Bulletin of the
Korean Mathematical Society BKMS

The unique positive zero of $F_m(z):=z^{2m}-z^{m+1}-z^{m-1}-1$ leads to analogues of $2 \binom {2n}k \, (k \, \text{even})$ by using hypergeometric functions. The minimal polynomials of these analogues are related to Chebyshev polynomials, and the minimal polynomial of an analogue of $2 \binom {2n}k$ $(k \, \text{even}>2)$ can be computed by using an analogue of $2 \binom {2n}2$. In this paper we show that the analogue of $2 \binom {2n}2$ is the only real zero of its minimal polynomial, and has a different representation, by using a polynomial of smaller degree than $F_m(z)$.

Keywords: binomial coefficients, analogues, minimal polynomial, Chebyshev polynomial

MSC numbers: Primary 11B65; Secondary 05A10