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\bf
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\centerline{ON AN IDENTITY OF DAUBECHIES}
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\noindent
\it
\centerline{Doron Zeilberger\footnote{$^1$}
{\eightrm 
Department of Mathematics, Temple University,
Philadelphia, PA 19122, USA. 
\break
{\eighttt zeilberg@euclid.math.temple.edu .}
Supported in part by the NSF.
}}
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(Appeared in the Amer. Math. Monthly 100(1993), bottom of p. 487.)
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\noindent
\rm
Tossing a coin (whose $Pr(head)=p$) until reaching $n$ heads or $n$ 
tails and equating
the probability,$1$, of finishing with the sum of the probabilities
of all the possible final outcomes leads to 
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$$
\sum_{i=0}^{n-1}{{n+i-1} \choose {i}} p^n (1-p)^i 
+\sum_{i=0}^{n-1} {{n+i-1} \choose {i}} p^i (1-p)^n  =1 \quad ,
$$
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which was proved in [1](pp. 167-171) and [2] using Bezout's theorem and 
induction respectively.
Rolling a $k$-faced die instead leads to the multivariate generalization
\smallskip
$$
\sum_{i=1}^k \sum_{ {0 \leq \alpha_j \leq n-1} \atop {j \not=i} }
{ 
{( \alpha_1 + ... + \alpha_{i-1}+ (n-1) + \alpha_{i+1} + \dots + \alpha_k )!}
\over
{ \alpha_1 ! \dots \alpha_{i-1}!\, (n-1)!\, \alpha_{i+1}! \dots \alpha_k ! } 
}
p_1^{\alpha_1} \dots p_{i-1}^{\alpha_{i-1}}\, p_i^n \,
p_{i+1}^{\alpha_{i+1}} \dots \, p_{k}^{\alpha_k} =1 \quad ,
$$
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provided $p_1 + ... + p_k =1$.
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\noindent
{\bf References}
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\noindent
1. Daubechies, Ingrid, ``{\it Ten Lectures on Wavelets}'',
SIAM, Philadelphia, 1992.
\par
2.\underbar {\qquad \qquad   }, {\it Orthogonal bases of compactly supported wavelets},
Comm. Pure Appl. Math., {\bf 41}(1988), 909-996.
\bye
 
