Mathematical Methods for Physicists: A concise introduction - Site Map

SPECIAL FUNCTIONS OF MATHEMATICAL PHYSICS
Then expanding coecients of z n in Eq. (7.15) leads to the recurrence relation
…2n ‡ 1†xP n …x† ˆ…n ‡ 1†P n‡1 …x†‡nP n1 …x†:
…7:16†
This gives P 4 ; P 5 ; P 6 , etc. very quickly in terms of P 0 ; P 1 , and P 3 .
Recurrence relations are very useful in simplifying work, helping in proofs
or derivations. We list four more recurrence relations below without proofs or
derivations:
xP 0 n…x†P 0 n1…x† ˆnP n …x†;
P 0 n…x†xP 0 n1…x† ˆnP n1 …x†;
…7:16a†
…7:16b†
…1 x 2 †P 0 n…x† ˆnP n1 …x†nxP n …x†; …7:16c†
…2n ‡ 1†P n …x† ˆP 0 n‡1…x†P 0 n1…x†:
…7:16d†
With the help of the recurrence formulas (7.16) and (7.16b), it is straightforward
to establish the other three. Omitting the full details, which are left for
the reader, these relations can be obtained as follows:
(i) di€erentiation of Eq. (7.16) with respect to x and the use of Eq. (7.16b) to
eliminate P 0 n‡1…x† leads to relation (7.16a);
(ii) the addition of Eqs. (7.16a) and (7.16b) immediately yields relation
(7.16d);
(iii) the elimination of P 0 n1…x† between Eqs. (7.16b) and (7.16a) gives relation
(7.16c).
Example 7.1
The physical signi®cance of expansion (7.14) is apparent in this simple example:
®nd the potential V of a point charge at point P due to a charge ‡q at Q.
Solution:
Suppose the origin is at O (Fig. 7.2). Then
V P ˆ q
R ˆ q…2 2r cos ‡ r 2 † 1=2 :
Figure 7.2.
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