Page 111 - Determinants and Their Applications in Mathematical Physics

96 4. Particular Determinants

The proof of (b) follows as a corollary since, differentiating B n by
columns,

n

(n)
B = − B .
n rr
r=1
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:
r
(r)
B = (−1) n!B n−r ,
(n − r)!
n
(r)
n
B n (0)
B n = x r
r!
r=0
n
n

= A n−r (−x) .
r
r
r=0
Put r = n − s and the given formula appears. It follows that B n is an
Appell polynomial for all values of n.
Exercises
1. Let
A n = |a ij | n ,
where
ψ j−i+1 ,j ≥ i,

a ij = j, j = i − 1,
0, otherwise.

Prove that if A n satisfies the Appell equation A = nA n−1 for small
n
values of n, then A n satisfies the Appell equation for all values of n and
that the elements must be of the form
ψ 1 = x + α 1 ,
ψ m = α m , m > 1,
where the α’s are constants.
2. If

A n = |a ij | n ,
where
φ j−i ,j ≥ i,

a ij = −j, j = i − 1,
0, otherwise,