By Louis Comtet

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**Additional resources for Analyse Combinatoire **

**Example text**

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.

### Analyse Combinatoire by Louis Comtet

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