# §29.6 Fourier Series

## §29.6(i) Function

When , where is a nonnegative integer, it follows from §2.9(i) that for any value of the system (29.6.4)–(29.6.6) has a unique recessive solution ; furthermore

In addition, if satisfies (29.6.2), then (29.6.3) applies.

In the special case , , there is a unique nontrivial solution with the property , . This solution can be constructed from (29.6.4) by backward recursion, starting with and an arbitrary nonzero value of , followed by normalization via (29.6.5) and (29.6.6). Consequently, reduces to a Lamé polynomial; compare §§29.12(i) and 29.15(i).

An alternative version of the Fourier series expansion (29.6.1) is given by

Here is as in §22.2, and

29.6.9
29.6.10,

with , and now defined by

29.6.11

and

29.6.13

Also,

where

29.6.24
29.6.25,

with

29.6.26

and

29.6.28

Also,

where

29.6.39
29.6.40,

with

29.6.41

and

29.6.43

Also,

where

29.6.54
29.6.55,

with

29.6.56

and

29.6.58