Page 137 - Determinants and Their Applications in Mathematical Physics

122 4. Particular Determinants

Theorem 4.36.
(n,r) (n,r+2)
D(T )= −(2n + r − 1)T .
11 n,n−1
Proof.
(n,r) (n−1,r+2)
T 11 = T .
The theorem follows by adjusting the parameters in Theorem 4.35.
Both these theorems are applied in Section 6.5.3 on the Milne–Thomson
equation.

4.9.3 Determinants with Simple Derivatives of All Orders
Let Z r denote the column vector with (n + 1) elements defined as

, 1 ≤ r ≤ n, (4.9.15)
T
n+1
Z r = 0 r φ 0 φ 1 φ 2 ··· φ n−r
where 0 r denotes an unbroken sequence of r zero elements and φ m is an
Appell polynomial.
Let

B = Z 1 C 0 C 1 C 2 ··· C n−1 , (4.9.16)

n+1
where C j is defined in (4.9.5). Differentiating B repeatedly, it is found that,
apart from a constant factor, only the first column changes:

r r , 0 ≤ r ≤ n − 1.
D (B)=(−1) r! Z r+1 C 0 C 1 C 2 ··· C n−1
n+1
Hence

D n−1 (B)=(−1) n−1 (n − 1)!φ 0 C 0 C 1 C 2 ··· C n−1

n
=(−1) n−1 (n − 1)!φ 0 |φ m | n , 0 ≤ m ≤ 2n − 2
= constant;
2
that is, B is a polynomial of degree (n − 1) and not (n − 1), as may
be expected by examining the product of the elements in the secondary
diagonal of B. Once again, the loss of degree due to cancellations is n(n−1).

Exercise
Let

S m = φ r φ s .
r+s=m
This function appears in Exercise 2 at the end of Appendix A.4 on Appell
polynomials. Also, let

T
C j = S j−1 S j S j+1 ··· S j+n−2 , 1 ≤ j ≤ n,
n

T
K = • S 0 S 1 S 2 ··· S n−2 ,
n
E = |S m | n , 0 ≤ m ≤ 2n − 2.