Hypergeometric[DefiniteSumAsymptotic] - 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 $S\left(n\right)=\sum _{k=l}^{u}T$ 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 $l..u$:
 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 $S\left(n\right)$ 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:

$\mathrm{Sgn}{n}^{{C}_{0}n+{C}_{1}{n}^{{a}_{1}}+\mathrm{...}+{C}_{m}{n}^{{a}_{m}}}{n}^{D}{ⅇ}^{Q\left({n}^{\frac{1}{b}}\right)}\left(1+\mathrm{O}\left(\frac{1}{{n}^{c}}\right)\right)$

 or

$\mathrm{Sgn}{n}^{{C}_{0}n+{C}_{1}{n}^{{a}_{1}}+\mathrm{...}+{C}_{m}{n}^{{a}_{m}}}{n}^{D}{ⅇ}^{Q\left({n}^{\frac{1}{b}}\right)}\left(1+\mathrm{O}\left(1\right)\right)$

 or

$\mathrm{Sgn}{n}^{{C}_{0}n+{C}_{1}{n}^{{a}_{1}}+\mathrm{...}+{C}_{m}{n}^{{a}_{m}}}{n}^{D}{ⅇ}^{Q\left({n}^{\frac{1}{b}}\right)}\left(1+\mathrm{O}\left(n\right)\right)$

 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
 1 ${n}_{0}$ is a lower bound for the samples w.r.t. $n$;
 2 ${n}_{1}$ 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 ${\mathrm{_s}}_{i}$.
 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 $1000\le {n}_{0}$, $2{n}_{0}\le {n}_{1}$,  $h=10$; $q=3$ if $c=1$ and $q=6$ if $c<1$. By default, calling $f\left(\right)$ without arguments is equivalent to $f\left(1000,2000,10,3\right)$.
 • If there is a conjecture for an exact value $\mathrm{s1}$ of ${\mathrm{_s}}_{1}$, then $f\left({n}_{0},{n}_{1},h,q,\left[\mathrm{s1}\right]\right)$ computes approximate values for the subsequent coefficients. Similarly, it is possible to call $f\left({n}_{0},{n}_{1},h,q,\left[\mathrm{s1},\mathrm{s2}\right]\right)$, $f\left({n}_{0},{n}_{1},h,q,\left[\mathrm{s1},\mathrm{s2},\mathrm{s3}\right]\right)$, 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.

