Concrete mathematics : a foundation for computer science

5.7 PARTIAL HYPERGEOMETRIC SUMS 229
We also computed 1 kzk 6k in Chapter 2. This summand is zero when
k = 0, so we get a more suitable hypergeometric term by considering the sum
1 (k + 1 )zk 6k instead. The appropriate formula turns out to be
(5.125)
in hypergeometric notation.
There’s also the formula 1 (k) 6k = (,:,), equation (5.10); we write it
I(
k+;+l) &k = (“‘,;t’) , to avoid division by zero, and get
,‘6k = &F(n+;‘l(‘)k, n # -1. ( 5 . 1 2 6 )
Identity (5.9) turns out to be equivalent to this, when we express it hypergeometrically.
In general if we have a summation formula of the form
then we also have
al, . . . . a,, 1
AI, . . . . AM, 1
z kbk = CF
h, . . . . b, 1) '5, . . . , BN k’
al, . . . . a,, 1
bl, . . . . bn k+l ’
for any integer 1. There’s a general formula for shifting the index by 1:
F
al, . . . , am i i
=
a, . . . a, z1 F al fl, . . . , a,+4 1
bl, . . . . b,
k+l b; . . . b, 1! bl+1, . . . , b,+l,l+l 1) ’ k ’
Hence any given identity (5.127) has an infinite number of shifted forms:
a1 +1, . . . , a,+4 1
z 6k
bltl, . . . . b,+l 1) k
=c”
bi ..bT, Ai...AT, F A1+1, . . ..AM+~. 1
a\ . . . a,
i B:. . . BL Blfl, . . . . BN+~ I>’ k’
(5.127)
(5.128)
There’s usually a fair amount of cancellation among the a’s, A’s, b’s, and
B’s here. For example, if we apply this shift formula to (5.126), we get the
general identity
k6k = sF(n+;';'lll)k, (5.129)