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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 which L m (x) is the simple Laguerre polynomial.

4.12.1 Orthogonal Polynomials
Theorem 4.54.
(−1) n(n−1)/2 0! 1! 2! ··· (n − 2)! n(n−1)
|L m (x)| n = x , n ≥ 2.
n!(n + 1)! (n + 2)! ··· (2n − 2)!
0≤m≤2n−2
Proof. Let

1
,
m
x
φ m (x)= x L m
then

φ (x)= mφ m−1 (x),
m
φ 0 =1. (4.12.1)
Hence, φ m is an Appell polynomial in which
(−1) m
φ m (0) = .
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,

1
m = |φ m (x)| n , 0 ≤ m ≤ 2n − 2
x L m x
n
= |φ m (0)| n

m
(−1)
=
m!
n

1
=
m!

n
(−1) n(n−1)/2 0! 1! 2! ··· (n − 2)!
= . (4.12.2)
n!(n + 1)! (n + 2)! ··· (2n − 2)!
But

1 1
m = x n(n−1) . (4.12.3)
x L m x L m x
n n