Determinants and Their Applications in Mathematical Physics

5 Further Determinant Theory

5.1 Determinants Which Represent Particular Polynomials

5.1.1 Appell Polynomial...................

Notes on Appell polynomials are given in Appendix A.4.

Let

ψn(x)=(−1)

n

n

∑

r=0

(

n

r

)

αn−r(−x)

r

. (5.1.1)

Theorem.

a.ψn(x)=

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

∣

∣

∣

α 0 α 1 α 2 α 3 ··· αn− 1 αn

1 xx

2

x

3

··· ···

(

n

0

)

x

n

12 x 3 x

2

··· ···

(

n

1

)

x

n− 1

13 x ··· ···

(

n

2

)

x

n− 2

··· ··· ··· ···

1 nx

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

∣

∣

∣

n+1

,