Reflection formulas for order derivatives of 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.