By Marko Petkovsek, Herbert S. Wilf, Doron Zeilberger
This booklet is of curiosity to mathematicians and laptop scientists operating in finite arithmetic and combinatorics. It offers a leap forward approach for studying advanced summations. superbly written, the booklet comprises functional purposes in addition to conceptual advancements that would have purposes in different components of mathematics.From the desk of contents: * evidence Machines * Tightening the objective * The Hypergeometric Database * The 5 easy Algorithms: Sister Celine's procedure, Gosper&'s set of rules, Zeilberger's set of rules, The WZ Phenomenon, set of rules Hyper * Epilogue: An Operator Algebra point of view * The WWW websites and the software program (Maple and Mathematica) each one bankruptcy comprises an creation to the topic and ends with a collection of routines.
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D. dissertation of Sister Mary Celine Fasenmyer, in 1945. It showed how recurrences for certain polynomial sequences could be found algorithmically. ) • [Gosp78], by R. W. , is the discovery of the algorithmic solution of the problem of indefinite hypergeometric summation (see Chapter 5). Such a summation is of the form f (n) = nk=0 F (k), where F is hypergeometric. • [Zeil82], of Zeilberger, recognized that Sister Celine’s method would also be the basis for proving combinatorial identities by recurrence.
A − b − c)! 6 1 a ;z = − (1 − z)a Γ( 12 )Γ(c + 12 )Γ( 12 + a2 + 2b )Γ( 12 − a2 − a, b, c ; 1 = 1+a+b , 2c Γ( 12 + a2 )Γ( 12 + 2b )Γ( 12 − a2 + c)Γ( 12 − 2 b 2 b 2 + c) + c) π21−2c (d − 1)! (2c + d)! a, 1 − a, c ; 1 = a−d−1 a+d d, 1 + 2c − d ( 2 )! ( 2 − 1)! (c − a+d )! ( d−a−1 )! 2 2 Using the database Let’s review where we are. In this chapter we have seen how to take a sum and identify it, when possible, as a standard hypergeometric sum. We have also seen a list of many of the important hypergeometric sums that can be expressed in simple, closed form.
K, 0, n}], we find that Mathematica is very well trained indeed, since it gives LaguerreL[n, 0, 1] which means that it recognizes our sum as a Laguerre polynomial! The trick of inserting xk won’t change this behavior, so there isn’t any way to adapt this routine to the present example. ) and then ask for3 the term ratio, FactorialSimplify[t[k+1]/t[k]]. We would obtain the term ratio in the nicely factored form k−n . 4, −n ;1 . 1F1 1 3 Read in DiscreteMath‘RSolve‘ before attempting to FactorialSimplify something.
A=B by Marko Petkovsek, Herbert S. Wilf, Doron Zeilberger