Concrete mathematics : a foundation for computer science

How do you write
2 to the W power,
when W is the
complex conjugate
of w ?
pl
I see, the lower
index arrives at
its limit first.
That’s why (;)
is zero when w is
a negative integer.
following formulas:
5.5 HYPERGEOMETRIC FUNCTIONS 211
T(z+l) = z!; (5.86)
(-z)! T(z) = -T-.
sin 712
(5.87)
We can use these generalized factorials to define generalized factorial
powers, when z and w are arbitrary complex numbers:
+= z! .
(z-w)! ’
w= ryz + w)
z
r(z) .
The only proviso is that we must use appropriate limiting values when these
formulas give CXI/OO. (The formulas never give O/O, because factorials and
Gamma-function values are never zero.) A binomial coefficient can be written
0
z
= lim lim
L!
W L-+2 w-w w! (< - w)!
(5.90)
when z and w are any complex numbers whatever.
Armed with generalized factorial tools, we can return to our goal of reducing
the identities derived earlier to their hypergeometric essences. The
binomial theorem (5.13) turns out to be neither more nor less than (5.77),
as we might expect. So the next most interesting identity to try is Vandermonde’s
convolution (5.27):
integer n.
$)(n”k) = (‘i”)~
The kth term here is
T! s!
tk = (r-k)!k! (s-n+k)!(n-k)! ’
and we are no longer too shy to use generalized factorials in these expressions.
Whenever tk contains a factor like (LX + k)!, with a plus sign before
the k, we get (o1+ k + l)!/(a + k)! = k + a + 1 in the term ratio tk+j/tk,
by (5.85); this contributes the parameter ‘a+ 1’ to the corresponding hypergeometric-as
an upper parameter if ( cx + k)! was in the numerator of tk,
but as a lower parameter otherwise. Similarly, a factor like (LX - k)! leads to
(a - k - l)!/(a - k)! = (-l)/(k - a); this contributes ‘-a’ to the opposite
set of parameters (reversing the roles of upper and lower), and negates the
hypergeometric argument. Factors like r!, which are independent of k, go