reduce-historical/r36/xlog/SUM.LOG

REDUCE 3.6, 15-Jul-95, patched to 6 Mar 96 ...


% Tests of the SUM package.

% Author: Fujio Kako (kako@kako.math.sci.hiroshima-u.ac.jp)

% 1) Summations.

sum(n,n);


 n*(n + 1)
-----------
     2


for i:=2:10 do write sum(n**i,n);


       2
 n*(2*n  + 3*n + 1)
--------------------
         6

  2   2
 n *(n  + 2*n + 1)
-------------------
         4

       4       3       2
 n*(6*n  + 15*n  + 10*n  - 1)
------------------------------
              30

  2     4      3      2
 n *(2*n  + 6*n  + 5*n  - 1)
-----------------------------
             12

       6       5       4      2
 n*(6*n  + 21*n  + 21*n  - 7*n  + 1)
-------------------------------------
                 42

  2     6       5       4      2
 n *(3*n  + 12*n  + 14*n  - 7*n  + 2)
--------------------------------------
                  24

        8       7       6       4       2
 n*(10*n  + 45*n  + 60*n  - 42*n  + 20*n  - 3)
-----------------------------------------------
                      90

  2     8       7       6       4       2
 n *(2*n  + 10*n  + 15*n  - 14*n  + 10*n  - 3)
-----------------------------------------------
                      20

       10       9       8       6       4       2
 n*(6*n   + 33*n  + 55*n  - 66*n  + 66*n  - 33*n  + 5)
-------------------------------------------------------
                          66


sum((n+1)**3,n);


     3      2
 n*(n  + 6*n  + 13*n + 12)
---------------------------
             4


sum(x**n,n);


  n
 x *x
-------
 x - 1


sum(n**2*x**n,n);


  n     2  2      2      2
 x *x*(n *x  - 2*n *x + n  - 2*n*x + 2*n + x + 1)
--------------------------------------------------
                3      2
               x  - 3*x  + 3*x - 1


sum(1/n,n);


     1
sum(---,n)
     n


sum(1/n/(n+2),n);


   n*(3*n + 5)
------------------
     2
 4*(n  + 3*n + 2)


sum(log (n/(n+1)),n);


       1
log(-------)
     n + 1


% 2) Expressions including trigonometric functions.

sum(sin(n*x),n);


         2*n*x + x
  - cos(-----------)
             2
---------------------
            x
     2*sin(---)
            2


sum(n*sin(n*x),n,1,k);


 sin(k*x + x)*k - sin(k*x)*k - sin(k*x)
----------------------------------------
             2*(cos(x) - 1)


sum(cos((2*r-1)*pi/n),r);


      2*pi*r
 sin(--------)
        n
---------------
         pi
  2*sin(----)
         n


sum(cos((2*r-1)*pi/n),r,1,n);


0


sum(cos((2*r-1)*pi/(2*n+1)),r);


       2*pi*r
  sin(---------)
       2*n + 1
------------------
          pi
 2*sin(---------)
        2*n + 1


sum(cos((2*r-1)*pi/(2*n+1)),r,1,n);


       2*n*pi
  sin(---------)
       2*n + 1
------------------
          pi
 2*sin(---------)
        2*n + 1


sum(sin((2*r-1)*x),r,1,n);


  - cos(2*n*x) + 1
-------------------
     2*sin(x)


sum(cos((2*r-1)*x),r,1,n);


 sin(2*n*x)
------------
  2*sin(x)


sum(sin(n*x)**2,n);


  - sin(2*n*x + x) + 2*sin(x)*n
--------------------------------
            4*sin(x)


sum(cos(n*x)**2,n);


 sin(2*n*x + x) + 2*sin(x)*n
-----------------------------
          4*sin(x)


sum(sin(n*x)*sin((n+1)*x),n);


  - sin(2*n*x + 2*x) + sin(2*x)*n
----------------------------------
             4*sin(x)


sum(sec(n*x)*sec((n+1)*x),n);


sum(sec(n*x + x)*sec(n*x),n)


sum(1/2**n*tan(x/2**n),n);


          x
     tan(----)
           n
          2
sum(-----------,n)
         n
        2


sum(sin(r*x)*sin((r+1)*x),r,1,n);


