Some Hankel Determinants After invoking a recursion relation for Hankel determinants in my answer to a (mostly unrelated) question, I started wondering what else I could use this recursion for, and stumbled upon some results that surprised me.  The proofs are purely computational, and I'm hoping someone can provide a more conceptual understanding.
So:  Let $f(m)$ be a function defined on positive integers.  The corresponding Hankel matrices are:  
$${\cal H}(m,n)=\pmatrix{ f(m)&f(m+1)&\ldots&f(m+n)\cr
f(m+1)&f(m+2)&\ldots &f(m+n+1)\cr
\vdots&\vdots&&\vdots\cr
f( m+n)&f( m+n+1)&\ldots&f(m+2n)\cr}$$
and the Hankel determinants are
$H(m,n)=det({\cal H}(m,n))$.
In particular, set:
$$H(m,n)=\hbox{ the Hankel determinant associated to $f(m)=1/m$}$$
$$J(m,n)=\hbox{ the Hankel determinant associated to $f(m)=m!$}$$
$$K(m,n)=\hbox{ the Hankel determinant associated to $f(m)=1/m!$}$$
Also, let $c(n)=\prod_{i=1}^{n-1}i!$
Then I can show that:
$$H(m,n)={c(m+n-1)^2c(n)^2\over c(m-1)c(m+2n-1)}$$
$$J(m,n)={c(n)c(m+n)\over c(m)}$$
$$K(m,n)=\pm{c(n)c(m+n-1)\over c(m+2n-1)}$$
where the $\pm$ sign on $K(m,n)$ is $+$ or $-$ depending on whether $n$ is a square mod $4$.
These calculations immediately yield two theorems, each of which seems to cry out for a deeper explanation:
Theorem 1. $H(m,n)=J(m-1,n)K(m,n)$
Theorem 2. The $H(m,n)$ and $K(m,n)$ are all reciprocals of integers.
Question:  Aside from the fact that they fall out of a calculation, why should we expect these theorems to be true? 
For example, for Theorem 1, is there some natural interpretation  of the product of two Hankel matrices that puts this in context?  Or for Theorem 2, are the reciprocals of the $H(m,n)$ and $K(m,n)$ counting something?
Remark 1: Theorem 2 is certainly well-known for the $H(1,n)$, which are the determinants of the Hilbert matrices.  I'm not sure whether it's well-known for the rest of the $H(m,n)$ or for the $K(m,n)$.
Remark 2:  Theorem 2 would follow from the stronger statement that the Hankel matrices ${\cal H}(m,n)$ and  ${\cal K}(m,n)$ (that is, the matrices of which the $H(m,n)$ and $K(m,n)$ are the determinants) have inverses with all integer entries.  This, again, is well-known for $H(1,n)$ at least.  I have both the outline of an argument and extensive numerical evidence to suggest that it is true for all the ${\cal H}(m,n)$ and all the ${\cal K}(m,n)$ but I fear that even if the argument pans out, it won't provide much conceptual understanding.
 A: After comparing Steven's original example with his new example, I believe I have an interesting generalization of both. Let $c \in \mathbb{C}$. Define the following three functions: 
$$h(m)=\frac{1}{m-1+c},k(m) = \frac{1}{\Gamma(m+c)},$$
$$j(m)=\frac{h(m+1)}{k(m+1)}=\Gamma(m+c).$$
Let $\mathcal{H}(m,n),\mathcal{J}(m,n),\mathcal{K}(m,n)$ be the Hankel matrices corresponding to $h,j,k$, respectively. Let $H(m,n),J(m,n),K(m,n)$ be the respective determinants. Then the following holds:
$$H(m,n) =\pm J(m-1,n) K(m,n).$$


*

*When $c=1$, we recover the example from the original post.

*When $c=\frac{1}{2}$, we recover Steven's second example, from his answer to the post (up to some normalization).


Now I turn to prove the generalization. 
Lemma 1: Let $V$ be an inner-product space. Let $\{v_i\}_{i=1}^{n},\{\tilde{v}_i\}_{i=1}^{n}$ be two pairs of sequences of linearly independent vectors in $V$, spanning the same subspace. Similarly, let $\{u_i\}_{i=1}^{n},\{\tilde{u}_i\}_{i=1}^{n}$ be two pairs of sequences of linearly independent vectors in $V$, spanning the same subspace.
There exist unique scalars $x_{i,j}, y_{i,j}$ such that $\tilde{v}_i = \sum x_{i,k} v_k$ and $\tilde{u}_i = \sum y_{i,k} u_k$.
Define the following 4 matrices: 


*

*$A_{i,j} = \langle \tilde{v}_i, \tilde{u}_j \rangle$, $B_{i,j} = \langle v_i, u_j \rangle$

