asymptotic expansion of a definite hypergeometric sum
DefiniteSumAsymptotic(T, n, k, l..u, f)
algebraic expression representing a hypergeometric term of both n and k
range for k
(optional) unevaluated name
For a hypergeometric term T of n and k over the real number field, the DefiniteSumAsymptotic(T,n,k,l..u) command computes the asymptotic expansion of the definite sum S⁡n=∑k=lu⁡T with respect to the variable n (as n approaches ∞), where l=r⁢n+s and u=t⁢n+v for some real numbers r, s, t, v.
The routine returns an error if T does not satisfy the following conditions for all large enough n and for all k in the range l..u:
T is defined;
T has constant sign.
In trivial cases (for example, when T is a rational function in k and polynomial in n) the procedure returns an asymptotic expansion of S⁡n 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:
Sgn is 1 or -1,
C0, C1, ..., Cm, D are constants,
a1, ..., am are positive rational numbers ≤1,
c is a positive rational number,
b is a positive integer, and
Q is a polynomial of degree ≤b.
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 O⁡1nc, 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 _s1, _s2, ... The second element is a list of equations _s1=s1, _s2=s2, ... where s1, s2, ... are floating-point numbers approximating the values of _s1, _s2, ...
The typical calling sequence of the auxiliary procedure is f⁡n0,n1,h,q, where
n0 is a lower bound for the samples w.r.t. n;
n1 is an upper bound for the samples w.r.t. n;
h is the step size for the samples w.r.t. n;
q is the desired number of coefficients _si.
These parameters should satisfy the following constraints:
h is a positive integer,
The recommended values for the parameters are 1000≤n0, 2⁢n0≤n1, h=10; q=3 if c=1 and q=6 if c<1. By default, calling f⁡ without arguments is equivalent to f⁡1000,2000,10,3.
If there is a conjecture for an exact value s1 of _s1, then f⁡n0,n1,h,q,s1 computes approximate values for the subsequent coefficients. Similarly, it is possible to call f⁡n0,n1,h,q,s1,s2, f⁡n0,n1,h,q,s1,s2,s3, 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 s1, s2, ... can be increased by calling f with higher values of n0, n1, and Digits. Generally, the values si are less accurate the higher the index i is.
T ≔ binomial⁡2⁢n,2⁢k3:
res ≔ f⁡
Digits ≔ 20:
res ≔ f⁡1000,2000,10,3,−16
Ryabenko, A.A., and Skorokhodov, S.L. "Asymptotics of Sums of Hypergeometric Terms." Programming and Computer Software. Vol. 31, (2005): 65-72.
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