  - sin(2*n*x + 2*x) + sin(2*x)*n + sin(2*x)
---------------------------------------------
                  4*sin(x)


sum(sec(r*x)*sec((r+1)*x),r,1,n);


sum(sec(r*x + x)*sec(r*x),r,1,n)


sum(1/2**r*tan(x/2**r),r,1,n);


          x
     tan(----)
           r
          2
sum(-----------,r,1,n)
         r
        2


sum(k*sin(k*x),k,1,n - 1);


  - sin(n*x - x)*n + sin(n*x)*n - sin(n*x)
-------------------------------------------
              2*(cos(x) - 1)


sum(k*cos(k*x),k,1,n - 1);


  - cos(n*x - x)*n + cos(n*x)*n - cos(n*x) + 1
-----------------------------------------------
                2*(cos(x) - 1)


sum(sin((2k - 1)*x),k,1,n);


  - cos(2*n*x) + 1
-------------------
     2*sin(x)


sum(sin(x + k*y),k,0,n);


         2*n*y + 2*x + y          2*x - y
  - cos(-----------------) + cos(---------)
                2                    2
--------------------------------------------
                        y
                 2*sin(---)
                        2


sum(cos(x + k*y),k,0,n);


      2*n*y + 2*x + y          2*x - y
 sin(-----------------) - sin(---------)
             2                    2
-----------------------------------------
                      y
               2*sin(---)
                      2


sum((-1)**(k - 1)*sin((2k - 1)*x),k,1,n + 1);


       n
 ( - 1) *sin(2*n*x + 2*x)
--------------------------
         2*cos(x)


sum((-1)**(k - 1)*cos((2k - 1)*x),k,1,n + 1);


       n
 ( - 1) *cos(2*n*x + 2*x) + 1
------------------------------
           2*cos(x)


sum(r**k*sin(k*x),k,1,n - 1);


     n                   n
  - r *sin(n*x - x)*r + r *sin(n*x) - sin(x)*r
-----------------------------------------------
                            2
              2*cos(x)*r - r  - 1


sum(r**k*cos(k*x),k,0,n - 1);


     n                   n
  - r *cos(n*x - x)*r + r *cos(n*x) + cos(x)*r - 1
---------------------------------------------------
                              2
                2*cos(x)*r - r  - 1


sum(sin(k*x)*sin((k + 1)*x),k,1,n);


  - sin(2*n*x + 2*x) + sin(2*x)*n + sin(2*x)
---------------------------------------------
                  4*sin(x)


sum(sin(k*x)*sin((k + 2)*x),k,1,n);


  - sin(2*n*x + 3*x) + sin(3*x)*n + sin(3*x) - sin(x)*n
--------------------------------------------------------
                        4*sin(x)


sum(sin(k*x)*sin((2k - 1)*x),k,1,n);


         6*n*x + x          2*n*x - 3*x          2*n*x - x          2*n*x + x
( - sin(-----------) + sin(-------------) + sin(-----------) + sin(-----------)
             2                   2                   2                  2

         3*x          x            3*x
  + sin(-----) + sin(---))/(4*sin(-----))
          2           2             2



% The next examples cannot be summed in closed form.

sum(1/(cos(x/2**k)*2**k)**2,k,1,n);


            1
sum(-----------------,k,1,n)
      2*k      x   2
     2   *cos(----)
                k
               2


sum((2**k*sin(x/2**k)**2)**2,k,1,n);


     2*k      x   4
sum(2   *sin(----) ,k,1,n)
               k
              2


sum(tan(x/2**k)/2**k,k,0,n);


          x
     tan(----)
           k
          2
sum(-----------,k,0,n)
         k
        2


sum(cos(k**2*2*pi/n),k,0,n - 1);


            2
         2*k *pi
sum(cos(---------),k,0,n - 1)
            n


sum(sin(k*pi/n),k,1,n - 1);


         2*n*pi - pi          pi
  - cos(-------------) + cos(-----)
             2*n              2*n
------------------------------------
                   pi
            2*sin(-----)
                   2*n


% 3) Expressions including the factorial function.

for all n,m such that fixp m let
        factorial(n+m)=if m > 0 then factorial(n+m-1)*(n+m)
                   else factorial(n+m+1)/(n+m+1);



sum(n*factorial(n),n);


factorial(n)*(n + 1)


sum(n/factorial(n+1),n);


