ABSTRACT

Definitions (special cases): Ckn =

(n k

) =

n! k! (n – k)! , where k = 1, . . . ,n;

C0a = 1, Cka = (a k

) = (–1)k (–a)k

k! =

a(a – 1) . . . (a – k + 1) k!

, where k = 1, 2, . . .

Here a is an arbitrary real number. Definition (general case):

Cba = Γ(a + 1)

Γ(b + 1)Γ(a – b + 1) , where Γ(x) is the gamma function. Properties:

C0a = 1, Ckn = 0 for k = –1, –2, . . . or k > n,

Cb+1a = a

b + 1 Cba-1 =

a – b

b + 1 Cba, C

b a + C

b+1 a = C

Cn –1/2 =

(–1)n 22n

Cn2n = (–1)n (2n – 1)!! (2n)!! ,

Cn1/2 = (–1)n-1 n22n-1

Cn-12n-2 = (–1)n-1

n

(2n – 3)!! (2n – 2)!! ,

C2n+1n+1/2 = (–1)n2-4n-1Cn2n, Cn2n+1/2 = 2-2nC2n4n+1,

C1/2n = 22n+1

πCn2n , Cn/2n =

22n

π C(n-1)/2n ,

1 + C1n + C2n + · · · + Cnn = 2n, 1 – C1n + C2n – · · · + (–1)nCnn = 0.