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LEGENDRE FUNCTIONS AND OTHER ORTHOGONAL FUNCTIONS
[CHAP. 11
RECURRENCE FORMULAS FOR LEGENDRE POLYNOMIALS 11.5. Prove that From the generating function of Problem 11.4 we have (1) Differentiating with respect to t, Multiplying by 1 - 2xt + t2, (2)
Now the left side of (2) can be written in terms of (1) and we have
i.e. Equating the coefficients of t" on each side, we find which yields the required result.
11.6.
Given that P0(x) = 1, Pi(x) = x, find (a) P2(x) and (b) P3(x). Using the recurrence formula of Problem 11.5, we have on letting n = 1,
Similarly letting n — 2,
LEGENDRE FUNCTIONS OF THE SECOND KIND 11.7. Obtain the results (5) and (6), page 243, for the Legendre functions of the second kind in the case where n is a non-negative integer. The Legendre functions of the second kind are the series solutions of Legendre's equation which do not terminate. From Problem 11.1, equation (3), we see that if n is even the series which does not terminate is
while if n is odd the series which does not terminate is
These series solutions, apart from multiplicative constants, provide definitions for Legendre functions of the second kind and are given by (5) and (6) on page 243.