Determinants and Their Applications in Mathematical Physics

A.4 Appell Polynomials 319

These polynomials can be displayed in matrix form as follows:

Let

U(x)=







u 00 u 01 u 02 ···

u 10 u 11 u 12 ···

u 20 u 21 u 22 ···

··· ··· ··· ···





.

Then,

U(x)=e

xQ

U(0)

(

e

xQ

)T

.

Hence,

U(0) =e

−xQ

U(x)

(

e

−xQ

)T

,

that is,

αij=

i

∑

r=0

j

∑

s=0

(

i

r

)(

j

s

)

urs(−x)

i+j−r−s

,i,j=0, 1 , 2 ,....

Other solutions of (A.4.12) can be expressed in terms of simple Appell

polynomials; for example,

uij=φiφj,

uij=

∣

∣

∣

∣

φi φj

φi+1 φj+1

∣

∣

∣

∣

.

Solutions of the three-parameter Appell equation, namely

u

′

ijk=iui−^1 ,j,k+jui,j−^1 ,k+kuij,k−^1 ,

include

uijk=φiφjφk,

uijk=

∣ ∣ ∣ ∣ ∣ ∣

φi φj φk

φi+1 φj+1 φk+1

φi+2 φj+2 φk+2

∣ ∣ ∣ ∣ ∣ ∣

.

Carlson has studied polynomialsφm(x,y,z,...) which satisfy the relation

(Dx+Dy+Dz+···)φm=mφm− 1 ,Dx=

∂

∂x

,etc.,

and Carlitz has studied polynomialsφmnp...(x,y,z,...) which satisfy the

relations

Dx(φmnp...)=mφm− 1 ,np...,

Dy(φmnp...)=nφm,n− 1 ,p...,

Dz(φmnp...)=pφmn,p− 1 ,....