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 $S\left(n\right)=\sum _{k=l}^{u}T$ with respect to the variable $n$ (as $n$ approaches $\mathrm{\infty }$), where $l=rn+s$ and $u=tn+v$ 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 $S\left(n\right)$ 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

$\mathrm{Sgn}$ is 1 or -1,

$C_0$, $C_1$, ..., $C_m$, $D$ are constants,

$a_1$, ..., $a_m$ are positive rational numbers $\le 1$,

$c$ is a positive rational number,

$b$ is a positive integer, and

$Q$ is a polynomial of degree $\le b$.

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 $\mathrm{O}\left(\frac{1}{n^c}\right)$, 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 ${\mathrm{\_s}}_1$, ${\mathrm{\_s}}_2$, ... The second element is a list of equations ${\mathrm{\_s}}_1=\mathrm{s1}$, ${\mathrm{\_s}}_2=\mathrm{s2}$, ... where $\mathrm{s1}$, $\mathrm{s2}$, ... are floating-point numbers approximating the values of ${\mathrm{\_s}}_1$, ${\mathrm{\_s}}_2$, ...

The typical calling sequence of the auxiliary procedure is $f\left(n_0\,n_1\,h\,q\right)$, where

$n_0$ is a lower bound for the samples w.r.t. $n$;

$n_1$ 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 ${\mathrm{\_s}}_i$.

These parameters should satisfy the following constraints:

$100\le n_0$,

$h$ is a positive integer,

$n_0+10h\le n_1$, and

$3\le q$.

The recommended values for the parameters are $1000\le n_0$, $2n_0\le n_1$,  $h=10$; $q=3$ if $c=1$ and $q=6$ if $c<1$. By default, calling $f\left(\right)$ without arguments is equivalent to $f\left(1000\&comma;2000\&comma;10\&comma;3\right)$.

If there is a conjecture for an exact value $\mathrm{s1}$ of ${\mathrm{\_s}}_1$, then $f\left(n_0\&comma;n_1\&comma;h\&comma;q\&comma;\left[\mathrm{s1}\right]\right)$ computes approximate values for the subsequent coefficients. Similarly, it is possible to call $f\left(n_0\&comma;n_1\&comma;h\&comma;q\&comma;\left[\mathrm{s1}\&comma;\mathrm{s2}\right]\right)$, $f\left(n_0\&comma;n_1\&comma;h\&comma;q\&comma;\left[\mathrm{s1}\&comma;\mathrm{s2}\&comma;\mathrm{s3}\right]\right)$, 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 $\mathrm{s1}$, $\mathrm{s2}$, ... can be increased by calling f with higher values of $n_0$, $n_1$, and Digits. Generally, the values $\mathrm{si}$ are less accurate the higher the index $i$ is.

$\mathrm{with}\left(\mathrm{SumTools}\left[\mathrm{Hypergeometric}\right]\right)\&colon;$

$\mathrm{DefiniteSumAsymptotic}\left(\mathrm{binomial}\left(n\&comma;k\right)\&comma;n\&comma;k\&comma;0..n\right)$

$2^n\left(1+\mathrm{O}\left(\frac{1}{n}\right)\right)$

$\mathrm{DefiniteSumAsymptotic}\left(\mathrm{binomial}\left(2n\&comma;n-k\right)k\&comma;n\&comma;k\&comma;0..n\right)$

$\mathrm{O}\left(n\right){\left(2^n\right)}^2$

$T≔{\mathrm{binomial}\left(2n\&comma;2k\right)}^3\&colon;$

$\mathrm{DefiniteSumAsymptotic}\left(T\&comma;n\&comma;k\&comma;0..n\&comma;'f'\right)$

$\frac{{64}^n\sqrt{3}\left(1+\mathrm{O}\left(\frac{1}{n}\right)\right)}{6\mathrm{\pi }n}$

$\mathrm{res}≔f\left(\right)$

$\mathrm{res}≔\frac{{64}^n\sqrt{3}\left(1+\frac{{\mathrm{\_s}}_1}{n}+\frac{\frac{{\mathrm{\_s}}_1^2}{2}+{\mathrm{\_s}}_2}{n^2}+\frac{{\mathrm{\_s}}_3+{\mathrm{\_s}}_1{\mathrm{\_s}}_2+\frac{1}{6}{\mathrm{\_s}}_1^3}{n^3}+\mathrm{O}\left(\frac{1}{n^4}\right)\right)}{6\mathrm{\pi }n},\left[{\mathrm{\_s}}_1=\mathrm{-0.1666666667}\&comma;{\mathrm{\_s}}_2=\mathrm{-0.004629630518}\&comma;{\mathrm{\_s}}_3=0.001544419223\right]$

$\mathrm{convert}\left(\mathrm{res}\left[2\right]\left[1\right]\&comma;\mathrm{rational}\&comma;9\right)$

${\mathrm{\_s}}_1=-\frac{1}{6}$

$\mathrm{Digits}≔20\&colon;$

$\mathrm{res}≔f\left(1000\&comma;2000\&comma;10\&comma;3\&comma;\left[-\frac{1}{6}\right]\right)$

$\mathrm{res}≔\frac{{64}^n\sqrt{3}\left(1-\frac{1}{6n}+\frac{\frac{1}{72}+{\mathrm{\_s}}_1}{n^2}+\frac{{\mathrm{\_s}}_2-\frac{{\mathrm{\_s}}_1}{6}-\frac{1}{1296}}{n^3}+\frac{{\mathrm{\_s}}_3-\frac{1}{6}{\mathrm{\_s}}_2+\frac{1}{72}{\mathrm{\_s}}_1+\frac{1}{2}{\mathrm{\_s}}_1^2+\frac{1}{31104}}{n^4}+\mathrm{O}\left(\frac{1}{n^5}\right)\right)}{6\mathrm{\pi }n},\left[{\mathrm{\_s}}_1=\mathrm{-0.0046296296295318913642}\&comma;{\mathrm{\_s}}_2=0.0015432094765491628609\&comma;{\mathrm{\_s}}_3=0.00053636958191821916288\right]$

$\mathrm{convert}\left(\mathrm{res}\left[2\right]\left[1\right]\&comma;\mathrm{rational}\&comma;10\right)$

${\mathrm{\_s}}_1=-\frac{1}{216}$

$\mathrm{Sum}\left(T\&comma;k=0..n\right)=\mathrm{eval}\left(\mathrm{eval}\left(\mathrm{res}\left[1\right]\&comma;\right)\&comma;\mathrm{res}\left[2\right]\right)$

$\sum _{k=0}^n{\left(\genfrac{}{}{0ex}{}{2n}{2k}\right)}^3=\frac{{64}^n\sqrt{3}\left(1-\frac{1}{6n}+\frac{1}{108n^2}+\frac{0.0015432094765491628609}{n^3}+\frac{0.00025773453198581410444}{n^4}+\mathrm{O}\left(\frac{1}{n^5}\right)\right)}{6\mathrm{\pi }n}$

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