Page 334 - Determinants and Their Applications in Mathematical Physics

A.4 Appell Polynomials 319

These polynomials can be displayed in matrix form as follows:
Let
 
u 00 u 01 u 02 ···
 u 10 u 11 u 12 ··· 
 .
U(x)= 
u 20 u 21 u 22 ···
··· ··· ··· ···
Then,
xQ
U(x)= e U(0) e xQ T .
Hence,
−xQ −xQ T

U(0) = e U(x) e ,
that is,

j
i
i j
α ij = u rs (−x) i+j−r−s , i, j =0, 1, 2,... .
r s
r=0 s=0
Other solutions of (A.4.12) can be expressed in terms of simple Appell
polynomials; for example,
u ij = φ i φ j ,

u ij = φ i φ j .
φ i+1 φ j+1

Solutions of the three-parameter Appell equation, namely
u = iu i−1,j,k + ju i,j−1,k + ku ij,k−1 ,
ijk
include

u ijk = φ i φ j φ k ,

φ i φ j φ k
u ijk = φ i+1 φ j+1 .

φ k+1
φ i+2 φ j+2 φ k+2

Carlson has studied polynomials φ m (x,y,z,...) which satisfy the relation
∂
(D x + D y + D z + ···)φ m = mφ m−1 , D x = , etc.,
∂x
and Carlitz has studied polynomials φ mnp... (x,y,z,...) which satisfy the
relations

D x (φ mnp... )= mφ m−1,np... ,
D y (φ mnp... )= nφ m,n−1,p... ,
D z (φ mnp... )= pφ mn,p−1,... .