5.1 BASIC IDENTITIES 171
So much for Table 169. What about sums with three or more binomial coefficients? If the index of summation is spread over all the coefficients, our chances of finding a closed form aren’t great: Only a few closed forms are known for sums of this kind, hence the sum we need might not match the given specs. One of these rarities, proved in exercise 43, is
r
s
=( m )On’
integers m,n 3 0.
(5.28)
Here’s another, more symmetric example:
= (a+b+c)! a’b’c’ . . . ’
integers a, b, c 3 0.
(5.29)
This one has a two-coefficient counterpart, ~(~~~)(~:~)(-l)k = w, integersa,b>O,
( 5 . 3 0 )
which incidentally doesn’t appear in Table 169. The analogous four-coefficient sum doesn’t have a closed form, but a similar sum does:
= (a+b+c+d)! (a+b+c)! (a+b+d)! (a+c+d)! (b+c+d)! (2a+2b+2c+2d)! (a+c)! (b+d)! a! b! c! d! integers a, b, c, d 3 0. This was discovered by John Dougall [69] early in the twentieth century. Is Dougall’s identity the hairiest sum of binomial coefficients known? No! The champion so far is
=(
al +...+a, al,az,...,a, 1 '
integers al, al,. . . , a, > 0.
(5.31)
Here the sum is over (“r’) index variables kii for 1 < i < j < n. Equation (5.29) is the special case n = 3; the case n = 4 can be written out as follows,
