Concrete mathematics : a foundation for computer science

218 BINOMIAL COEFFICIENTS
It’s almost unbelievable, but true, for all b. (Except when a factor in the
denominator vanishes.)
This is fun; let’s try again. Maybe we’ll find a formula that will really
astonish our friends. What Idoes Pfaff’s reflection law tell us if we apply it to
the strange form (s.gg), where z = 2? In this case we set a = -m, b = 1,
and c = -2mf e, obtaining
(-m)“(-2m- 1 + e)” 2k
= lim x
E'O
k>O (-2m + c)k ii
because none of the limiting terms is close to zero. This leads to another
miraculous formula,
(-2)k = (-,yy2,
-l/2
=l/( m > ’
When m = 3, for example, the sum is
integer m 3 0. (5.104)
and (-y2) is indeed equal to -&.
When we looked at our binomial coefficient identities and converted them
to hypergeometric form, we overlooked (5.19) because it was a relation between
two sums instead of a closed form. But now we can regard (5.19) as
an identity between hypergeometric series. If we differentiate it n times with
respect to y and then replace k by m - n - k, we get
m+r n+k
k
Xm-n-k Y
EC k>O m - n - k)( n
/
)
=
-r nfk
n
m - n - k >( >
(-X)m-n-k(X + y)k.
This yields the following hypergeometric transformation:
F
a, -n
( 1)
2. =--
(a-c:)“F a, -n integer
C (-cp ( 1 -n+a-c 1 ‘-’ ) ’ n>O (5.105)
/ .