Hypergeometric identities

The hypergeometric identities are results on sums of terms of a hypergeometric series. These identities frequently appear in problems of Combinatoire and of algorithm analyzes. The first identities were found with the hand by brilliant mathematicians like Carl Friedrich Gauss or Ernst Kummer. Now, the objective is to obtain algorithms which automate the demonstrations of these inequalities.

The list of the hypergeometric identities is sometimes called list of Bailey following the work of Bailey.

Among the hypergeometric identities most traditional

\ sum_ {i=0} ^ {N} {N \ choose I} = 2^ {N}, \ qquad \ sum_ {i=0} ^ {N} {N \ choose I} ^2 = {2n \ choose N} \ qquad \ mbox {and} \ qquad \ sum_ {K} K {N \ choose K} = n2^ {n-1}.

Hypergeometric series.

With term tk has hypergeometric term yew

\ frac {t_ {k+1}} {t_k}

has rational function in K .

With term F (N, K) has hypergeometric term yew

\ frac {F (N, k+1)}{F (N, K)}

has rational function in K .

There exist two standard off sums over hypergeometric terms, the definite and indefiniteness sums. With definite sum is off the form

\ sum_ {K} t_k. \,

The indefiniteness sum is off the form

\ sum_ {k=0} ^ {N} F (N, K). -->

Automation of the proof

The automated proof rests on two stages:
  • to find an expression simple of the sum hypergeometric, in best of the cases a closed form;
  • to show by A=B that this expression is quite equal to the initial sum.

For each type of hypergeometric nap, there exist many methods to find an expression simple. These methods offer also a proof of the equality. One can name:

  • for the definite sums: method of sister Celine Fasenmyer, the algorithm of Zeilberger
  • for the indefinite sums: the algorithm of Gosper.

The methods employed often call upon results of the formal Calcul.

External bonds

  • Marko Petkovšek, Herbert Wilf and Doron Zeilberger, '' has = B ''. It is about a work clarifying an algorithm to find a relation of recurrence starting from an identity. Calling upon a formal computation software, such Maple and Mathematica, the algorithm put an end to the need to hold a catalog of identities and relations of recurrence.
  • On the algorithm of sister Celine
  • Examples of special functions

Reference

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