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

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

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

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

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

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.

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

$\mathrm{extended\_gosper}\left(\frac{{\left(-1\right)}^{k+1}\left(4k+1\right)\left(2k\right)!}{k!4^k\left(2k-1\right)\left(k+1\right)!}\,k\right)$

$-\frac{2\left(k+1\right){\left(\mathrm{-1}\right)}^{k+1}\left(2k\right)!}{k!4^k\left(2k-1\right)\left(k+1\right)!}$

$\mathrm{extended\_gosper}\left(\frac{\mathrm{binomial}\left(n\,k\right)}{2^n}-\frac{\mathrm{binomial}\left(n-1\,k\right)}{2^{n-1}}\,k\right)$

$-\frac{k\left(\frac{\left(\genfrac{}{}{0ex}{}{n}{k}\right)}{2^n}-\frac{\left(\genfrac{}{}{0ex}{}{n-1}{k}\right)}{2^{n-1}}\right)}{2k-n}$

$\mathrm{extended\_gosper}\left(\frac{\mathrm{pochhammer}\left(b\,\frac{k}{2}\right)}{\left(\frac{k}{2}\right)!}\,k\right)$

$\frac{k\mathrm{pochhammer}\left(b\,\frac{k}{2}\right)}{2b\left(\frac{k}{2}\right)!}+\frac{\left(k+1\right)\mathrm{pochhammer}\left(b\,\frac{k}{2}+\frac{1}{2}\right)}{2b\left(\frac{k}{2}+\frac{1}{2}\right)!}$

$\mathrm{extended\_gosper}\left(\left(\frac{k}{2}\right)!\,k\right)$

$\mathrm{FAIL}$

$\mathrm{extended\_gosper}\left(k\left(\frac{k}{2}\right)!\,k\right)$

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

$\mathrm{extended\_gosper}\left(k\left(\frac{k}{2}\right)!\,k\,2\right)$

$2\left(\frac{k}{2}\right)!$

$\mathrm{extended\_gosper}\left(k\left(\frac{k}{2}\right)!\,k=1..n\right)$

$2\left(\frac{n}{2}+\frac{1}{2}\right)!+2\left(\frac{n}{2}+1\right)!-2\left(\frac{1}{2}\right)!-21!$