Examples

 > $\mathrm{with}\left(\mathrm{SumTools}\left[\mathrm{Hypergeometric}\right]\right):$
 > $\mathrm{DefiniteSumAsymptotic}\left(\mathrm{binomial}\left(n,k\right),n,k,0..n\right)$
 ${{2}}^{{n}}{}\left({1}{+}{\mathrm{O}}{}\left(\frac{{1}}{{n}}\right)\right)$ (1)
 > $\mathrm{DefiniteSumAsymptotic}\left(\mathrm{binomial}\left(2n,n-k\right)k,n,k,0..n\right)$
 ${\mathrm{O}}{}\left({n}\right){}{\left({{2}}^{{n}}\right)}^{{2}}$ (2)
 > $T≔{\mathrm{binomial}\left(2n,2k\right)}^{3}:$
 > $\mathrm{DefiniteSumAsymptotic}\left(T,n,k,0..n,'f'\right)$
 $\frac{{{64}}^{{n}}{}\sqrt{{3}}{}\left({1}{+}{\mathrm{O}}{}\left(\frac{{1}}{{n}}\right)\right)}{{6}{}{\mathrm{\pi }}{}{n}}$ (3)
 > $\mathrm{res}≔f\left(\right)$
 ${\mathrm{res}}{≔}\frac{{{64}}^{{n}}{}\sqrt{{3}}{}\left({1}{+}\frac{{{\mathrm{_s}}}_{{1}}}{{n}}{+}\frac{\frac{{{\mathrm{_s}}}_{{1}}^{{2}}}{{2}}{+}{{\mathrm{_s}}}_{{2}}}{{{n}}^{{2}}}{+}\frac{{{\mathrm{_s}}}_{{3}}{+}{{\mathrm{_s}}}_{{1}}{}{{\mathrm{_s}}}_{{2}}{+}\frac{{1}}{{6}}{}{{\mathrm{_s}}}_{{1}}^{{3}}}{{{n}}^{{3}}}{+}{\mathrm{O}}{}\left(\frac{{1}}{{{n}}^{{4}}}\right)\right)}{{6}{}{\mathrm{\pi }}{}{n}}{,}\left[{{\mathrm{_s}}}_{{1}}{=}{-0.1666666667}{,}{{\mathrm{_s}}}_{{2}}{=}{-0.004629630518}{,}{{\mathrm{_s}}}_{{3}}{=}{0.001544419222}\right]$ (4)
 > $\mathrm{convert}\left(\mathrm{res}\left[2\right]\left[1\right],\mathrm{rational},9\right)$
 ${{\mathrm{_s}}}_{{1}}{=}{-}\frac{{1}}{{6}}$ (5)
 > $\mathrm{Digits}≔20:$
 > $\mathrm{res}≔f\left(1000,2000,10,3,\left[-\frac{1}{6}\right]\right)$
 ${\mathrm{res}}{≔}\frac{{{64}}^{{n}}{}\sqrt{{3}}{}\left({1}{-}\frac{{1}}{{6}{}{n}}{+}\frac{\frac{{1}}{{72}}{+}{{\mathrm{_s}}}_{{1}}}{{{n}}^{{2}}}{+}\frac{{{\mathrm{_s}}}_{{2}}{-}\frac{{{\mathrm{_s}}}_{{1}}}{{6}}{-}\frac{{1}}{{1296}}}{{{n}}^{{3}}}{+}\frac{{{\mathrm{_s}}}_{{3}}{-}\frac{{1}}{{6}}{}{{\mathrm{_s}}}_{{2}}{+}\frac{{1}}{{72}}{}{{\mathrm{_s}}}_{{1}}{+}\frac{{1}}{{2}}{}{{\mathrm{_s}}}_{{1}}^{{2}}{+}\frac{{1}}{{31104}}}{{{n}}^{{4}}}{+}{\mathrm{O}}{}\left(\frac{{1}}{{{n}}^{{5}}}\right)\right)}{{6}{}{\mathrm{\pi }}{}{n}}{,}\left[{{\mathrm{_s}}}_{{1}}{=}{-0.0046296296295318913642}{,}{{\mathrm{_s}}}_{{2}}{=}{0.0015432094765491628609}{,}{{\mathrm{_s}}}_{{3}}{=}{0.00053636958191821916288}\right]$ (6)
 > $\mathrm{convert}\left(\mathrm{res}\left[2\right]\left[1\right],\mathrm{rational},10\right)$
 ${{\mathrm{_s}}}_{{1}}{=}{-}\frac{{1}}{{216}}$ (7)
 > $\mathrm{Sum}\left(T,k=0..n\right)=\mathrm{eval}\left(\mathrm{eval}\left(\mathrm{res}\left[1\right],\right),\mathrm{res}\left[2\right]\right)$
 ${\sum }_{{k}{=}{0}}^{{n}}{}{\left(\genfrac{}{}{0}{}{{2}{}{n}}{{2}{}{k}}\right)}^{{3}}{=}\frac{{{64}}^{{n}}{}\sqrt{{3}}{}\left({1}{-}\frac{{1}}{{6}{}{n}}{+}\frac{{1}}{{108}{}{{n}}^{{2}}}{+}\frac{{0.0015432094765491628609}}{{{n}}^{{3}}}{+}\frac{{0.00025773453198581410444}}{{{n}}^{{4}}}{+}{\mathrm{O}}{}\left(\frac{{1}}{{{n}}^{{5}}}\right)\right)}{{6}{}{\mathrm{\pi }}{}{n}}$ (8)

References

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