Two Hankel Determinants dear to Volodia Retakh Shalosh B. EKHAD and Doron ZEILBERGER

Dedicated to Vladimir Retakh (b. May 20, 1948) on his 70th birthday

Let B_{s} := \frac{(2s)!}{s!^{2}}, then

det (B_{r+i+j})_{0≤i,j≤n−1} =

(\frac{(2 r)!}{(r!)^{2}})^{n} ∏_{n_{1}=1}^{n} ( ∏_{n_{2}=2}^{n_{1}} 4 \begin{matrix}(n_{2} − 1) & (2 n_{2} − 3) & (2 n_{2} − 3 + 2 r) & (n_{2} − 2 + r) \\ (2 n_{2} − 2 + r) & (2 n_{2} − 4 + r) & (2 n_{2} − 3 + r) & 2 \end{matrix}) .

Let C_{s} := \frac{(2s)!}{s!(s+1)!}, then

det (C_{r+i+j})_{0≤i,j≤n−1} =
(\frac{(2 r)!}{r! (r + 1)!})^{n} ∏_{n_{1}=1}^{n} ( ∏_{n_{2}=2}^{n_{1}} 4 \begin{matrix}(n_{2} − 1) & (2 n_{2} − 1) & (n_{2} − 1 + r) & (2 n_{2} − 3 + 2 r) \\ (2 n_{2} − 1 + r) & (2 n_{2} − 3 + r) & (2 n_{2} − 2 + r) & 2 \end{matrix})

Proofs First Proof of both determinants (by SBE). Go to the Maple package

http://sites.math.rutgers.edu/~zeilberg/tokhniot/CLD ,

and type

EvalHpaper((2*r)!/(r!*r!),n,r,4); ,

and

EvalHpaper((2*r)!/(r!*(r+1)!),n,r,4); ,

respectively, getting, ab initio, the above expressions, together with proofs, using Dodgson’s rule and induction. See [Z]. Sketch of the Second Proof of the First determinant (by DZ) We use the fact that

\frac{(2r)!}{r!^{2}} = ConstantT erm_{x}(\frac{1}{x} + 2 + x)^{r} .

Let f(x) = \frac{1}{x} + 2 + x. The determinant is the Constant term, in x_{1}, …, x_{n}, of

det (f(x_{i})^{r+i+j−2})_{1≤i,j≤n} .

= ConstantT erm_{x_{1},...,x_{n}} ∏_{i=1}^{n} f(x_{i})^{r} det (f(x_{i})^{i+j−2})_{1≤i,j≤n} .

By Vandermonde (also dear to Volodia Retakh!), this equals

ConstantT erm_{x_{1},...,x_{n}} ∏_{i=1}^{n} f(x_{i})^{r} ∏_{i=1}^{n} f(x_{i})^{i−1} ∏_{1≤i<j≤n}(f(x_{i}) − f(x_{j})) .

Symmetrizing, and using Vandermonde one more time, this equals

\frac{1}{n!} CT_{x_{1},...,x_{n}} ∏_{i=1}^{n} f(x_{i})^{r} ∏_{1≤i<j≤n}(f(x_{i}) − f(x_{j}))^{2} .

Now, lo and behold, this is a special case k_{1} = r, k_{2} = 1, k_{3} = 0, of the BC_{n} Macdonald Constant-Term Identity (see, e.g., [FW], top of p. 501).

Sketch of the Second Proof of the Second determinant (by DZ)

We use the fact that

\frac{(2r)!}{r!(r + 1)!} = ConstantT erm_{x}(1 − x)(\frac{1}{x} + 2 + x)^{r} .

The determinant is the constant term, in x_{1}, …, x_{n}, of

∏_{i=1}^{n}(1 − x_{i}) det (f(x_{i})^{r+i+j−2})_{1≤i,j≤n} .

This equals

ConstanT erm_{x_{1},...,x_{n}} ∏_{i=1}^{n}(1 − x_{i}) ∏_{i=1}^{n} f(x_{i})^{r} det (f(x_{i})^{i+j−2})_{1≤i,j≤n} .

By Vandermonde, this equals

ConstantT erm_{x_{1},...,x_{n}} ∏_{i=1}^{n}(1 − x_{i}) ∏_{i=1}^{n} f(x_{i})^{r} ∏_{i=1}^{n} f(x_{i})^{i−1} ∏_{1≤i<j≤n}(f(x_{i}) − f(x_{j})) .

Symmetrizing this equals

\frac{1}{n!} ConstantT erm_{x_{1},...,x_{n}} ∏_{i=1}^{n}(1 − x_{i}) ∏_{i=1}^{n} f(x_{i})^{r} ∏_{1≤i<j≤n}(f(x_{i}) − f(x_{j}))^{2} .

Now, lo and behold, this is a special case k_{1} = r − \frac{1}{2}, k_{2} = \frac{1}{2}, k_{3} = 1, of the BC_{n} Macdonald Constant-Term Identity mentioned above ([FW], top of p. 501).

Challenges for Volodia

In the beautiful article [BR], the authors prove a non-commutative analog, using quasi-determinants, of the special cases r = 0 and r = 1 of the second determinant (where the determinant happens to be 1, in the commutative case). What is the non-commutative analog, for general r, of the second determinant? The first determinant?

References

[BR] Arkady Berenstein and Vladimir Retakh, Noncommutative Catalan Numbers,

https://arxiv.org/abs/1708.03316 .

[FW] Peter J. Forrester and S. Ole Warnaar , The importance of the Selberg integral, Bull. Amer. Math. Soc. 45 (2008), 489-534

http://www.ams.org/journals/bull/2008-45-04/S0273-0979-08-01221-4/ .

[Z] Doron Zeilberger, Lieber Opa Paul, Ich Bin Auch Ein Experimental Scientist!, Adv. Applied Math. 31 (2003), 532-543,

http://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/opa.html .

Shalosh B. Ekhad, c/o D. Zeilberger, Department of Mathematics, Rutgers University (New Brunswick), Hill Center-Busch Campus, 110 Frelinghuysen Rd., Piscataway, NJ 08854-8019, USA.

Email: ShaloshBEkhad at gmail dot com .

Doron Zeilberger, Department of Mathematics, Rutgers University (New Brunswick), Hill Center-Busch Campus, 110 Frelinghuysen Rd., Piscataway, NJ 08854-8019, USA.

Email: DoronZeil at gmail dot com .

First Written: May 20, 2018. This version: July 5, 2018.