The function $\text{sinc}(x)=\frac{\sin x}x$ permeates mathematics and physics in several aspects, and it carries multiple presentations/formulations. My interest is to inject yet another one of such.


Let's understand the determinant $\det(M\_{ij})\_1^{\infty}$ to mean $\lim\_{n\rightarrow\infty}\det(M\_{ij})\_1^n$. The following has experimental basis, therefore I like to ask:



> 
> **Question.** Is there a proof for the determinantal representation
>  $$\det\left[\frac{(i-1)!}{(2j-1)!}\binom{i^2-\theta^2}j\right]\_{i,j=1}^{\infty}=\text{sinc}(\pi\theta) \,\,\,?$$
> 
> 
> 



"sinc'n determinant"
Let's look at $a\_n=\det\left[\frac{(i-1)!}{(2j-1)!}\binom{i^2-\theta^2}j\right]\_{i,j=1}^{n}$. By taking out common factors from rows we can write
$$a\_n=\left(\prod\_{i=1}^n (i-1)!(i^2-\theta^2)\right)\det\left[\frac{1}{(2j-1)!}\frac{1}{(i^2-\theta^2)}\binom{i^2-\theta^2}{j}\right]\_{i,j=1}^{n}$$
$$=\left(\prod\_{i=1}^n (i-1)!(i^2-\theta^2)\right)\det\left[\frac{2}{2j!}\binom{i^2-\theta^2-1}{j-1}\right]\_{i,j=1}^{n}$$
$$=2^n \left(\prod\_{i=1}^n \frac{(i-1)!(i^2-\theta^2)}{2i!}\right)\det\left[\binom{i^2-\theta^2-1}{j-1}\right]\_{i,j=1}^{n}.$$
By the result I mentioned in [this answer](https://mathoverflow.net/questions/268859/almost-hankelized-numerical-vandermonde/268863#268863), with polynomials $P\_j(x)=\binom{x-\theta^2-1}{j-1}$, the last determinant evaluates as
$$\det\left[\binom{i^2-\theta^2-1}{j-1}\right]\_{i,j=1}^{n}=\left(\prod\_{j=1}^n\frac{1}{(j-1)!}\right)\left(\prod\_{i<j}(j^2-i^2)\right)=\left(\prod\_{j=1}^n\frac{1}{(j-1)!}\right)\left(\prod\_{j=1}^n \frac{(2j-1)!}{j}\right).$$
Putting everything together we have
$$a\_n=\frac{1}{n!^2}\prod\_{i=1}^n(i^2-\theta^2)=\prod\_{i=1}^n\left(1-\frac{\theta^2}{i^2}\right),$$
which implies that
$$\lim\_{n\to \infty}a\_n=\prod\_{i\geq 1}\left(1-\frac{\theta^2}{i^2}\right)=\operatorname{sinc}(\pi\theta).$$


