Reflection Formulas for Order Derivatives of Bessel Functions

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 derivatives of J(v) (t) and Y-v (t) for integral order, and for the n-th order derivatives of I-v (t) and K-v (t) 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.

Automated summary

From new integral representations of the n-th derivative of Bessel functions with respect to the order, this paper derives reflection formulas for the first and second order derivatives of J(v)(t) and Y-v(t) for integral order, and for the n-th order derivatives of I-v(t) and K-v(t) for arbitrary real order. These reflection formulas are applied to extend some formulas given in the literature to negative integral order, and also lead to the calculation of an integral that has not been reported before.