001
published in: Applied Mathematics and Computation
publication date: 2007-06-13T00:00:00Z
review text: The authors consider a new expansion of an analytic function \(f(z)\) in \(\mathbb R\) in terms of a power series \(\theta(t) = tq(t)\), where \(q(t)\) is any regular function and is equal to the quadratic function \(\left[ {\left( {z - z_1 } \right)\left( {z - z_2 } \right)} \right], z_1 \neq z_2.\) It is shown that the coefficient of \((z - z_1 )^n (z - z_2 )^n\) in the power series of the function \((z - z_1 )^\alpha (z - z_2 )^\beta f(z)\) is \[ D_{z_1 - z_2 }^{ - \alpha + n} \left[ {f(z_1 )(z_1 - z_2 )^{\beta - n - 1} (z_1 - z_2 + z - w)} \right]\left. {} \right|_{w = z_1 } /\Gamma (1 - \alpha + n), \] where \(\alpha\) and \(\beta\) are arbitrary complex numbers. The derivation of the main results makes use of the multi-loop contour integration (Pochhammer contour), and several special forms are mentioned giving some new identities involving special functions and integrals. The paper contains a comprehensive list of references which would be of immense benefit to the interested reader.
Mathematics Subject Classification ID: 26A33
Mathematics Subject Classification ID: 30B10
Mathematics Subject Classification ID: 33C99
zbMATH DE Number: 5163959
zbMATH Keywords: Fractional derivatives
zbMATH Keywords: Taylor's theorem
zbMATH Keywords: Laurent series
zbMATH Keywords: Power series
zbMATH Keywords: Special functions
author: R. Tremblay
author: Benoît-Jean Fugère
reviewed by: Ravinder Krishna Raina
MaRDI profile type: Publication
full work available at URL: j.amc.2006.09.076
OpenAlex ID: W2022110967
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DOI: 10.1016/J.AMC.2006.09.076
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