Each optional argument is of the form keyword = value. The following options are supported.

'output'

Specifies the desired form of representations of sequences in the summable space. Possible values:

'RESol'

Indicates that the sequences are to be represented by an RESol data structure, of the form $\mathrm{RESol}\left(\left\{\mathrm{reqn}\right\}\,\left\{v\left(n\right)\right\}\,\mathrm{inits}\right)$, where inits is a set of initial conditions.

'piecewise'

Indicates that the sequences are to be represented by an explicit expression depending on $n$, which in general is a piecewise expression.

This argument is ignored in the AccurateSummation case, and an RESol data structure is returned always. In the Gosper case, the default is piecewise.

'range'=a..b

Specify an interval $R=a..b$ with integer or infinite bounds ($-\mathrm{\infty }..\mathrm{\infty }$ by default). If this option is given then it is assumed that $v\left(n\right)$ is determined only for $n\in R$ and satisfies reqn for all integers $n$ such that both $n$ and $n+1$ are in $R$. Moreover, the discrete Newton-Leibniz formula should be valid for any integers $n_1,n_2\in R$.

'primitive'=truefalse

If this option is given, the command returns a pair $V,T$ where $V$ represents the summable space of all $v\left(n\right)$ and $T$ represents the space of all primitives $u\left(n\right)$. In the Gosper case, both are returned in the form specified by the option 'output'. In the AccurateSummation case, $T$ is returned as an expression in terms of $n$ and $v$ and is typically a piecewise expression. The default is false.

The command SummableSpace(reqn, fcn) or SummableSpace[Gosper](reqn, fcn) constructs the space of all Gosper definite summable sequences $v\left(n\right)$ satisfying the given homogeneous first order linear recurrence reqn with polynomial coefficients, of the form $a_1\left(n\right)v\left(n+1\right)+a_0\left(n\right)v\left(n\right)=0$, for all integers $n$.

The command SummableSpace[AccurateSummation](reqn, fcn) constructs the space of accurate summation definite summable sequences satisfying a given homogeneous linear recurrence reqn of arbitrary order with polynomial coefficients.

The form in which the result is returned is determined by the output option; see below for details. The output may contain placeholders of the form $v\left(0\right),v\left(1\right),\mathrm{...}$ representing initial conditions or free parameters of the resulting space.

Instead of the recurrence, a certificate cert can be specified, in which case the recurrence is taken as $\mathrm{denom}\left(\mathrm{cert}\right)v\left(n+1\right)-\mathrm{numer}\left(\mathrm{cert}\right)v\left(n\right)=0$.

A sequence satisfying a first order linear recurrence is called hypergeometric. A hypergeometric sequence $v\left(n\right)$ is called Gosper indefinite summable if there is another hypergeometric sequence $u\left(n\right)$ such that $v\left(n\right)=u\left(n+1\right)-u\left(n\right)$. The sequence $u\left(n\right)$ is called a primitive for $v\left(n\right)$. A Gosper indefinite summable sequence is called Gosper definite summable if the discrete Newton-Leibniz formula

is valid for any integers $n_1,n_2$.

A sequence $v\left(n\right)$ satisfying a homogeneous linear recurrence with polynomial coefficients of order $d$ is called accurate summation indefinite summable if there is a sequence $u\left(n\right)$ such that $v\left(n\right)=u\left(n+1\right)-u\left(n\right)$ and $u\left(n\right)$ satisfies another homogeneous linear recurrence if the same order $d$. The sequence $u\left(n\right)$ is called a primitive for $v\left(n\right)$. An accurate summation indefinite summable sequence is called accurate summation definite summable if the discrete Newton-Leibniz formula is valid for any integers $n_1,n_2$.

The primitive $u\left(n\right)$ is a linear combination of $v\left(n\right),v\left(n+1\right),\mathrm{...},v\left(n+d-1\right)$ with rational function coefficients, where $d$ is the order of reqn, with the possible exception of finitely many values $n$. In particular, in the Gosper case the primitive is a rational function multiple of $v\left(n\right)$.

If no nonzero summable sequences for reqn exist, then the command returns $\mathrm{FAIL}$.

$\mathrm{with}\left(\mathrm{SumTools}\left[\mathrm{DefiniteSum}\right]\right)\:$

$\mathrm{rec}≔kv\left(k+1\right)-{\left(k+1\right)}^{2}v\left(k\right)=0$

$\mathrm{rec}≔kv\left(k+1\right)-{\left(k+1\right)}^{2}v\left(k\right)=0$

$\mathrm{SummableSpace}\left(\mathrm{rec}\,v\left(k\right)\,'\mathrm{output}'='\mathrm{RESol}'\right)$

$\mathrm{RESol}\left(\left\{\left(-k^2-2k-1\right)v\left(k\right)+kv\left(k+1\right)=0\right\}\,\left\{v\left(k\right)\right\}\,\left\{v\left(\mathrm{-1}\right)=v\left(\mathrm{-1}\right)\,v\left(0\right)=0\,v\left(1\right)=0\right\}\,\mathrm{INFO}\right)$

$V,T≔\mathrm{SummableSpace}\left['\mathrm{Gosper}'\right]\left(\mathrm{rec}\,v\left(k\right)\,'\mathrm{output}'='\mathrm{piecewise}'\,'\mathrm{primitive}'\right)$

