Determinants and Their Applications in Mathematical Physics

A.4 Appell Polynomials 317

Appell Sets

Any sequence of polynomials{φm(x)}whereφm(x) is of exact degreem

and satisfies the Appell equation (A.4.5) is known as an Appell set.

The sequence in which

φm(x)=

(

m+s

s

)− 1

(x+c)

m+s

,s=1, 2 , 3 ,...,

satisfies (A.4.5), but its members are not of degreem. The sequence in

which

φm(x)=

2

2 m+1

m!(m+ 1)! (x+c)

m+(1/2)

(2m+ 2)!

satisfies (A.4.5), but its members are not polynomials. Hence, neither

sequence is an Appell set.

Carlson proved that if{φm}and{ψm}are each Appell sets and

θm=2

−m

m

∑

r=0

(

m

r

)

φrψm−r,

then{θm}is also an Appell set.

In a paper on determinants with hypergeometric elements, Burchnall

proved that if{φm}and{ψm}are each Appell sets and

θm=

n

∑

r=0

(−1)

r

(

n

r

)

φm+n−rψr,n=0, 1 , 2 ,...,

then{θm}is also an Appell set for each value ofn. Burchnall’s formula can

be expressed in the form

θm=

n

∑

r=0

(−1)

r

∣

∣

∣

∣

ψr φm+n−r

ψr+1 φm+n−r+1

∣

∣

∣

∣

,n=0, 1 , 2 ,....

The generalized Appell equation

θ

′

m=mf

′

θm− 1 ,f=f(x),

is satisfied by

θm=φm(f),

whereφm(x) is any solution of (A.4.5). For example, the equation

θ

′

m=

mθm− 1

(1 +x)

2

is satisfied by

θm=φm

(

−

1

1+x

)

.