On the $n$-th derivative and the fractional integration of Bessel functions with respect to the order

TL;DR

This paper derives integral representations for the derivatives and fractional derivatives of Bessel functions with respect to their order, enabling efficient numerical evaluation and introducing new integral formulas.

Contribution

It provides new integral representations for derivatives and fractional derivatives of Bessel functions, along with efficient numerical methods and novel integral formulas.

Findings

Integral representations for derivatives of Bessel functions obtained

Efficient double exponential integration strategy developed

New integral expression for the Macdonald function derived

Abstract

We obtain integral representations of the $n$-th derivatives of the Bessel functions with respect to the order. The numerical evaluation of these expressions is very efficient using a double exponential integration strategy. Also, from the integral representation corresponding to the Macdonald function, we have calculated a new integral. Finally, we calculate integral expressions for the fractional derivatives of the Bessel functions with respect to the order. Simple proofs for some particular cases given in the literature are provided as well.