Concrete mathematics : a foundation for computer science

228 BINOMIAL COEFFICIENTS
negative integer. What if it is? Let’s set n = -N, where N is positive. Then
the term ratio for x (z) 6k is
t(k+ 1) -(k+N) p&S ‘I q(k)
___ zz
t(k) (k+l) = ~p(k) r(k+ ‘I
and it should be represented by p(k) = (k+ l)Npl, q(k) = -1, r(k) = 1.
Gosper’s method now tells us to look for a polynomial s(k) of degree d = N -1;
maybe there’s hope after all. For example, when N = 2 we want to solve
k+ 1 = -((k+ l)cxl + LXO) - (km, + Q) .
Equating coefficients of k and 1 tells us that
1 = -a1 - oL1; 1 = -cc~-cx~-cQ;
hence s(k) = -ik - i is a solution, and
CT(k) =
l+;k-$(,2)
k+l
Can this be the desired sum? Yes, it checks out:
= (-l)k(k+l) = i2 .
( >
We can write the summation formula in another form,
= (-‘y-l y . 11
This representation conceals the fact that ( ,‘) is summable in hypergeometric
terms, because [m/21 is not a hypergeometric term.
A catalog of summable hypergeometric terms makes a useful addition
to the database of hypergeometric sums mentioned earlier in this chapter.
Let’s try to compile a list of the sums-in-hypergeometric-terms that we know.
The geometric series x zk 6k is a very special case, which can be written
tzk6k=(z-l))‘zk+Cor
~F(l;‘+)*,k = -&F(l;‘l~k+C. (5.124)
“Excellent,
Holmes!”
“Elementary, my
dear Wa hon. ”