*$X_{i,j} = x_{i,j}, Y_{i,j} = y_{i,j}$


Then
$$A = X B Y^{T}.$$
Lemma 2: $\det(\binom{i+j+m}{i}_{0\le i,j \le n}) = 1$ for any $m \in \mathbb{C}$.
We work in the inner-product space $L^2([0,1])$.
Note that 
$$\mathcal{H}(m,n)_{i,j} = \langle x^{i+m+c-2}, x^{j} \rangle$$
and that
$$\mathcal{K}(m,n)_{i,j} = \langle \frac{x^{i+m+c-2}}{\Gamma(i+m-1+c)}, \frac{(1-x)^j}{\Gamma(j+1)} \rangle.$$
We are in a situation where we may apply Lemma 1 with $A=\mathcal{H}(m,n)$ and $ B=\mathcal{K}(m,n)$, which yields
$$\mathcal{H}(m,n)= \text{Diag}((\Gamma(i+m-1+c))_{0\le i\le n}) \times \mathcal{K}(m,n) \times (\Gamma(i+1)\binom{j}{i}(-1)^i)_{0\le i,j \le n},$$
hence
$$(*) \frac{H(m,n)}{K(m,n)} = \det \text{Diag}((\Gamma(i+m-1+c))_{0\le i\le n}) \det (\Gamma(i+1)\binom{j}{i}(-1)^i)_{0\le i,j \le n}.$$
On the other hand,
$$\mathcal{J}(m-1,n) = (\Gamma(m-1+c+i+j))_{0 \le i,j \le n}) = \text{Diag}((\Gamma(i+m-1+c))_{0\le i\le n})  (\binom{m-2+i+j+c}{j})_{0 \le i,j \le n}) \text{Diag}(\Gamma(j+1)_{0\le j\le n}),$$
which, combined with Lemma 2, yields
$$(**) J(m-1,n) = \det \text{Diag}((\Gamma(i+m-1+c))_{0\le i\le n}) \det\text{Diag}(\Gamma(j+1)_{0\le j\le n}).$$
We finish by noting that
$$\det (\Gamma(i+1)\binom{j}{i}(-1)^i)_{0\le i,j \le n} = (-1)^{\binom{n+1}{2}} \prod_{i=0}^{n} \Gamma(i+1) =  (-1)^{\binom{n+1}{2}} \det \text{Diag}(\Gamma(j+1)_{0\le j\le n}),$$
which implies that $(*)$ and $(**)$ coincide up to sign. $\blacksquare$
A: In as far as the computation goes, these determinants all fall out easily from the method shown in this paper.
When inductive proofs work, this technique is simpler to use than any other listed in this paper.
On the basis of the solution and comments above, it only remains to verify Theorem 2 to the effect that $\frac1{\det(H^{-1})}$ is integral.
Proof of Theorem 2. If the entries of the "Hilbert matrix" are given by
$$H_{ij}=\frac{1}{i+j+m-1}$$
then its inverse has an explicit form given by:
\begin{align*} (H^{-1})_{ij} &= (-1)^{i+j}(m+i+j-1) \times \\
&{{n+m+i-1}\choose{n-j}}{{n+m+j-1}\choose{n-i}}{{m+i+j-2}\choose{i-1}}{{m+i+j-2}\choose{j-1}}.
\end{align*}
It follows that the entries in the inverse matrix are all integers. Therefore, $$\det(H)=\frac1{\det(H^{-1})}$$
explains why these are reciprocals of integers. The proof follows. $\square$
A: Not yet a fully satisfying answer, but Theorem 1 does generalize.  
Given two sequences $j$ and $k$, define the sequence $h$ by
$$h(m)=j(m-1)k(m)$$
Let $H$, $J$ and $K$ be the corresponding families of Hankel determinants.   Say that $(j,k)$ is a good example if, for all $m$ and $n$,
$$H(m,n)=\pm J(m-1,n)K(m,n)$$
Theorem 1 above is the statement that if we set  $j(m)=m!$, and $k(m)=1/m!$, then $(j,k)$ is a good example. This continues to surprise me, because it requires a lot of seemingly miraculous cancellation.
I now have other examples of good pairs, but  not yet a good way to describe them all.  However, the following will give the flavor:  Let
$$j(m)=(2m)!/(m!)\qquad k(m)=\prod_{i=1}^m 1/(2i-1)$$
Then $(j,k)$ a good example.
My hope continues to be that the existence of this and related good examples will shed some light on Theorem 1.
A: IMHO Theorem 1 is a lucky coincidence. These Hankel determinants do not so much depend on the values of their entries but rather on the linear functionals which they generate.  
More precisely: Let $a(n)$ be a sequence of real numbers with $a(0)=1$  such that no Hankel determinant vanishes. Define a linear functional $F$  on the polynomials by $F(x^n)=a(n).$ 
Let $p(n,x)$ be the uniquely determined set of monic  polynomials which are orthogonal with respect to $F.$ 
By Favard’s theorem they satisfy $p(n,x)=(x-s(n-1))p(n-1,x)-t(n-2)p(n-2,x).$ 
Then the Hankel determinant  $\det(a(i+j))_{i,j=0}^n= \prod_{i=1}^n\prod_{j=0}^{i-1}t(j)$ only depends on the numbers $t(n).$ These well-known results can for example be found here.
For $a_{1}(n,m)=\frac{m}{n+m}$, $a_{2}(n,m)=\frac{(n+m)!}{m!}$ and $a_{3}(n,m)=\frac{m!}{(n+m)!}$ the corresponding $t’s$ are
$t_{1}(n,m)=\frac{(n+1)^2(n+m)^2}{(2n+m)(2n+m+1)^2(2n+m+2)}$, 
$t_{2}(n,m-1)=(n+1)(n+m)$
and
$t_{3}(n,m)=\frac{(n+1)(n+m)}{(2n+m)(2n+m+1)^2(2n+m+2)}.$
Therefore
$t_{1}(n,m)=t_2(n,m-1) t_{3}(n,m),$ which is the reason for Theorem 1.
