For a hypergeometric term  of  and  over the real number field, the DefiniteSumAsymptotic(T,n,k,l..u) command computes the asymptotic expansion of the definite sum  with respect to the variable  (as  approaches ), where  and  for some real numbers , , , .

The routine returns an error if   does not satisfy the following conditions for all large enough  and for all  in the range :

 is defined;

 has constant sign.

In trivial cases (for example, when  is a rational function in  and polynomial in ) the procedure returns an asymptotic expansion of  with a truncation order specified by the global variable Order. Otherwise, if possible, the procedure returns the main part of an asymptotic expansion of the form:

or

or

where

 is 1 or -1,

, , ..., ,  are constants,

, ...,  are positive rational numbers ,

 is a positive rational number,

 is a positive integer, and

 is a polynomial of degree .

The procedure can compute the asymptotics of most frequently used binomial sums. In case it cannot compute one, it returns FAIL.

If the optional argument f is specified, the input is not trivial, and the main part of the asymptotic expansion was computed to be , then f will be assigned an auxiliary procedure. This procedure computes approximate values for the next coefficients in the asymptotic expansion, by treating an experimental sample for large n statistically, using the least-squares method.

The procedure assigned to f returns a sequence of two elements. The first element is the asymptotic expansion, which contains placeholder names , , ... The second element is a list of equations , , ... where , , ... are floating-point numbers approximating the values of , , ...

The typical calling sequence of the auxiliary procedure is , where

 is a lower bound for the samples w.r.t. ;

 is an upper bound for the samples w.r.t. ;

 is the step size for the samples w.r.t. ;

 is the desired number of coefficients .

These parameters should satisfy the following constraints:

,

 is a positive integer,

, and

.

The recommended values for the parameters are , ,  ;  if  and  if . By default, calling  without arguments is equivalent to .

If there is a conjecture for an exact value  of , then  computes approximate values for the subsequent coefficients. Similarly, it is possible to call , , etc.

Note that the value of Digits controls only the working precision, i.e., the number of digits that f uses when it calculates the experimental sample and runs the least-squares method. The accuracy of , , ... can be increased by calling f with higher values of , , and Digits. Generally, the values  are less accurate the higher the index  is.

Ryabenko, A.A., and Skorokhodov, S.L. "Asymptotics of Sums of Hypergeometric Terms." Programming and Computer Software. Vol. 31, (2005): 65-72.