Determinants and Their Applications in Mathematical Physics

A.4 Appell Polynomials 315

TABLE A.1. Particular Appell Polynomials and Their Generating Functions

αr G(t)=

∑∞

r=0

αrt

r

r!

φm(x)=

∑m

r=0

(

m

r

)

αrx

m−r

1 δr 1 x

m

21 e

t

(1 +x)

m

3 rte

t

m(1 +x)

m− 1

4

1

r+1

et− 1

t

(1+x)m+1−xm+1

m+1

5

(−1)r

r!

J 0 (2

√

t) (Bessel) x

m

Lm

(

1

x

)

(Laguerre)

6

α 2 r =

(−1)

r

(2r)!

2

2 r

r!

α 2 r+1=0

}

e

−t^2

2

−m

Hm(x) (Hermite)

7

t

et− 1

Bm(x) (Bernoulli)

8

2

et+1

Em(x) (Euler)

Note:Further examples are given by Carlson.

The first four polynomials are

φ 0 (x)=α 0 ,

φ 1 (x)=α 0 x+α 1 ,

φ 2 (x)=α 0 x

2

+2α 1 x+α 2 ,

φ 3 (x)=α 0 x

3

+3α 1 x

2

+3α 2 x+α 3. (A.4.7)

Particular cases of these polynomials and their generating functions are

given in Table 1. When expressed in matrix form, equations (A.4.7) become











φ 0 (x)

φ 1 (x)

φ 2 (x)

φ 3 (x)

···











=











1

x 1

x

2

2 x 1

x

3

3 x

2

3 x 1

................





















α 0

α 1

α 2

α 3

···











. (A.4.8)