4.5. Consider Hermite's differential equation (4.5) and determine two linearly independent solutions. Show that when v = n > 0 just one element of this pair is a polynomial of degree n. Denote this polynomial by Hn(z), the Hermite polynomial. Normalize Hn(z) such that the coefficient of zn equals 2n and verify the following special cases:
H0(z) = 1, H1(z) = 2z, H2(z) = 4z^2 - 2, H3(z) = 8z^3 - 12z, H4(z) = 16z^4 - 48z^2 + 12, H5(z) = 32z^5 - 160z^3 + 120z