Page 330 - Determinants and Their Applications in Mathematical Physics

A.4 Appell Polynomials 315

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

r
m !
∞ !
α r G(t)= α r t φ m(x)= m α rx m−r
r!
r=0 r=0 r
1 δ r 1 x m
t m
2 1 e (1 + x)
3 r te t m(1 + x) m−1

t
1 e −1 (1+x) m+1 −x m+1
4 r+1 t m+1
(−1) r √ m
1
5 r! J 0(2 t) (Bessel) x L m x (Laguerre)
r
(−1) (2r)!
α 2r = −t 2 −m
2r
6 2 r! e 2 H m(x) (Hermite)
α 2r+1 =0
t
7 e t −1 B m(x) (Bernoulli)
8 2 E m(x) (Euler)
e t +1
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 ,
2
φ 2 (x)= α 0 x +2α 1 x + α 2 ,
3
2
φ 3 (x)= α 0 x +3α 1 x +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) 1 α 0
     
 x 1
 φ 1 (x)    α 1 
 φ 2 (x)  =  x 2 2x 1   α 2  . (A.4.8)
 




φ 3 (x) x 3x 3x 1 α 3
   3 2   
··· ................ ···