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 extended_gosper
 Gosper's algorithm for summation

 Calling Sequence extended_gosper(f, k) extended_gosper(f, k=m..n) extended_gosper(f, k, j)

Parameters

 f - expression k - name, summation variable m, n - expressions, representing upper and lower summation bounds j - integer

Description

 • This function is an implementation of an extension of Gosper's algorithm, and calculates a closed-form (upward) antidifference of a j-fold hypergeometric expression f whenever such an antidifference exists. In this case, the procedure can be used to calculate definite sums

$\sum _{k=m}^{n}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}f\left(k\right)$

 whenever f does not depend on variables occurring in m and n.
 • An expression f is called a j-fold hypergeometric expression with respect to k if

$\frac{f\left(k+j\right)}{f\left(k\right)}$

 is rational with respect to k. This is typically the case for ratios of products of rational functions, exponentials, factorials, binomial coefficients, and Pochhammer symbols that are rational-linear in their arguments. The implementation supports this type of input.
 • An expression g is called an upward antidifference of f if

$f\left(k\right)=g\left(k+1\right)-g\left(k\right)$

 • An expression g is called j-fold upward antidifference of f if

$f\left(k\right)=g\left(k+j\right)-g\left(k\right)$

 • If the second argument k is a name, and extended_gosper is invoked with two arguments, then extended_gosper returns the closed form (upward) antidifference of f with respect to k, if applicable.
 • If the second argument has the form $k=m..n$ then the definite sum

$\sum _{k=m}^{n}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}f\left(k\right)$

 is determined if Gosper's algorithm applies.
 • If extended_gosper is invoked with three arguments then the third argument is taken as the integer j, and a j-fold upward antidifference of f is returned whenever it is a j-fold hypergeometric term.
 • If the result FAIL occurs, then the implementation has proved either that the input function f is no j-fold hypergeometric term, or that no j-fold hypergeometric antidifference exists.
 • The command with(sumtools,extended_gosper) allows the use of the abbreviated form of this command.

Examples

 > $\mathrm{with}\left(\mathrm{sumtools}\right):$

see (SIAM Review, 1994, Problem 94-2)

 > $\mathrm{extended_gosper}\left(\frac{{\left(-1\right)}^{k+1}\left(4k+1\right)\left(2k\right)!}{k!{4}^{k}\left(2k-1\right)\left(k+1\right)!},k\right)$
 ${-}\frac{{2}{}\left({k}{+}{1}\right){}{\left({-}{1}\right)}^{{k}{+}{1}}{}\left({2}{}{k}\right){!}}{{k}{!}{}{{4}}^{{k}}{}\left({2}{}{k}{-}{1}\right){}\left({k}{+}{1}\right){!}}$ (1)
 > $\mathrm{extended_gosper}\left(\frac{\mathrm{binomial}\left(n,k\right)}{{2}^{n}}-\frac{\mathrm{binomial}\left(n-1,k\right)}{{2}^{n-1}},k\right)$
 ${-}\frac{{k}{}\left(\frac{{\mathrm{binomial}}{}\left({n}{,}{k}\right)}{{{2}}^{{n}}}{-}\frac{{\mathrm{binomial}}{}\left({n}{-}{1}{,}{k}\right)}{{{2}}^{{n}{-}{1}}}\right)}{{2}{}{k}{-}{n}}$ (2)
 > $\mathrm{extended_gosper}\left(\frac{\mathrm{pochhammer}\left(b,\frac{k}{2}\right)}{\left(\frac{k}{2}\right)!},k\right)$
 $\frac{{1}}{{2}}{}\frac{{k}{}{\mathrm{pochhammer}}{}\left({b}{,}\frac{{1}}{{2}}{}{k}\right)}{{b}{}\left(\frac{{1}}{{2}}{}{k}\right){!}}{+}\frac{{1}}{{2}}{}\frac{\left({k}{+}{1}\right){}{\mathrm{pochhammer}}{}\left({b}{,}\frac{{1}}{{2}}{}{k}{+}\frac{{1}}{{2}}\right)}{{b}{}\left(\frac{{1}}{{2}}{}{k}{+}\frac{{1}}{{2}}\right){!}}$ (3)
 > $\mathrm{extended_gosper}\left(\left(\frac{k}{2}\right)!,k\right)$
 ${\mathrm{FAIL}}$ (4)
 > $\mathrm{extended_gosper}\left(k\left(\frac{k}{2}\right)!,k\right)$
 ${2}{}\left(\frac{{1}}{{2}}{}{k}\right){!}{+}{2}{}\left(\frac{{1}}{{2}}{}{k}{+}\frac{{1}}{{2}}\right){!}$ (5)
 > $\mathrm{extended_gosper}\left(k\left(\frac{k}{2}\right)!,k,2\right)$
 ${2}{}\left(\frac{{1}}{{2}}{}{k}\right){!}$ (6)
 > $\mathrm{extended_gosper}\left(k\left(\frac{k}{2}\right)!,k=1..n\right)$
 ${2}{}\left(\frac{{1}}{{2}}{}{n}{+}\frac{{1}}{{2}}\right){!}{+}{2}{}\left(\frac{{1}}{{2}}{}{n}{+}{1}\right){!}{-}{2}{}\left(\frac{{1}}{{2}}\right){!}{-}{2}{\cdot }{1}{!}$ (7)