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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
lag(n,alpha,x):=rat(%e^x*x^(-alpha)*diff(x^(n+alpha)*%e^(-x),x,n)/n!);
                                x  - alpha       n + alpha   - x
                              %e  x        diff(x          %e   , x, n)
(%o1) lag(n, alpha, x) := rat(-----------------------------------------)
                                                 n!
(%i2) nn:2;
(%o2)                                  2
(%i3) gen:_laguerre(nn,a,x);
(%o3)                         _laguerre(2, a, x)
(%i4) lag(nn,a,x);
                        2                    2
                       x  + (- 2 a - 4) x + a  + 3 a + 2
(%o4)/R/               ---------------------------------
                                       2
(%i5) 
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) 

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