Concrete mathematics : a foundation for computer science

232 BINOMIAL COEFFICIENTS
18 Find an alternative form analogous to (5.35) for the product
(;) (r-y) (r-y).
19 Show that the generalized binomials of (5.58) obey the law
2&(z) = tBp,(-z)-‘.
20 Define a “generalized bloopergeometric series” by the formula
G
al, . . . , am a!. . , at zk
z =
bl, . . . . b, 1) k>O = b+...b$ k!’
using falling powers inst,ead of the rising ones in (5.76). Explain how G is
related to F.
21 Show that Euler’s definition of factorials is consistent with the ordinary
definition, by showing that the limit in (5.83) is 1/ ((m - 1) . . . (1)) when
2 = m is a positive integer.
22 Use (5.83) to prove the factorial duplication formula:
x! (x - i)! = (2x)! (-;)!/22”.
23 What is the value of F(-n, 1; ; 1 )?
24 Find tk (,,,tk) (“$“)4” by using hypergeometric series.
25 Show that
(a1 - bl) F
= alF
al, a2, . . . . a,
bl+1, bz, . . . . b,
al+l, al, . . . . a,
bl+l, b2, . . . . b,
14 --b,F(“d~:~~::;~:“bniL).
Find a similar relation between the hypergeometrics F( al, al, a3 . . . , a,;
bl,... ,bn;z), F(al + ‘l,az,as . . . . a,;bl,..., b,;z), and F(al,az + 1,
as.. . , a,; bl,. . . , b,;z).
26 Express the function G(z) in the formula
F
al, . . . . a,
z = 1 + G(z)
bl, . . . . b, 1)
as a multiple of a hypergeometric series.
By the way,
(-i)! = fi.