Determinants and Their Applications in Mathematical Physics

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

.