174 5. Further Determinant Theory
5.1.3 Orthogonal Polynomials
Determinants which represent orthogonal polynomials (Appendix A.5)
have been constructed using various methods by Pandres, R ̈osler, Yahya,
Stein et al., Schleusner, and Singhal, Frost and Sackfield and others. The
following method applies the Rodrigues formulas for the polynomials.
Let
An=|aij|n,
where
aij=
(
j− 1
i− 1
)
u
(j−i)
−
(
j− 1
i− 2
)
v
(j−i+1)
,u
(r)
=D
r
(u),etc.,
u=
vy
′
y
=v(logy)
′
. (5.1.9)
In some detail,
An=
∣
∣
∣
∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣
uu
′
u
′′
u
′′′
··· u
(n−2)
u
(n−1)
−vu−v
′
2 u
′
−v
′′
3 u
′′
−v
′′′
··· ··· ···
−vu− 2 v
′
3 u
′
− 3 v
′′
··· ··· ···
−vu− 3 v
′
··· ··· ···
−v ··· ··· ···
............................
−vu−(n−1)v
′
∣
∣
∣
∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ n
.
(5.1.10)
Theorem.
a.A
(n+1)
n+1,n=−A
′
n,
b.An=
v
n
D
n
(y)
y
.
Proof. ExpressAnin column vector notation:
An=
∣
∣C
1 C 2 C 3 ···Cn
∣
∣
n
,
where
Cj=
[
a 1 ja 2 ja 3 j···aj+1,jOn−j− 1
]T
n
(5.1.11)
whereOrrepresents an unbroken sequence ofrzero elements.
LetC
∗
j
denote the column vector obtained by dislocating the elements
ofCjone position downward, leaving the uppermost position occupied by
a zero element:
C
∗
j=
[
Oa 1 ja 2 j···ajjaj+1,jOn−j− 2
]T
n
. (5.1.12)
Then,
C
′
j
+C
∗
j
=
[
a
′
1 j
(a
′
2 j
+a 1 j)(a
′
3 j
+a 2 j)···(a
′
j+1,j
+ajj)aj+1,jOn−j− 2
]T
n