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

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.