Indefinite summation of rational functions                                

   Calling sequence:
   RationalSum( f , x );                                                     

   Input:                                                                    
     f - rational function in x                                              
     x - name                                                                

   Output:                                                                   
     [ g , r ]                                                               
   where g and r are rational functions in x such that                       
     g(x+1) - g(x) + r = f                                                   
   and r has minimal degree of denominator                                   

   Optional arguments:                                                       

   'factorization' = false                                                 
      - full factorization of denominator of f should not be used           

   'method' = <integer>                                                    
      - specifies which summation algorithm should be used:                 
        0: factorization-based [Polyakov, 2011] (default one)                
        1: recursive [Abramov, 1975]                                         
        2: linear-algebraic [Abramov, 1995]                                  
        3: GGSZ [Gerhard et al, 2003]                                        

   'normalize'= true                                                       
      - for method 0, the output should be in normalized form               

   'minimize' = "sum denominator"                                          
      - g should have the denominator of minimal degree                     

   'minimize' = "numerator"                                                
      - r should have the numerator of minimal degree (not guaranteed)  
                                                  
   'minimized' = <name>                                                    
      - if 'minimize' = "numerator" option is used, <name> is               
        assigned true if the degree of numerator is guaranteed              
        to be minimal and false otherwise                                   

   'alignment' = "left"                                                    
      - for methods 0 and 1, remainder should be aligned to the             
        left side of f(x) (i.e. if p(x) is an irreducible factor of         
        the remainder's denominator, then it is also divides the            
        denominator of f(x), and p(x-k) does not divide the denominator     
        of f(x) for any positive integer k)                                 

   'alignment' = "right"                                                   
      - for methods 0 and 1, remainder should be aligned to the             
        right side of f(x) (i.e. if p(x) is an irreducible factor of        
        the remainder's denominator, then it is also divides the            
        denominator of f(x), and p(x+k) does not divide the denominator     
        of f(x) for any positive integer k)                                 

   'alignment' = "peak degree"                                             
      - for methods 0 and 1, remainder should be aligned to the             
        peak degrees of the denominator of f(x) [Pirastu, 1995]             

   'dispersion' = <nonnegative integer>                                    
      - for methods 1, 2, and 3, the given integer should be used           
        as a dispersion value (this may result in mistakes if the actual    
        dispersion is greater)                                              
                                                                            

   References:
     Polyakov S.P. Indefinite summation of rational functions with
     factorization of denominators. 
     Programming and Computer Software, 2011, vol. 37, no. 6, pp. 322-325.

     Abramov S.A. The rational component of of the solution of a first-order
     linear recurrence relation with rational right-hand side.
     U.S.S.R. Computational Mathematics and Mathematical Phisycs, 1975, 
     15(4), pp. 216-221.

     Abramov S.A. Indefinite sums of rational functions.
     Proceedings of ISSAC'95, 1995, pp. 303-308.
 
     Gerhard J., Giesgrecht M., Storjohann A., Zima E.V. Shiftless
     decomposition and polynomial-time rational summation.
     Proceedings of ISSAC'03, 2003, pp. 119-126.

     Pirastu R. Algorithms for indefinite summation of rational functions
     in Maple.
     The Maple Technical Newsletter, 1995, vol. 2, no. 1, pp. 1-12.