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### 77.1 Introduction to zeilberger

`zeilberger` is an implementation of Zeilberger’s algorithm for definite hypergeometric summation, and also Gosper’s algorithm for indefinite hypergeometric summation. `zeilberger` makes use of the "filtering" optimization method developed by Axel Riese. `zeilberger` was developed by Fabrizio Caruso. `load(zeilberger)` loads this package.

#### 77.1.1 The indefinite summation problem

`zeilberger` implements Gosper’s algorithm for indefinite hypergeometric summation. Given a hypergeometric term F_k in k we want to find its hypergeometric anti-difference, that is, a hypergeometric term f_k such that F_k = f_(k+1) - f_k.

#### 77.1.2 The definite summation problem

`zeilberger` implements Zeilberger’s algorithm for definite hypergeometric summation. Given a proper hypergeometric term (in n and k)

F_(n,k) and a positive integer d we want to find a d-th order linear recurrence with polynomial coefficients (in n) for F_(n,k) and a rational function R in n and k such that

a_0 F_(n,k) + ... + a_d F_(n+d),k = Delta_k(R(n,k) F_(n,k)),

where Delta_k is the k-forward difference operator, i.e., Delta_k(t_k) := t_(k+1) - t_k.

#### 77.1.3 Verbosity levels

There are also verbose versions of the commands which are called by adding one of the following prefixes:

`Summary`

  Just a summary at the end is shown

`Verbose`

  Some information in the intermidiate steps

`VeryVerbose`

  More information

`Extra`

  Even more information including information on the linear system in Zeilberger’s algorithm

For example:

`GosperVerbose`, `parGosperVeryVerbose`, `ZeilbergerExtra`, `AntiDifferenceSummary`.

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