(a) Laguerre’s differential equation
ty” + (1 – t)y’ + ny = 0
is known to possess polynomial solutions when n is a nonnegative integer. These solutions are naturally called Laguerre polynomials and are denoted by Ln(t). Find y = Ln(t), for n = 0, 1, 2, 3, 4 if it is known that Ln(0) = 1.
(b) Show that
Where Y(s) = ℒ{y} and y = Ln(t) is a polynomial solution of the DE in part (a). Conclude that
This last relation for generating the Laguerre polynomials is the analogue of Rodrigues’ formula for the Legendre polynomials. See (33) in Section 6.4.
