Page 171 - Determinants and Their Applications in Mathematical Physics

156 4. Particular Determinants

n+r
n n
= u v
σ n−σ+r
n − σ n − σ + r
σ=r
n+r
n n
=(uv) r u σ−r n−σ . (4.12.8)
v
σ σ − r
σ=r
Part (a) follows after interchanging u and v and replacing n by i. Part (b)
then follows easily.
It follows from Lemma 4.56 that P i+j (x) is equal to the coefficient of
t i+j in the expansion of the polynomial
[(u + t)(v + t)] i+j =[(u + t)(v + t)] [(u + t)(v + t)] j
i
2j
2i

= λ ir t r λ js t . (4.12.9)
s
r=0 s=0
Each sum consists of an odd number of terms, the center terms being λ ii t i
and λ jj t respectively. Hence, referring to Lemma 4.57,
j
min(i,j) min(i,j)

P i+j (x)= λ i,i−r λ j,j+r + λ ii λ jj + † λ i,i+r λ j,j−r
r=1 r=1
min(i,j)

=2 λ i,i+r λ j,j−r , (4.12.10)
r=0
where the symbol † denotes that the factor 2 is omitted from the r =0
term. Replacing i by i − 1 and j by j − 1,
min(i,j)

P i+j−2 (x)=2 † λ i−1,i−1+r λ j−1,j−1−r . (4.12.11)
r=0
Preparations for the second proof are now complete. Adjusting the dummy
variable and applying, in reverse, the formula for the product of two
determinants (Section 1.4),

min(i,j)

|P i+j−2 | n = 2 †

λ i−1,i+s−2 λ j−1,j−s
s=1

n

= 2λ ∗ λ j−1,j−i , (4.12.12)

n
i−1,i+j−2 n
where the symbol ∗ denotes that the factor 2 is omitted when j = 1. Note
that λ np =0 if p< 0or p> 2n. The first determinant is lower triangular
and the second is upper triangular so that the value of each determinant
is given by the product of the elements in its principal diagonal:
n

n−1
|P i+j−2 | n =2 λ i−1,2i−2 λ j−1,0
i=1