The SumDecomposition(T, n, k) command constructs two hypergeometric terms  and  such that  and the certificate  has a rational normal form  with  of minimal degree.

The output from SumDecomposition is a list of two elements . Both are represented in the form

for some integer . The form shown above is called a multiplicative decomposition of the hypergeometric term .

If the third optional argument k is not specified, the first unused name in the sequence  is used.

If the fourth optional argument newT is specified, it will be assigned an expression in terms of inert Products of the same form as for  above that is equivalent to .

If  is a rational function of , then  and  will be rational functions as well, and the denominator of  is of smallest possible degree. In that case,  and  are not unique, however, and you can use the minimize=v option to impose some additional conditions on  and  (see below).

Note: If you set infolevel[SumDecomposition] to , Maple prints diagnostics.

The following optional arguments can be used if  is a rational function of .

minimize=v, where v is either a numeric value between  and  or one of "numerator", "sum denominator", "combined", "left", "right".

If v="sum denominator", then the degree  of the denominator of  will be minimized.

If v="numerator", then the degree  of the numerator of  will be minimized.

If v="combined", then then the sum of the degrees  will be  minimized.

If v is  a numeric constant between  and , then the weighted sum of the degrees  will be minimized.

Note that small values of v may lead to time-consuming search; the option maxiterations (see below) can be used to restrict it.

If v="left", then the remainder of the result will be aligned such that  for all integers .

If v="right", then the remainder of the result will be aligned such that  for all integers .

maxiterations=integer

This option can be used to restrict the number of iterations performed by the command when the option minimize=v is used with a small positive numeric value v. The default value is .

SumDecomposition:   "calling dterm"
SumDecomposition:   "construct the RCF_1 for the certificate of T"
SumDecomposition:   "construct a regular description of T"
SumDecomposition:   "calling dcert"
SumDecomposition:   "using factorization method"
SumDecomposition:   "construct a regular description of T1"
SumDecomposition:   "construct a regular description of T2"
SumDecomposition:   "T2 is not summable"
SumDecomposition:   "An attempt to control the degree of the numerator"
SumDecomposition:   "construct a triple that regularly describes T2"

Abramov, S.A. "Indefinite Sums of Rational Functions." Proceedings ISSAC'95, pp. 303-308. 1995.

Abramov, S.A., and Petkovsek, M. "Minimal Decomposition of Indefinite Hypergeometric Sums." Proceedings ISSAC'2001, pp. 7-14. 2001.

Abramov, S.A., and Petkovsek, M. "Rational Normal Forms and Minimal Decompositions of Hypergeometric Terms." Journal of Symbolic Computation. Vol. 33 No. 5. (2002): 521-543.

Polyakov, S.P. "Symbolic Additive Decomposition of Rational Functions." Programming and Computer Software, Vol. 31 No. 2. (2005): 60-64.

Polyakov, S.P. "Indefinite Summation of Rational Functions with Additional Minimization of the Summable Part." Programming and Computer Software 34 No. 2, (2008): 95-100.