Reflection formulas for order derivatives of Bessel functions

TL;DR

This paper derives new reflection formulas for derivatives of Bessel functions with respect to order, extending existing results and calculating a previously unreported integral, with applications to negative integral orders.

Contribution

It introduces new integral representations and reflection formulas for derivatives of Bessel functions, extending known formulas to negative orders and providing novel integral evaluations.

Findings

Reflection formulas for first and second derivatives of Bessel functions at integral orders.

Extension of formulas to negative integral orders.

Calculation of a new integral related to Bessel functions.

Abstract

From new integral representations of the $n$-th derivative of Bessel functions with respect to the order, we derive some reflection formulas for the first and second order derivative of $J_{\nu }\left( t\right) $ and $% Y_{\nu }\left( t\right) $ for integral order, and for the $n$-th order derivative of $I_{\nu }\left( t\right) $ and $K_{\nu }\left( t\right) $ for arbitrary real order. As an application of the reflection formulas obtained for the first order derivative, we extend some formulas given in the literature to negative integral order. Also, as a by-product, we calculate an integral which does not seem to be reported in the literature.