REPRESENTATION OF SOME BINOMIAL COEFFICIENTS BY POLYNOMIALS

REPRESENTATION OF SOME BINOMIAL COEFFICIENTS BY POLYNOMIALS

The unique positive zero of $F_m(z):=z^{2m}-z^{m+1}-z^{m-1}-1$ leads to analogues of $2(\array{2n\\k}\)$(k 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(\array{2n\\k}\)$(k even>2) can be computed by using an analogue of $2(\array{2n\\k}\)$. In this paper we show that the analogue of $2(\array{2n\\2}\)$. In this paper we show that the analygue $2(\array{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).