Determinants and Their Applications in Mathematical Physics

96 4. Particular Determinants

The proof of (b) follows as a corollary since, differentiating Bn by

columns,

B

′

n

=−

n

∑

r=1

B

(n)

rr

.

The given result follows from (4.6.11).

To prove (c), differentiate (a) repeatedly, apply the Maclaurin formula,

and refer to (4.6.11) again:

B

(r)

n =

(−1)

r

n!Bn−r

(n−r)!

,

Bn=

n

∑

r=0

B

(r)

n (0)

r!

x

r

=

n

∑

r=0

(

n

r

)

An−r(−x)

r

.

Putr=n−sand the given formula appears. It follows thatBnis an

Appell polynomial for all values ofn.

Exercises

1.Let

An=|aij|n,

where

aij=

{

ψj−i+1,j≥i,

j, j=i−1,

0 , otherwise.

Prove that ifAnsatisfies the Appell equationA

′

n

=nAn− 1 for small

values ofn, thenAnsatisfies the Appell equation for all values ofnand

that the elements must be of the form

ψ 1 =x+α 1 ,

ψm=αm,m> 1 ,

where theα’s are constants.

2.If

An=|aij|n,

where

aij=

{

φj−i,j≥i,

−j, j=i−1,

0 , otherwise,