An Umbral Approach to the Hankel Transform for Sequences

Doron ZEILBERGER^{1}

Recall that the binomial transform of a sequence {a_{n}}_{n=0}^{∞}, let’s call it BIN(a) = {b_{n}}_{n=0}^{∞}, is defined by

b_{n} = ∑_{k=0}^{n} \binom{n}{k}a_{k} ,

and the Hankel transform ([L][SpSt]), let’s call it H(a) = {h_{n}}_{n=0}^{∞}, is defined by

h_{n} = det(a_{i+j})_{0≤i,j≤n} .

In [L] (see also [SpSt]) it was proved that H ∘ BIN = H, in other words, for any sequence {a_{n}}_{n=0}^{∞}, H(BIN(a)) = H(a). Let’s prove it à la Gian-Carlo Rota, using umbras. For any sequence {c_{n}}_{n=0}^{∞} associate an umbra (linear functional on K[z], K any commutative ring), defined by C(z^{i}) := c_{i}(0 ≤ i < ∞), and extended linearly. Let A and B be the umbras associated with {a_{n}}_{n=0}^{∞}, and b := BIN(a) respectively. Then

B(z^{n}) = b_{n} = ∑_{k=0}^{n} \binom{n}{k}a_{k} = ∑_{k=0}^{n} \binom{n}{k}A(z^{k}) = A (∑_{k=0}^{n} \binom{n}{k}z^{k}) = A ((1 + z)^{n}) ,

for every n ≥ 0, and hence by linearity

B(p(z)) = A(p(z + 1)) , for every p(z) ∈ K[z] .

We can also express the Hankel sequence transform umbrally. If {h_{n}}_{n=0}^{∞} is the Hankel transform of {a_{n}}_{n=0}^{∞} then

h_{n} = det(a_{i+j})_{0≤i,j≤n} = det(A_{x_{j}}(x_{j}^{i+j}))_{0≤i,j≤n} = A_{x_{0}}A_{x_{1}} … A_{x_{n}} (det(x_{j}^{i+j})_{0≤i,j≤n}) =

A_{x_{0}}A_{x_{1}} … A_{x_{n}} (x_{0}^{0}x_{1}^{1} ⋯ x_{n}^{n} ∏_{0≤i<j≤n}(x_{i} − x_{j})) = \frac{1}{n!}A_{x_{0}}A_{x_{1}} … A_{x_{n}} ( ∏_{0≤i<j≤n}(x_{i} − x_{j})^{2}) ,

using multi-linearity, the Vandermonde determinant, and symmetrizing (plus Vandermonde again), respectively. Equating left and right, we have:

H(a)_{n} = \frac{1}{n!}A_{x_{0}}A_{x_{1}} … A_{x_{n}} ( ∏_{0≤i<j≤n}(x_{i} − x_{j})^{2}) .

1 Department of Mathematics, Rutgers University (New Brunswick), Hill Center-Busch Campus, 110 Frelinghuysen

Rd., Piscataway, NJ 08854-8019, USA. zeilberg at math dot rutgers dot edu , http://www.math.rutgers.edu/~zeilberg . Exclusively published in the Personal Journal of Ekhad and Zeilberger http://www.math.rutgers.edu/~zeilberg/pj.html . First version: Nov. 8, 2005. Supported in part by the NSF.

This immediately implies that the Hankel transform is preserved under the binomial transform, since the discriminant is invariant under x_{i} → 1 + x_{i}.

Remarks. 1. Similarly one can prove umbrally the various analogs in [L] and [SpSt]. 2. One can deduce explicit Hankel evaluations for a_{n} = n! (and hence for any sequence (like the derangement numbers,) binomially related to it), by using A(z^{n}) = ∫_{0}^{∞} e^{−z}z^{n} dz, and invoking specializations of Mehta’s integral. Ditto for Macmahon and Carlitz determinants, and their generalizations, using Selberg’s integral. 3. One can view the Hankel transform without using determinants explicitly, just as a certain multi-linear form. It may be fun to talk about analogs using other ‘semi-invariants’ (in the sense of Cayley and Sylvester), as well as analogs of the discriminant for other root systems (the square of the Weyl denominator), and then use Macdonald integrals.

References

[L] John W. Layman, The Hankel transform and some of its properties, Journal of Integer Sequences 4:01.1.5, 2001.

[SpSt] Michael Z. Spivey and Laura L. Steil, The k-binomial transforms and the Hankel transform, preprint. See also the abstract to Spivey’s talk at the Integers 2005 Ron Graham Conference.