Concrete mathematics : a foundation for computer science

27 Prove that
F
al, al+;, . . . . a,, a,+;
b,,b,+; ,..., b,,b,+;,;
28 Prove Euler’s identity
2a1,...,2am
2b1,...,2b,
(2m-n-1 z)2
= (, +-a-bF (c-a;-blg
by applying Pfaff’s reflection law (5.101) twice.
29 Show that confluent hypergeometrics satisfy
e’F(;i-z) = F(b;aiz).
30 What hypergeometric series F satisfies zF’(z) + F(z) = l/(1 - z)?
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5 EXERCISES 233
31 Show that if f(k) is any function summable in hypergeometric terms,
then f itself is a multiple of a hypergeometric term. In other words, if
x f(k) 6k = cF(A,, . . . ,AM; Bl,. . . , BN; Z)k + C, then there exist constants
al, . . . , a,, bl, . . . , b,, and z such that f(k) is a constant times
F( al, . . . , a,; bl , . . . , b,; z)k.
32 Find t k2 6k by Gosper’s method.
33 Use Gosper’s method to find t 6k/(k2 - 1).
34 Show that a partial hypergeometric sum can always be represented as a
limit of ordinary hypergeometrics:
k
= F.o F
E-C, bl, . . . , b,
when c is a nonnegative integer. Use this idea to evaluate xkbm (E) (-1 )k.
Homework exercises
35 The notation tkG,, (;)2”-” is ambiguous without context. Evaluate it
a as a sum on k;
b as a sum on n.
36 Let pk be the largest power of the prime p that divides (“‘z”), when m
and n are nonnegative integers. Prove that k is the number of carries
that occur when m is added to n in the radix p number system. Hint:
Exercise 4.24 helps here.