### 95.1 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.

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#### 95.1.1 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

$$ F_k = f_{k+1} - f_k. $$

#### 95.1.2 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

$$ a_0 \, F_{n,k} + \ldots + a_d \, F_{n+d,k} = \Delta_K \left(R\left(n,k\right) F_{n,k}\right), $$

where \(\Delta_k\) is the k-forward difference operator, i.e., \(\Delta_k \left(t_k\right) \equiv t_{k+1} - t_k.\)

#### 95.1.3 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 intermediate steps

`VeryVerbose`

  More information

`Extra`

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

For example:
`GosperVerbose`, `parGosperVeryVerbose`, `ZeilbergerExtra`, `AntiDifferenceSummary`.