Representation of some binomial coefficients by polynomials
Bull. Korean Math. Soc. 2007 Vol. 44, No. 4, 677-682
Published online December 1, 2007
Seon-Hong Kim
Sookmyung Women's University
Abstract : 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
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