          - 1
----------------------
 factorial(n)*(n + 1)


sum((n**2+n-1)/factorial(n+2),n);


          - 1
----------------------
 factorial(n)*(n + 2)


sum(n*2**n/factorial(n+2),n);


                 n
            - 2*2
-----------------------------
                2
 factorial(n)*(n  + 3*n + 2)


sum(n*x**n/factorial(n+2),n);


                             n
                            x *n
sum(-----------------------------------------------------,n)
                   2
     factorial(n)*n  + 3*factorial(n)*n + 2*factorial(n)


for all n,m such that fixp m and m > 3 let
        factorial((n+m)/2)= factorial((n+m)/2-1)*((n+m)/2),
        factorial((n-m)/2)= factorial((n-m)/2+1)/((n-m)/2+1);



sum(factorial(n-1/2)/factorial(n+1),n);


                    2*n - 1
         factorial(---------)
                       2
sum(-------------------------------,n)
     factorial(n)*n + factorial(n)


for all n,m such that fixp m and m > 3 clear factorial((n+m)/2);



for all n,m such that fixp m and m > 3 clear factorial((n-m)/2);



% 4) Expressions including combination.

operator comb;

          % Combination function.

for all n ,m let comb(n,m)=factorial(n)/factorial(n-m)/factorial(m);



sum((-1)**k*comb(n,k),k,1,m);


          m                        m
( - ( - 1) *factorial(n)*m + ( - 1) *factorial(n)*n

  - factorial( - m + n)*factorial(m)*n)/(factorial( - m + n)*factorial(m)*n)


sum(comb(n + p,q)/comb(n + r,q + 2),n,1,m);


( - factorial( - q + r)*factorial(m + p - q)*factorial(m + r)*factorial(p)*m*p*q

  - 2*factorial( - q + r)*factorial(m + p - q)*factorial(m + r)*factorial(p)*m*p

  - factorial( - q + r)*factorial(m + p - q)*factorial(m + r)*factorial(p)*m*q

  - 2*factorial( - q + r)*factorial(m + p - q)*factorial(m + r)*factorial(p)*m

                                                                              2
  + factorial( - q + r)*factorial(m + p - q)*factorial(m + r)*factorial(p)*p*q

  - factorial( - q + r)*factorial(m + p - q)*factorial(m + r)*factorial(p)*p*q*r

  + 2*factorial( - q + r)*factorial(m + p - q)*factorial(m + r)*factorial(p)*p*q

  - 2*factorial( - q + r)*factorial(m + p - q)*factorial(m + r)*factorial(p)*p*r

                                                                            2
  + factorial( - q + r)*factorial(m + p - q)*factorial(m + r)*factorial(p)*q

  - factorial( - q + r)*factorial(m + p - q)*factorial(m + r)*factorial(p)*q*r

  + 2*factorial( - q + r)*factorial(m + p - q)*factorial(m + r)*factorial(p)*q

  - 2*factorial( - q + r)*factorial(m + p - q)*factorial(m + r)*factorial(p)*r

                                                                           2
  - factorial(m - q + r)*factorial(m + p)*factorial(p - q)*factorial(r)*m*q

  + factorial(m - q + r)*factorial(m + p)*factorial(p - q)*factorial(r)*m*q*r

  - 2*factorial(m - q + r)*factorial(m + p)*factorial(p - q)*factorial(r)*m*q

  + 2*factorial(m - q + r)*factorial(m + p)*factorial(p - q)*factorial(r)*m*r

                                                                           2
  - factorial(m - q + r)*factorial(m + p)*factorial(p - q)*factorial(r)*p*q

  + factorial(m - q + r)*factorial(m + p)*factorial(p - q)*factorial(r)*p*q*r

  - 2*factorial(m - q + r)*factorial(m + p)*factorial(p - q)*factorial(r)*p*q

  + 2*factorial(m - q + r)*factorial(m + p)*factorial(p - q)*factorial(r)*p*r

                                                                         2
  - factorial(m - q + r)*factorial(m + p)*factorial(p - q)*factorial(r)*q

  + factorial(m - q + r)*factorial(m + p)*factorial(p - q)*factorial(r)*q*r

  - 2*factorial(m - q + r)*factorial(m + p)*factorial(p - q)*factorial(r)*q

  + 2*factorial(m - q + r)*factorial(m + p)*factorial(p - q)*factorial(r)*r)/(

   factorial(m + p - q)*factorial(m + r)*factorial(p - q)*factorial(r)*(m*p*q

                                  2            2                2    2      2
       - m*p*r - m*q*r + m*q + m*r  - m*r - p*q  + 2*p*q*r - p*r  + q *r - q

