Exact results for a fractional derivative of elementary functions

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Dear Editors,

We want to thank the Referee for reviewing our manuscript and providing valuable feedback.
The Referee concluded that our manuscript provides ``an interesting connection between fractional derivatives and special functions, specifically some algebraic functions.'' The Referee requested clarification of the paper regarding three major points: the justification of the semigroup property for the Caputo fractional derivative, inclusion of a proof of the relation between the Hermite polynomials of non-integer order $\alpha$ and the Tricomi confluent hypergeometric function, and the resolution of the connection between the Caputo fractional derivative of integer order $\alpha$ and the regular integer order derivative.
Following the Referee’s suggestions, we have modified the manuscript by addressing the raised points. Moreover, we have added an additional section on the exact evaluation of the Caputo derivative of a shifted polynomial and fixed numerous typos throughout the paper and references specified by the Referee.

As we have addressed all of the comments of the Referee, we hope our work will now be found suitable for SciPost. Please find below a detailed list of changes.

Sincerely,
Gavriil Shchedrin,
Nathanael C. Smith,
Anastasia Gladkina,
Lincoln D. Carr


{\itResponse to Referee}

1. ``The definition given by Eq.(1) is valid for $0<\alpha<1$ with $\alpha\in{\mathbb{R}}$. The cases $\alpha=0$ and $\alpha=1$ must be treat as separated cases. See Kilbas, Srivastava and Trujillo, reference [1], page 92. This occurs
also on pages 4, 7, 10, 11, 12, 13, specifically to obtain the graphics.''


Following the Referee’s suggestion we have included a paragraph right after Eq.(1) that now reads: ``Strictly speaking, the Caputo fractional derivative is given only for non-integer values of the fractional order, i.e., $\alpha\notin{\mathbb{N}}$ [1,2]. In the special case of the integer values of the parameter $\alpha=n\in{\mathbb{N}}$, the Caputo fractional derivative is {\it defined} in terms of the integer order derivative of the $n^{\rm th}$ order. We will find that except for the special case $\alpha=0$ the definition of the Caputo fractional derivative given by Eq.(1) for integer values of $\alpha=n\in{\mathbb{N}}$ is in fact consistent with the integer $n^{\rm th}$ order derivative. However, in the case of $\alpha=0$, the Caputo fractional derivative given by Eq.(1) results in a shift of the function by its value at the origin, i.e., ${}^{\rm C}D^{\alpha=0}_{x}f(x) = f(x)-f(0)$. Based on multiple examples of elementary functions ranging from trigonometric and inverse trigonometric functions to Gaussian and Lorentzian functions, we find that the Caputo fractional derivative of the order $0\leq{}\alpha\leq{1}$ {\it continuously} deforms the curve given by the Caputo fractional derivative of the $\alpha=0^{\rm th}$ order into $\alpha=1^{\rm st}$ order derivative. However, except for the special case $f(0)=0$, the requirement ${}^{\rm C}D^{\alpha=0}_{x}f(x) \equiv{} f(x)$ results in a discontinuous jump for the Caputo fractional derivative in the vicinity of $\alpha=0$. In order to avoid such a discontinuity, we impose the definition given by Eq.(1) for all $0\leq{}\alpha\leq{1}$.''


2. ``The authors must justify the validity of the semigroup property for Caputo fractional derivative.''

Following the Referee’s suggestion we have included a paragraph at the end of the second section that now reads:
`` The semi-group property for the Caputo fractional derivative of a factional order $0\leq{}\alpha\leq{1}$ can be directly established by expressing it in terms of the Riemann--Liouville fractional derivative,
$$
{}^{\rm C}D^{\alpha}_{x}[f(x)] =
{}^{\rm RL}D^{\alpha}_{x}[f(x) - f(0)]
$$
Thus, the semi-group property of the {\Cp} derivative follows directly from the
semi-group property of the Riemann--Liouville derivative [2]"

3.``The authors must justify the relation given by Eq.(57), because this relation is valid for n an integer number (polynomial case) and α is not an integer number."

Following the Referee’s suggestion we have included a paragraph at the end of Eq.(60) that now reads:
``
The Hermite polynomials, $H_{\alpha}(x) $, of a positive non-integer order $\alpha$ are directly related to the parabolic cylinder function, $D_{\alpha}(z)$. This correspondence can be shown via the integral representation of the Hermite polynomials, [41,42,50,51]
$$
H_{\alpha}(x) = \frac{2^{\alpha + 1}}{\sqrt{\pi}}e^{x^{2}}\int^{\infty}_{0}
e^{-t^{2}}t^{\alpha}\cos\left(2xt - \frac{\pi\alpha}{2}\right)dt =
\exp\left(\frac{x^2}{2}\right) 2^{\alpha /2} D_{\alpha }\left(\sqrt{2} x\right)
.
$$
On the order hand, the parabolic cylinder function can be expressed in terms of the Tricomi confluent hypergeometric function [41,42,50,51],
%
$$
D_{\alpha}(x) = \exp\left(-\frac{x^2}{4}\right) 2^{\alpha /2} U\left(-\frac{\alpha }{2},\frac{1}{2},\frac{x^2}{2}\right)
.
$$
Thus, the Hermite polynomials of non-integer order $\alpha$ are directly related to the Tricomi confluent hypergeometric function,
$$
H_{\alpha}(x) = 2^{\alpha}\,U\left(-\frac{\alpha}{2},\frac{1}{2},x^2\right).
$$
"

4. The Referee pointed out a number of misprints that have been promptly fixed.

5. Finally, we have added section 8 where we report the Caputo fractional derivative of a shifted polynomial.