# Article ID: MTJPAM-D-20-00021

## Title: Fractional Derivatives of Logarithmic Singular Functions and Applications to Special Functions

Montes Taurus J. Pure Appl. Math. / ISSN: 2687-4814

Article ID: MTJPAM-D-20-00021; Volume 3 / Issue 1 / Year 2021, Pages 7-37

Document Type: Research Paper

Author(s): Richard Tremblay a

aDépartement d’Informatique et Mathématique, Université du Québec à Chicoutimi, Chicoutimi, Qué., Canada G7H 2B1

Received: 26 July 2020, Accepted: 14 October 2020, Available online: 7 January 2021.

Corresponding Author: Richard Tremblay (Email address: rtrembla@uqac.ca)

Full Text: PDF

Abstract

In 1974, Lavoie, Tremblay and Osler (Fundamental properties of fractional derivatives via Pochhammer integrals in ‘Fractional calculus and its applications’, Lecture Notes in Mathematics No.457, Springer-Verlag, (1974), 323-356) introduced a Pochhammer integral representation in the complex plane for the fractional derivative $D_{z}^{\alpha}\, z^{p} (\ln \ z)^{\delta} f(z)$ where $\delta = 0$ or 1. In the same vein, we present integral representations for the fractional derivative of functions with multiple branch-points (complex power, logarithm and their product) $D_{z-z_{0}}^{\alpha}U_{ \delta,\, \theta; \, p, \, q} \, (z-z_{0}, w-z)\big|_{w=z}^{*}$ where $U_{\delta,\, \theta; \, p, \, q} \, (z-z_{0}, w-z)= f(z-z_{0},w-z)(z-z_{0})^{p}(w-z)^{q} \ [\ln(z – z_{0}) ]^{\delta} \[\ln(w-z)]^{\theta}$ for value $\delta, \, \theta = 0$ or 1 using a Pochhammer contour integral enclosing the singularity points $z_{0}$, $z$ and $w$. The symbol (*) indicates that $w \rightarrow z$ inside the Pochhammer contour used for the representation. The transformation formula for the fractional operator $D_{z-z_{0}}^{\alpha}U_{ \delta,\, \theta; \, p+r, \, q}\,(z-z_{0},w-z)\big|_{w=z}^{*}=\frac{\Gamma(1+p)}{\Gamma(-\alpha)} D_{z-z_{0}^{}}^{-p-1} U_{ \delta,\,\theta; r; \, q-\alpha-1} \, (w-z,z-z_{0})\big|_{w=z}^{*}$ is derived. Some applications to special functions are given; in particular, a new form of the Leibniz rule is obtained. Another application includes many summation formulas involving the orthogonal polynomials and deduced from the Christoffel-Darboux identity for orthogonal polynomials.

Keywords: Fractional derivatives, Pochhammer contour, Transformation formulas, Special functions, Leibniz rules, Christoffel-Darboux formula, Logarithmic singular function

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