$V,T≔\left\{\begin{array}{cc}\frac{v\left(\mathrm{-1}\right){\left(\mathrm{-1}\right)}^{-k}k}{\mathrm{\Gamma }\left(-k\right)}& k\le \mathrm{-1}\\ 0& 0\le k\end{array}\right.,\left\{\begin{array}{cc}\frac{v\left(\mathrm{-1}\right){\left(\mathrm{-1}\right)}^{-k}}{\mathrm{\Gamma }\left(-k\right)}& k\le \mathrm{-1}\\ 0& 0\le k\end{array}\right.$

$\mathrm{add}\left(\mathrm{eval}\left(V\,k=i\right)\,i=-100..100\right)=\mathrm{eval}\left(T\,k=101\right)-\mathrm{eval}\left(T\,k=-100\right)$

$-\frac{v\left(\mathrm{-1}\right)}{933262154439441526816992388562667004907159682643816214685929638952175999932299156089414639761565182862536979208272237582511852109168640000000000000000000000}=-\frac{v\left(\mathrm{-1}\right)}{933262154439441526816992388562667004907159682643816214685929638952175999932299156089414639761565182862536979208272237582511852109168640000000000000000000000}$

$\mathrm{SummableSpace}\left['\mathrm{Gosper}'\right]\left(\mathrm{rec}\,v\left(k\right)\,'\mathrm{range}'=0..\mathrm{\infty }\right)$

$v\left(1\right)\mathrm{\Gamma }\left(k+1\right)k$

$\mathrm{cert}≔\frac{k}{k+2}$

$\mathrm{cert}≔\frac{k}{k+2}$

$\mathrm{SummableSpace}\left['\mathrm{Gosper}'\right]\left(\mathrm{cert}\,k\,v\right)$

$\left\{\begin{array}{cc}0& k\le \mathrm{-2}\\ v\left(\mathrm{-1}\right)& k=\mathrm{-1}\\ -v\left(\mathrm{-1}\right)& k=0\\ 0& 1\le k\end{array}\right.$

$\mathrm{SummableSpace}\left['\mathrm{Gosper}'\right]\left(\mathrm{cert}\,k\,v\,'\mathrm{range}'=1..\mathrm{\infty }\right)$

$\frac{2v\left(1\right)}{\left(k+1\right)k}$

$\mathrm{SummableSpace}\left['\mathrm{Gosper}'\right]\left(2\left(k^2-4\right)\left(k-9\right)v\left(k+1\right)-\left(2k-3\right)\left(k-1\right)\left(k-8\right)v\left(k\right)=0\,v\left(k\right)\right)$

$\left\{\begin{array}{cc}0& k\le \mathrm{-2}\\ 2v\left(1\right)& k=\mathrm{-1}\\ -3v\left(1\right)& k=0\\ v\left(1\right)& k=1\\ -\frac{8v\left(3\right)\mathrm{\Gamma }\left(k-\frac{3}{2}\right)\left(k-2\right)\left(k-9\right)}{\sqrt{\mathrm{\pi }}\mathrm{\Gamma }\left(k+2\right)}& 2\le k\end{array}\right.$

$L≔\left(k-3\right)\left(k-2\right)\left(k+1\right)v\left(k+2\right)-\left(k-3\right)\left(k^2-2k-1\right)v\left(k+1\right)-{\left(k-2\right)}^{2}v\left(k\right)=0$

$L≔\left(k-3\right)\left(k-2\right)\left(k+1\right)v\left(k+2\right)-\left(k-3\right)\left(k^2-2k-1\right)v\left(k+1\right)-{\left(k-2\right)}^{2}v\left(k\right)=0$

$\mathrm{SummableSpace}\left['\mathrm{AccurateSummation}'\right]\left(L\,v\left(k\right)\,\mathrm{primitive}\right)$

$\mathrm{RESol}\left(\left\{\left(-k^2+4k-4\right)v\left(k\right)+\left(-k^3+5k^2-5k-3\right)v\left(k+1\right)+\left(k^3-4k^2+k+6\right)v\left(k+2\right)=0\right\}\,\left\{v\left(k\right)\right\}\,\left\{v\left(2\right)=v\left(2\right)\,v\left(3\right)=0\,v\left(4\right)=v\left(4\right)\,v\left(5\right)=-\frac{v\left(4\right)}{4}\right\}\,\mathrm{INFO}\right),\left\{\begin{array}{cc}\frac{v\left(k\right)}{k-3}+kv\left(k+1\right)& k\le 2\\ 0& k=3\\ \frac{v\left(k\right)}{k-3}+kv\left(k+1\right)& 4\le k\end{array}\right.$

$\mathrm{SummableSpace}\left[\mathrm{AccurateSummation}\right]\left(L\,v\left(k\right)\,\mathrm{range}=4..\mathrm{\infty }\,\mathrm{primitive}\right)$

$\mathrm{RESol}\left(\left\{\left(-k^2+4k-4\right)v\left(k\right)+\left(-k^3+5k^2-5k-3\right)v\left(k+1\right)+\left(k^3-4k^2+k+6\right)v\left(k+2\right)=0\right\}\,\left\{v\left(k\right)\right\}\,\left\{v\left(4\right)=v\left(4\right)\,v\left(5\right)=v\left(5\right)\right\}\,\mathrm{INFO}\right),\frac{v\left(k\right)}{k-3}+kv\left(k+1\right)$

S.A. Abramov. "On the summation of P-recursive sequences." Proc. of ISSAC'06, (2006): 17-22.

The SumTools[DefiniteSum][SummableSpace] command was introduced in Maple 15.

For more information on Maple 15 changes, see Updates in Maple 15.