A new Leibniz rule and its integral analogue for fractional derivatives

In 1972, T.J. Osler proposed a generalization of the Leibniz rule for the fractional derivatives of the product of two functions with respect to an arbitrary function. This new rule was based on one of his own result on Taylor's series for the fractional derivative he obtained in 1971. Later, h...

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Published inIntegral transforms and special functions Vol. 24; no. 2; pp. 111 - 128
Main Authors Tremblay, R., Gaboury, S., Fugère, B.-J.
Format Journal Article
LanguageEnglish
Published Abingdon Taylor & Francis 01.02.2013
Taylor & Francis Ltd
Subjects
Analogue
Derivatives
fractional derivatives
integral analogue
Integrals
Leibniz rule
power series
Series expansion
special functions
Taylor series
Transforms
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Summary:In 1972, T.J. Osler proposed a generalization of the Leibniz rule for the fractional derivatives of the product of two functions with respect to an arbitrary function. This new rule was based on one of his own result on Taylor's series for the fractional derivative he obtained in 1971. Later, he gave the integral analogue of that new Leibniz rule. In this paper, we present a new Leibniz rule for the fractional derivatives of the product of two functions with respect to an arbitrary function and we give its integral analogue. Finally, new series expansions and definite integrals involving special functions are derived as special cases of the new Leibniz rule and the corresponding integral analogue.
Bibliography:ObjectType-Article-2
SourceType-Scholarly Journals-1
ObjectType-Feature-1
content type line 23
ISSN:1065-2469
1476-8291
DOI:10.1080/10652469.2012.668904