              2            3    2
       - 2*q*r  + 2*q*r + r  - r ))


sum((-1)**(k + 1)*comb(n,k)/(k + 1),k,1,n);


   n
-------
 n + 1


for all n ,m clear comb(n,m);



for all n,m such that fixp m clear factorial(n+m);



% 3) Examples taken from
%         "Decision procedure for indefinite hypergeometric summation"
%          Proc. Natl. Acad. Sci. USA  vol. 75, no. 1 pp.40-42 (1978)
%          R. William Gosper, Jr.
%

%            n
%           ____     2
%       f =  ||  (b*k +c*k+d)
%           k=1
%
%            n
%           ____     2
%       g =  ||  (b*k +c*k+e)
%           k=1
%
operator f,gg;

  % gg used to avoid possible conflict with high energy
                % physics operator.

for all n,m such that fixp m let
        f(n+m)=if m > 0 then f(n+m-1)*(b*(n+m)**2+c*(n+m)+d)
                   else f(n+m+1)/(b*(n+m+1)**2+c*(n+m+1)+d);



for all n,m such that fixp m let
        gg(n+m)=if m > 0 then gg(n+m-1)*(b*(n+m)**2+c*(n+m)+e)
                   else gg(n+m+1)/(b*(n+m+1)**2+c*(n+m+1)+e);



sum(f(n-1)/gg(n),n);


     f(n)
---------------
 gg(n)*(d - e)


sum(f(n-1)/gg(n+1),n);


          2  2      2        2
(f(n)*(2*b *n  + 4*b *n + 2*b  + 2*b*c*n + 2*b*c + 2*b*d*n + 3*b*d - 2*b*e*n

                             2            2            3    2      3        3
        - b*e + c*d - c*e + d  - 2*d*e + e ))/(gg(n)*(b *d*n  + 2*b *d*n + b *d

          3    2      3        3      2          2        2          2
       - b *e*n  - 2*b *e*n - b *e + b *c*d*n + b *c*d - b *c*e*n - b *c*e

            2  2  2      2  2        2  2    2          2  2  2      2  2
       + 2*b *d *n  + 4*b *d *n + 2*b *d  + b *d*e - 2*b *e *n  - 4*b *e *n

            2  2      2    2        2          2        2    2        2
       - 3*b *e  - b*c *d*n  - 2*b*c *d*n - b*c *d + b*c *e*n  + 2*b*c *e*n

            2            2            2          2            2      3  2
       + b*c *e + 2*b*c*d *n + 2*b*c*d  - 2*b*c*e *n - 2*b*c*e  + b*d *n

              3        3        2    2        2          2            2  2
       + 2*b*d *n + b*d  - 3*b*d *e*n  - 6*b*d *e*n - b*d *e + 3*b*d*e *n

                2            2      3  2        3          3    3        3
       + 6*b*d*e *n + 3*b*d*e  - b*e *n  - 2*b*e *n - 3*b*e  - c *d*n - c *d

          3        3      2        2  2      3        3        2            2
       + c *e*n + c *e - c *d*e + c *e  + c*d *n + c*d  - 3*c*d *e*n - 3*c*d *e

                2            2      3        3    3        2  2        3    4
       + 3*c*d*e *n + 3*c*d*e  - c*e *n - c*e  + d *e - 3*d *e  + 3*d*e  - e ))


for all n,m such that fixp m clear f(n+m);



for all n,m such that fixp m clear gg(n+m);



clear f,gg;



% 4) Products.

prod(n/(n+2),n);


      2
--------------
  2
 n  + 3*n + 2


prod(x**n,n);


   2
 (n  + n)/2
x


prod(e**(sin(n*x)),n);


                1
----------------------------------
  cos((2*n*x + x)/2)/(2*sin(x/2))
 e


end;
(TIME:  sum 3470 3740)