Determinants and Their Applications in Mathematical Physics

306 Appendix

Other applications are found in Appendix A.4 on Appell polynomials.

Γ(x)=

∫∞

0

e

−t

t

x− 1

dt,

Γ(x+1)=xΓ(x)

Γ(n+1)=n!,n=1, 2 , 3 ,....

The Legendre duplication formula is

√

πΓ(2x)=2

2 x− 1

Γ(x)Γ

(

x+

1

2

)

,

which is applied in Appendix A.8 on differences.

Stirling Numbers

The Stirling numbers of the first and second kinds, denoted bysijandSij,

respectively, are defined by the relations

(x)r=

r

∑

k=0

srkx

k

,sr 0 =δr 0 ,

x

r

=

r

∑

k=0

Srk(x)k,Sr 0 =δr 0 ,

where (x)ris 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

sij=si− 1 ,j− 1 −(i−1)si− 1 ,j

Sij=Si− 1 ,j− 1 +jSi− 1 ,j.

Some values of these numbers are given in the following short tables:

sij

j

i 12345

1 1

2 − 11

3 2 − 31

4 − 611 − 61

5 24 − 50 35 −10 1