Page 772 - Handbook Of Integral Equations

Supplement 10

Special Functions and Their Properties

Throughout Supplement 10 it is assumed that n is a positive integer, unless otherwise specified.

10.1. Some Symbols and Coefficients

Factorial
0! =1!=1, n!=1 ⋅ 2 ⋅ 3 ... (n – 1)n, n =2, 3, ... ,
n
(2n)!!=2 ⋅ 4 ⋅ 6 ... (2n – 2)(2n)=2 n!,
2 n+1 3
(2n + 1)!! = 1 ⋅ 3 ⋅ 5 ... (2n – 1)(2n +1) = √ Γ n + ,
π 2

(2k)!! if n =2k,
n!! = 0!!=1.
(2k + 1)!! if n =2k +1,
Binomial coefficients
n!
k
C = , where k =1, ... , n,
n
k!(n – k)!
k k (–a) k a(a – 1) ... (a – k +1)
C =(–1) = , where k =1, 2, ...
a
k! k!
General case:
Γ(a +1)
b
C = , where Γ(x) is the gamma function.
a
Γ(b +1)Γ(a – b +1)
Properties:
0
k
C =1, C = 0 for k = –1, –2, ... or k > n,
a n
a b a – b b b b+1 b+1
b+1
C = C = C , C + C = C ,
a a–1 a a a a+1
b +1 b +1
(–1) n n n (2n – 1)!!
n
C = C =(–1) ,
–1/2 2n 2n
2 (2n)!!
(–1) n–1 n–1 (–1) n–1 (2n – 3)!!
n
C = C = ,
1/2 2n–1 2n–2
n2 n (2n – 2)!!
n
n
2n
n –4n–1
2n+1
C n+1/2 =(–1) 2 C , C 2n+1/2 =2 –2n C 4n+1 ,
2n
2 2n+1 n/2 2 2n (n–1)/2
1/2
C n = n , C n = C n .
πC π
2n
© 1998 by CRC Press LLC
© 1998 by CRC Press LLC
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