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  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.