SPECIAL FUNCTIONS
randr′must be exchanged in (18.22) or the series would not converge. This
result may be used, for example, to write down the electrostatic potential at a
pointrdue to a chargeqat the pointr′. Thus, in the caser′<r, this is given by
V(r)=q
4 π 0 r∑∞=0(
r′
r)
P(cosθ).We note that in the special case where the charge is at the origin, andr′=0,
only the= 0 term in the series is non-zero and the expression reduces correctly
to the familiar formV(r)=q/(4π 0 r).
Recurrence relationsIn our discussion of the generating function above, we derived several useful
recurrence relations satisfied by the Legendre polynomialsPn(x). In particular,
from (18.18), we have the four-term recurrence relation
Pn′+1+Pn′− 1 =Pn+2xPn′.Also, from (18.19)–(18.21), we have the three-term recurrence relations
Pn′+1=(n+1)Pn+xPn′, (18.23)Pn′− 1 =−nPn+xPn′, (18.24)(1−x^2 )Pn′=n(Pn− 1 −xPn), (18.25)(2n+1)Pn=Pn′+1−Pn′− 1 , (18.26)where the final relation is obtained immediately by subtracting the second from
the first. Many other useful recurrence relations can be derived from those given
above and from the generating function.
Prove the recurrence relation
(n+1)Pn+1=(2n+1)xPn−nPn− 1. (18.27)Substituting from (18.15) into (18.17), we find
(x−h)∑
Pnhn=(1− 2 xh+h^2 )∑
nPnhn−^1.Equating coefficients ofhnwe obtain
xPn−Pn− 1 =(n+1)Pn+1− 2 xnPn+(n−1)Pn− 1 ,which on rearrangement gives the stated result.
The recurrence relation derived in the above example is particularly useful inevaluatingPn(x) for a given value ofx. One starts withP 0 (x)=1andP 1 (x)=x
and iterates the recurrence relation untilPn(x) is obtained.