#### 6.65 Associated Legendre functions (24.9.99)

There seem to be something curious about the associated Legendre function of the ﬁrst kind,
LegendreP(v,u,x) (I am using version 5.1 on a PC under NT 4.0):

> g1 := sum('binomial(2*n+1,2*k+1)*2^(3*k)','k'=0..n);
(3/4) n
g1 := 1/8 (2 n + 1) sqrt(Pi) 8 (-7) LegendreP(n, -1/2, -9/7)

let us see g1 for n=0 and 1

> for n from 0 to 1 do g1 od;
(3/4)
-2 (-1)
(3/4)
- 21/8 sqrt(Pi) 8 LegendreP(1, -1/2, -9/7)

ﬁrst surprise, we would have expected 1 and 11, respectively, and certainly not a complex
number in the case n=0. I found no way to obtain the answers expressed as integers, so I
decided to force the ﬂoating point evaluation of g1

> for n from 0 to 3 do evalf(g1) od;
1.414213562 - 1.414213562 I
22.00000000
298.0000001
4286.000004

Hmmm, truncation errors... AND, apart from the spurious complex value for n=0, what
seems to be TWICE the correct result.

Let us try something else, using the inert operator Sum.

> g1b := Sum(binomial(2*n+1,2*k+1)*2^(3*k),k=0..n):
> for n from 0 to 3 do value(g1b) od;
1
11
149
2143

correct results at last!

In fact, the result is the sum of the nth powers of `9 +- 4sqrt(2)`

with appropriate
coeﬃcients. It can also be calculated using

h := proc(n) option remember;
if n<2 then 10*n+1
else 18*h(n-1)-49*h(n-2)
fi
end;

I have two questions:

- why are the LegendreP functions used in the simpliﬁcation of sums when even
the Help about them acknowledges that their deﬁnition is a bit tricky, with
`_EnvLegendreCut being -1..1 or 1..infinity`

?

- is the numerical evaluation of these functions correctly implemented?
`LegendreP(0,-1/2,cos(phi))`

diﬀers from `2*sqrt(tan(phi/2)/Pi)`

, as given in the
`Gradshteyn-Ryzhik (eq. 8.753.1)`

.

Apparently there is a bug in Release 5 concerning Legendre functions. Release 4 uses
hypergeometric functions for the sum:

> restart;
> g1a:=sum(binomial(2*n+1,2*k+1)*2^(3*k),k=0..n);
g1a := (2 n + 1) hypergeom([-n + 1/2, -n], [3/2], 8)
> for n from 0 to 3 do simplify(g1a) od;
1
11
149
2143

This is the correct result.