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 k+;+l) &k = (“‘,;t’) , to avoid division by zero, and get I( ,‘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 al, . . . . a,, 1 z kbk = CF h, . . . . b, 1)
AI, . . . . AM, 1
'5, . . . , BN
(5.127)
k’
then we also have 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 bl, . . . . b,
k+l
i i = a, . . . a, z1 F b; . . . b, 1!
al fl, . . . , a,+4 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 bi ..bT, Ai...AT, F =c” i B:. . . BL a\ . . . a,
A1+1, . . ..AM+~. 1 Blfl,. . . . BN+~ I ’> k’
(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)
