Page 321 - Determinants and Their Applications in Mathematical Physics

306 Appendix

Other applications are found in Appendix A.4 on Appell polynomials.
∞
,
−t x−1
Γ(x)= e t dt,
0
Γ(x +1) = xΓ(x)
Γ(n +1) = n!, n =1, 2, 3,... .
The Legendre duplication formula is
√ 2x−1 1
π Γ(2x)=2 Γ(x)Γ x + ,
2
which is applied in Appendix A.8 on differences.

Stirling Numbers
The Stirling numbers of the first and second kinds, denoted by s ij and S ij ,
respectively, are defined by the relations
r

(x) r = s rk x , s r0 = δ r0 ,
k
k=0
r

x = S rk (x) k , S r0 = δ r0 ,
r
k=0
where (x) r is the falling factorial function defined as
(x) r = x(x − 1)(x − 2) ··· (x − r +1), r =1, 2, 3,... .
Stirling numbers satisfy the recurrence relations

s ij = s i−1,j−1 − (i − 1)s i−1,j
S ij = S i−1,j−1 + jS i−1,j .
Some values of these numbers are given in the following short tables:

j s ij
1 2 3 4 5
i
1 1
2 −1 1
3 2 −3 1
4 −6 11 −6 1
5 24 −50 35 −10 1