Question

(Bessel Functions) I have a question regarding the proof of a property of a Bessel Function, LaTeX of the relevant part posted below. After taking the derivative once w.r.t. x of the Bessel Function, the index shifts by one, and I can't figure out why-is taking the derivative of the function *causing* it to shift?

Answer 1

\[\frac{d}{dx}\bigg[x^{-n}J_n(x)\bigg]=\frac{d}{dx}\sum_{k=0}^{\infty}\frac{(-1)^k x^{2k}}{2^{n+2k}k! \ \Gamma(n+k+1)}\]\[=\sum_{k=1}^\infty \frac{(-1)^k(2k)x^{2k-1}}{2^{n+2k}k! \ \Gamma(n+k+1)}\]

What exactly is happening in between these two steps?

Answer 2

When \(k=0\), the numerator is zero, so we can skip ahead to \(k=1\).