# §30.8 Expansions in Series of Ferrers Functions

## §30.8(i) Functions of the First Kind

where is the Ferrers function of the first kind (§14.3(i)), , and the coefficients are given by

Let

Then the set of coefficients , is the solution of the difference equation

(note that ) that satisfies the normalizing condition

with

Also, as ,

and

## §30.8(ii) Functions of the Second Kind

where and are again the Ferrers functions and . The coefficients satisfy (30.8.4) for all when we set for . For they agree with the coefficients defined in §30.8(i). For they are determined from (30.8.4) by forward recursion using . The set of coefficients , , is the recessive solution of (30.8.4) as that is normalized by

with

It should be noted that if the forward recursion (30.8.4) beginning with , leads to , then is undefined for and does not exist.