Concrete mathematics : a foundation for computer science

224 BINOMIAL COEFFICIENTS
Therefore identity (5.114) corresponds to the indefinite summation formula
(-l)%k = (-l)k-’
and to the difference formula
A((-lik(;)) = (-l)k+l (;I;).
It’s easy to start with a function g(k) and to compute Ag(k) = f(k), a
function whose sum will be g(k) + C. But it’s much harder to start with f(k)
and to figure out its indefinite sum x f(k) 6k = g(k) + C; this function g
might not have a simple form. For example, there is apparently no simple
form for x (E) 6k; otherwise we could evaluate sums like xkSn,3 (z) , about
which we’re clueless.
In 1977, R. W. Gosper [124] discovered a beautiful way to decide whether
a given function is indefinitely summable with respect to a general class of
functions called hypergeometric terms. Let us write
F
i; i; k
al, . . . , am a, . . . a, 5
z =
b,, . . ..b., 1) k by. . . bi k!
(5.115)
for the kth term of the hypergeometric series F( al,. . . , a,,,; bl , . . . , b,; z). We
will regard F( al,. . . , a,; bl , . . . , b,; z)k as a function of k, not of z. Gosper’s
decision procedure allows us to decide if there exist parameters c, Al, . . . , AM,
BI, . . . . BN, and Z such that
al, . . . . a,
b,, .,., b,
AI, . . . , AM
BI, . . . , BN
(5.4
given al, . . . , a,, bl, . . . , b,, and z. We will say that a given function
F(al,. . . ,am;b,,. . . , bn;z)k is summable in hypergeometric terms if such
constants C, Al, . . . , AM, Bl, . . . , BN, Z exist.
Let’s write t(k) and T(k) as abbreviations for F(al , . . . , a,,,; bl, . . . , b,; z)k
and F(A,, . . . , AM; B,, . . . , BN; Z)k, respectively. The first step in Gosper’s
decision procedure is to express the term ratio
t(k+ 1) (k+al)...(k+a,)z
~ =
t(k) (k+b,)...(k+b,)(k+l)
in the special form
t(k+ 1) p(k+ 1) q(k)
-=-
0) p(k) r(k+
(5.117)