We can rewrite Bessel functions, \(J_{v}(x)\), in a form that will allow the order to be non-integer by using the gamma function. You will need the results from Problem \(12 b\) for \(\Gamma\left(k+\frac{1}{2}\right)\).

a. Extend the series definition of the Bessel function of the first kind of order \(v, J_{v}(x)\), for \(v \geq 0\) by writing the series solution for \(y(x)\) in Problem 13 using the gamma function.

b. Extend the series to \(J_{-v}(x)\), for \(v \geq 0\). Discuss the resulting series and what happens when \(v\) is a positive integer.

c. Use these results to obtain the closed form expressions

\[\begin{aligned} J_{1 / 2}(x) & =\sqrt{\frac{2}{\pi x}} \sin x \\ J_{-1 / 2}(x) & =\sqrt{\frac{2}{\pi x}} \cos x \end{aligned}\]

d. Use the results in part

c. with the recursion formula for Bessel functions to obtain a closed form for \(J_{3 / 2}(x)\).

Data from Problem 12

In Maple, one can type simplify(LegendreP(2*n-2,0)-LegendreP(2* \(\mathbf{n}, \mathbf{0})\) ); to find a value for \(P_{2 n-2}(0)-P_{2 n}(0)\). It gives the result in terms of Gamma functions. However, in Example 6. 10 for Fourier-Legendre series, the value is given in terms of double factorials! So, we have

\[P_{2 n-2}(0)-P_{2 n}(0)=\frac{\sqrt{\pi}(4 n-1)}{2 \Gamma(n+1) \Gamma\left(\frac{3}{2}-n\right)}=(-1)^{n-1} \frac{(2 n-3)!!}{(2 n-2)!!} \frac{4 n-1}{2 n}\]

Data from Problem 13

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 \(j=2 k\) :

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