Page 173 - Determinants and Their Applications in Mathematical Physics

158 4. Particular Determinants

H n = |S i+j−2 | n = λ n x n(n+1)/2 ,
J n = |A i+j−2 | n = λ n x n(n−1)/2 ,
where
2
λ n = 1! 2! 3! ··· (n − 1)!] .
The following proofs differ from the originals in some respects.
Proof. It is proved using a slightly different notation in Theorem 4.28
in Section 4.8.5 on Turanians that
2
E n G n − E n+1 G n−1 = F ,
n
which is equivalent to
2
E n−1 G n−1 − E n G n−2 = F n−1 . (4.12.16)
Put x = e in (4.12.5) so that
t
−t
D x = e D t
∂
D t = xD x , D x = , etc. (4.12.17)
∂x
Also, put
θ m (t)= ψ m (e )
t
∞

= r e
m rt
r=1
θ (t)= θ m+1 (t). (4.12.18)

m
Define the column vector C j (t) as follows:
T
C j (t)= θ j (t) θ j+1 (t) θ j+2 (t) ...
so that

C = C j+1 (t). (4.12.19)
j
The number of elements in C j is equal to the order of the determinant of
which it is a part, that is, n, n − 1, or n − 2 in the present context.
Let

Q n (t, τ)= C 0 (τ) C 1 (t) C 2 (t) ··· C n−1 (t) , (4.12.20)

n
where the argument in the first column is τ and the argument in each of
the other columns is t. Then,
Q n (t, t)= E n . (4.12.21)
Differentiate Q n repeatedly with respect to τ, apply (4.12.19), and put
τ = t.
D {Q n (t, t)} =0, 1 ≤ r ≤ n − 1, (4.12.22)
r
τ