8. Let n ~ 0 be a fixed nonnegative integer and recall that O! := 1. The Bessel function of order n is the function defined by 00 (-l)k (x)n+2k Bn(x) := {; (k!)(n + k)!"2 .

(a) show that Bn (x) converges pointwise on R and uniformly on any closed interval [a, b].

(b) Prove that y = Bn (x) satisfies the differential equation X2y" + xy' + (x2 - n2)y = 0 for x E R.

( c ) Prove that for n E N and x E R.