
%hankel.tex: a Plain TeX file by Doron Zeilberger (1 page)

%begin macros
\def\L{{\cal L}}
\def\Tilde{\char126\relax}
\baselineskip=14pt
\parskip=10pt
\def\halmos{\hbox{\vrule height0.15cm width0.01cm\vbox{\hrule height
  0.01cm width0.2cm \vskip0.15cm \hrule height 0.01cm width0.2cm}\vrule
  height0.15cm width 0.01cm}}
\font\eightrm=cmr8 \font\sixrm=cmr6
\font\eighttt=cmtt8
\magnification=\magstephalf
\def\P{{\cal P}}
\def\Q{{\cal Q}}
\def\1{{\overline{1}}}
\def\2{{\overline{2}}}
\parindent=0pt
\overfullrule=0in
\def\frac#1#2{{#1 \over #2}}
%\headline={\rm  \ifodd\pageno  \RightHead  \else  \LeftHead  \fi}
%\def\RightHead{\centerline{
%Title
%}}
%\def\LeftHead{ \centerline{Doron Zeilberger}}
%end macros
\bf
\centerline
{
An Umbral Approach to the Hankel Transform for Sequences
}
\rm
\bigskip
\centerline{ {\it
Doron 
ZEILBERGER}\footnote{$^1$}
{\eightrm  \raggedright
Department of Mathematics, Rutgers University (New Brunswick),
Hill Center-Busch Campus, 110 Frelinghuysen Rd., Piscataway,
NJ 08854-8019, USA.
%\break
{\eighttt zeilberg  at math dot rutgers dot edu} ,
\hfill \break
{\eighttt http://www.math.rutgers.edu/\~{}zeilberg} .
Exclusively published in the Personal Journal of
Ekhad and Zeilberger
{\eighttt http://www.math.rutgers.edu/\~{}zeilberg/pj.html} .
First version: Nov. 8, 2005.
Supported in part by the NSF.
}
}

Recall that the {\it binomial transform} of a sequence $\{a_n\}_{n=0}^{\infty}$,
let's call it $BIN(a)=\{b_n\}_{n=0}^{\infty}$,
is defined by
$$
b_n=\sum_{k=0}^{n} {{n} \choose {k}} a_k \quad,
$$
and the {\it Hankel transform} ([L][SpSt]),
let's call it $H(a)=\{h_n\}_{n=0}^{\infty}$, is defined by
$$
h_n=\det (a_{i+j})_{0 \leq i,j \leq n} \quad.
$$
In [L] (see also [SpSt]) it was proved that $H \circ BIN=H$, in other words, for any sequence
$\{a_n\}_{n=0}^{\infty}$, $H(BIN(a))=H(a)$. Let's prove it \`a la Gian-Carlo Rota,
using {\it umbras}.
For any sequence $\{c_n\}_{n=0}^{\infty}$ associate an {\it umbra}
(linear functional on K[z], K any commutative ring), defined by
$C(z^i):=c_i (0 \leq i < \infty)$, and {\bf extended linearly}.
Let $A$ and $B$ be the umbras associated with 
$\{a_n\}_{n=0}^{\infty}$, and $b:=BIN(a)$ respectively. Then
$$
B(z^n)=b_n=\sum_{k=0}^{n} {{n} \choose {k}} a_k=
\sum_{k=0}^{n} {{n} \choose {k}} A(z^k)=
A \left ( \sum_{k=0}^{n} {{n} \choose {k}} z^k \right )=
A \left ( (1+z)^n \right ) \quad,
$$
for every $n \geq 0$, and hence by {\bf linearity}
$$
B(p(z))=A(p(z+1)) \quad, \quad for \,\, every \,\, p(z) \in K[z] \quad .
$$
We can also express the Hankel sequence transform umbrally.
If $\{h_n\}_{n=0}^{\infty}$ is the Hankel transform of
$\{a_n\}_{n=0}^{\infty}$  then
$$
h_n=\det (a_{i+j})_{0 \leq i,j \leq n} =
\det (A_{x_j}(x_j^{i+j}))_{0 \leq i,j \leq n}=
A_{x_0} A_{x_1} \dots A_{x_n} \left ( \det (x_j^{i+j})_{0 \leq i,j \leq n} \right )=
$$
$$
A_{x_0} A_{x_1} \dots A_{x_n} 
\left ( x_0^{0} x_1^{1} \cdots x_n^{n} \prod_{0 \leq i < j \leq n} (x_i-x_j) \right )
=
{{1} \over {n!}} A_{x_0} A_{x_1} \dots A_{x_n} 
\left ( \prod_{0 \leq i < j \leq n} (x_i-x_j)^2 \right ) \quad,
$$
using multi-linearity, the Vandermonde determinant, and symmetrizing (plus Vandermonde again), respectively.
Equating left and right, we have:
$$
H(a)_n=
{{1} \over {n!}} A_{x_0} A_{x_1} \dots A_{x_n} 
\left ( \prod_{0 \leq i < j \leq n} (x_i-x_j)^2 \right ) \quad.
$$
This immediately implies that the Hankel transform is preserved under
the binomial transform, since the discriminant is invariant under
$x_i \rightarrow 1+ x_i$.

{\bf Remarks.} {\bf 1.} Similarly one can prove umbrally the various analogs in [L] and [SpSt].
{\bf 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
$A(z^n)=\int_{0}^{\infty} e^{-z} z^n \, dz$, and invoking specializations of Mehta's integral.
Ditto for Macmahon and Carlitz determinants, and their
generalizations, using Selberg's integral.
{\bf 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 {\it root systems} (the
square of the Weyl denominator), and then use Macdonald integrals.

{\bf References}

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

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



\end




