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gen_laguerre

Run Example
(%i1)? gen_laguerre;

 -- Function: gen_laguerre (<n>, <a>, <x>)
     The generalized Laguerre polynomial of degree <n>.

     Reference: Abramowitz and Stegun, equation 22.5.54, page 780.


(%o1)                                true
(%i2) 
Run Example
gamma_expand:true;
(%o1)                                true
(%i2) nn:2;
(%o2)                                  2
(%i3) rat(gen_laguerre(nn,a,x));
                        2                    2
                       x  + (- 2 a - 4) x + a  + 3 a + 2
(%o3)/R/               ---------------------------------
                                       2
(%i4) rat(sum(binomial(nn,kk)*gamma(nn+a+1)/gamma(kk+a+1)/nn!*(-x)^kk,kk,0,nn));
                        2                    2
                       x  + (- 2 a - 4) x + a  + 3 a + 2
(%o4)/R/               ---------------------------------
                                       2
(%i5) 
Run Example
evenn(n):=sum(binomial(n,2*kk)*a^(n-2*kk)*b^(2*kk),kk,0,floor(n/2));
                                         n - 2 kk  2 kk               n
(%o1) evenn(n) := sum(binomial(n, 2 kk) a         b    , kk, 0, floor(-))
                                                                      2
(%i2) oddn(n):=sum(binomial(n,2*kk+1)*a^(n-2*kk-1)*b^(2*kk+1),kk,0,floor(n/2));
                                            n - 2 kk - 1  2 kk + 1
(%o2) oddn(n) := sum(binomial(n, 2 kk + 1) a             b        , kk, 0, 
                                                                            n
                                                                      floor(-))
                                                                            2
(%i3) rat(gen_laguerre(1,x,a));
(%o3)/R/                           x - a + 1
(%i4) rat(gen_laguerre(2,x,a));
                        2                    2
                       x  + (- 2 a + 3) x + a  - 4 a + 2
(%o4)/R/               ---------------------------------
                                       2
(%i5) rat(gen_laguerre(3,x,a));
          3                2       2                   3      2
         x  + (- 3 a + 6) x  + (3 a  - 15 a + 11) x - a  + 9 a  - 18 a + 6
(%o5)/R/ -----------------------------------------------------------------
                                         6
(%i6) 

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