The analogues of some binomial coefficients (Q1430934)

The author investigates the irreducibility of the polynomials \[ F_m(u)= u^{2m}- u^{m+1}- u^{m-1}- 1 \] and the property of the unique positive zero \(r_m\) of \(F_m\). Using \(r_m\) a new analogue \(a(m,k)\) of the binomial coefficient \(2{2n\choose k}\) (\(k\) even) is defined as: For a positive even integer \(k\) \((k<n)\); \[ a(m,k)= 2{n\choose k/2}_2F_1\Biggl(- {k\over 2}- {1\over 2} (2n- k); {1\over 2}, {1\over 4} (r_m+ r_m^{-1})^2\Biggr)^2. \] The minimal polynomial of this analogue has been defined and shown that it is related to Chebyshev polynomials. The method of computation of the minimal polynomial of an analogue of \(2{2n\choose k}\) \((k> 2)\) by using an analogue of \(2{2n\choose 2}\) has been discussed.