Determinants and Their Applications in Mathematical Physics

4.12 Hankelians 5 153

4.12 Hankelians 5

Notes in orthogonal and other polynomials are given in Appendices A.5 and

A.6. Hankelians whose elements are polynomials have been evaluated by a

variety of methods by Geronimus, Beckenbach et al., Lawden, Burchnall,

Seidel, Karlin and Szeg ̈o, Das, and others. Burchnall’s methods apply the

Appell equation but otherwise have little in common with the proof of the

first theorem in whichLm(x) is the simple Laguerre polynomial.

4.12.1 Orthogonal Polynomials

Theorem 4.54.

|Lm(x)|n=

(−1)

n(n−1)/ 2

0! 1! 2!···(n−2)!

n!(n+ 1)! (n+ 2)!···(2n−2)!

x

n(n−1)

,n≥ 2.

0 ≤m≤ 2 n− 2

Proof. Let

φm(x)=x

m

Lm

(

1

x

)

,

then

φ

′

m(x)=mφm− 1 (x),

φ 0 =1. (4.12.1)

Hence,φmis an Appell polynomial in which

φm(0) =

(−1)

m

m!

.

Applying Theorem 4.33 in Section 4.9.1 on Hankelians with Appell polyno-

mial elements and Theorem 4.47b in Section 4.11.3 on determinants with

binomial and factorial elements,

∣

∣

∣

∣

x

m

Lm

(

1

x

)∣

∣

∣

∣

n

=|φm(x)|n, 0 ≤m≤ 2 n− 2

=|φm(0)|n

=

∣

∣

∣

∣

(−1)

m

m!

∣ ∣ ∣ ∣ n =

∣

∣

∣

∣

1

m!

∣ ∣ ∣ ∣ n =

(−1)

n(n−1)/ 2

0! 1! 2!···(n−2)!

n!(n+ 1)! (n+ 2)!···(2n−2)!

. (4.12.2)

But
∣
∣
∣
∣

x

m

Lm

(

1

x

)∣

∣

∣

∣

n

=x

n(n−1)

∣

∣

∣

∣

Lm

(

1

x

)∣

∣

∣

∣

n

. (4.12.3)