6.65 Associated Legendre functions (24.9.99)

6.65.1 TANGUY Christian

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).

6.65.2 Helmut Kahovec (30.9.99)

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.