1550251515-Classical_Complex_Analysis__Gonzalez_

Sequences, Series, and Special Functions 631

This function is a generalization of the usual factorial n! since

(1 )n = 1 · 2 · 3 · · · n = n!

(a )n is also called a 8hifted factorial, and it is sometimes denoted by the

Appell symbol (a,n).

Theorem 8.50 The factorial function has the following properties:

1. (a)-n = (-l)n /(1 - a)n


2. (a)m+n = (a)m(a + m)n


3. (a)n = r(a + n)/I'(a), Rea=/:-0, -1, -2, ... ; n ~ 1


4. (ahn = 22 n(J)n(~)n


5. (a+ b)n = L;k=O (~)(a)k(b)n-k (Vandermonde's theorem)
   Proofs Properties 1 and 2 follow easily from (8.21-1). Property 2 is also
   valid when m or n, or both, are negative integers.
   (3) From r(a + n) =(a+ n - 1) ... (a+ l)ar(a) = (a)nI'(a) it follows
   that (a)n = r(a + n)/I'(a).
   (4) We have
   (a)2n = a(a + l)(a + 2) .. ·(a+ 2n -1)
   = a(a + 2) ···(a+ 2n - 2) ·(a+ l)(a + 3) ···(a+ 2n -1)

= 2n ( ~) ( ~ + 1) ... ( ~ + n _ 1). 2n ( a ; l ) ( a ; l + 1) ...

. (a;l +n-1)

= 22n ( ~) ( ::.±_!_).

2 n 2 n

(5) We have

(a+ b)i =a+ b = (a) 1 + (b)i

and assuming that the formula holds for n = m, we get

(a+b)m+i = (a+b)m(a+b+m)

m'

=(a+ m - k + b + k) ~ (7)(a)m-k(b)k

= f (7) (a)m-k(a + m - k)(b)k + f (7)(a)m-k(b)k(b + k)

k=O. k=O

= f (7)(a)m-k+i(b)k + t (7)(a)m-k(b)k+i

k=O k=O