A solution of Bessel's equation, (x^{2} y^{prime prime}+x y^{prime}+left(x^{2}-n^{2} ight) y=0), can be found using the guess

A solution of Bessel's equation, \(x^{2} y^{\prime \prime}+x y^{\prime}+\left(x^{2}-n^{2}\right) y=0\), can be found using the guess \(y(x)=\sum_{j=0}^{\infty} a_{j} x^{j+n}\). One obtains the recurrence relation \(a_{j}=\frac{-1}{j(2 n+j)} a_{j-2}\). Show that for \(a_{0}=\left(n!2^{n}\right)^{-1}\), we get the Bessel function of the first kind of order \(n\) from the even values \(j=2 k\) :

\[J_{n}(x)=\sum_{k=0}^{\infty} \frac{(-1)^{k}}{k!(n+k)!}\left(\frac{x}{2}\right)^{n+2 k}\]