LEMMA 0.1   Let j, k and h be positive integers such that . Then

 (1)

where, in the j=1 case, .

Proof. We prove (1) using a generating-function approach. We begin by dividing both sides of (1) by (j-1)! and then rewriting the resulting left-hand side as

which, upon re-indexing, becomes

Replacing the binomial coefficient of right-hand side of (1) with the equivalent , and multiplying both sides by (-1)2k-h-j, we now prove (1) by establishing that

 (2)

The right-hand side (2) is simply the coefficient of xh-k in the polynomial (x-1)h-j. This must be true of the left-hand side as well. Expressing (x-1)h-j first as a rational function, and then as a series, we have

Thus, the coefficient of xh-k can be written as

Since r=h-k-s, we obtain, after simplifying, that the coefficient of xh-k can be written as

 (3)

But, noticing that (-1)k+j+s=(-1)k-j-s and that , we find that (3) can be rewritten as

which is the left hand side of (2). Thus both sides of (2) give the coefficient of xh-k in (x-1)h-j. height7pt width7pt depth0pt

Michael Prophet
1999-09-06