asymptotic expansion of a definite hypergeometric sum - Maple Help

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SumTools[Hypergeometric][DefiniteSumAsymptotic] - asymptotic expansion of a definite hypergeometric sum

Calling Sequence

DefiniteSumAsymptotic(T, n, k, l..u, f)

Parameters

T

-

algebraic expression representing a hypergeometric term of both n and k

n

-

name

k

-

name

l..u

-

range for k

f

-

(optional) unevaluated name

Description

• 

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 Sn=k=luT with respect to the variable n (as n approaches ), where l=nr+s and u=nt+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:

1. 

T is defined;

2. 

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 Sn 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:

SgnnC0n+C1na1+...+CmnamnD ⅇQn1b1+O1nc

  

or

SgnnC0n+C1na1+...+CmnamnD ⅇQn1b1+O1

  

or

SgnnC0n+C1na1+...+CmnamnD ⅇQn1b1+On

  

where

– 

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 O1nc, 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 fn0,n1,h,q, where

1. 

n0 is a lower bound for the samples w.r.t. n;

2. 

n1 is an upper bound for the samples w.r.t. n;

3. 

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

4. 

q is the desired number of coefficients _si.

  

These parameters should satisfy the following constraints:

– 

100n0,

– 

h is a positive integer,

– 

n0+10hn1, and

– 

3q.

  

The recommended values for the parameters are 1000n0, 2n0n1,  h&equals;10; q&equals;3 if c&equals;1 and q&equals;6 if c<1. By default, calling f without arguments is equivalent to f1000&comma;2000&comma;10&comma;3.

• 

If there is a conjecture for an exact value s1 of _s1, then fn0&comma;n1&comma;h&comma;q&comma;s1 computes approximate values for the subsequent coefficients. Similarly, it is possible to call fn0&comma;n1&comma;h&comma;q&comma;s1&comma;s2, fn0&comma;n1&comma;h&comma;q&comma;s1&comma;s2&comma;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.

Examples

withSumTools&lsqb;Hypergeometric&rsqb;&colon;

DefiniteSumAsymptoticbinomialn&comma;k&comma;n&comma;k&comma;0..n

2n1&plus;O1n

(1)

DefiniteSumAsymptoticbinomial2n&comma;nkk&comma;n&comma;k&comma;0..n

On2n2

(2)

T:=binomial2n&comma;2k3&colon;

DefiniteSumAsymptoticT&comma;n&comma;k&comma;0..n&comma;&apos;f&apos;

1664n31&plus;O1n&pi;n

(3)

res:=f

res:=1664n31&plus;_s1n&plus;12_s12&plus;_s2n2&plus;_s3&plus;_s1_s2&plus;16_s13n3&plus;O1n4&pi;n&comma;_s1&equals;0.1666666667&comma;_s2&equals;0.004629630518&comma;_s3&equals;0.001544419222

(4)

convertres21&comma;rational&comma;9

_s1&equals;16

(5)

Digits:=20&colon;

res:=f1000&comma;2000&comma;10&comma;3&comma;16

res:=1664n3116n&plus;172&plus;_s1n2&plus;_s216_s111296n3&plus;_s316_s2&plus;172_s1&plus;12_s12&plus;131104n4&plus;O1n5&pi;n&comma;_s1&equals;0.0046296296295318913642&comma;_s2&equals;0.0015432094765491628609&comma;_s3&equals;0.00053636958191821916288

(6)

convertres21&comma;rational&comma;10

_s1&equals;1216

(7)

k&equals;0nT&equals;evalevalres1&comma;&comma;res2

k&equals;0nbinomial2n&comma;2k3&equals;1664n3116n&plus;1108n2&plus;0.0015432094765491628609n3&plus;0.00025773453198581410444n4&plus;O1n5&pi;n

(8)

See Also

asympt, eulermac, LinearAlgebra[LeastSquares], SumTools[Hypergeometric], SumTools[Hypergeometric][DefiniteSum]

References

  

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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