ON AN IDENTITY OF DAUBECHIES Doron Zeilberger^{1}

(Appeared in the Amer. Math. Monthly 100(1993), bottom of p. 487.) Tossing a coin (whose P r(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

∑_{i=0}^{n−1} \binom{n + i − 1}{i}p^{n}(1 − p)^{i} + ∑_{i=0}^{n−1} \binom{n + i − 1}{i}p^{i}(1 − p)^{n} = 1 ,

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

∑_{i=1}^{k} ∑_{{0≤α_{j} ≤n−1 \atop j≠i}} \frac{(α_{1} + ... + α_{i−1} + (n − 1) + α_{i+1} + … + α_{k})!}{α_{1}! … α_{i−1}! (n − 1)! α_{i+1}! … α_{k}!}p_{1}^{α_{1}} … p_{i−1}^{α_{i−1}} p_{i}^{n} p_{i+1}^{α_{i+1}} … p_{k}^{α_{k}} = 1 ,

provided p_{1} + ... + p_{k} = 1. References

1. Daubechies, Ingrid, “Ten Lectures on Wavelets”, SIAM, Philadelphia, 1992.

2. , Orthogonal bases of compactly supported wavelets, Comm. Pure Appl. Math., 41(1988), 909-996.