The discussion on this page is motivated by Johann Cigler's MO question. My intention arose from a possible generalization of Cigler's matrix $$A_{n,m}=\left( \binom{2m}{j-i+m}-\binom{2m}{m-i-j-1} \right)_{0\leq {i,j} \leq {n-1}}.$$
Introduce a new parameter $x$ and the matrix $$T_{n,m}(x):=\left(\binom{x+m}{j-i+m}-\binom{x+m}{m-i-j-1}\right)_{i,j=0}^{n-1}.$$
QUESTION. Is this determinant evaluation true? $$\det T_{n,m}(x)=\prod_{i=1}^n\prod_{j=1}^m\frac{(x+i-j)(x+2i+j-2)}{(x+2i-j)(i+j-1)}.$$
Remark 1. The relation $\det T_{n,m}(m)=\det A_{n,m}=\prod_{1\leq i\leq j\leq m-1}\frac{2n+i+j}{i+j}$ recovers that of Cigler's determinant. The latter is connected to a host of other interpretations (see recent MO question).
Remark 2. I am hoping that the structure of the new determinant would possibly make it easier to resolve the earlier problem mentioned above.
