With Ea,,(lx I) denoting the error of best uniform approximation to Ixl by polynomials of degree at most 2n on the interval [-1, + 1J, the famous Russian mathematician S. Bernstein in 1914 established the existence of a positive constant fl for which lim 2nE2,,(rxJ) =: ft.Moreover, by means of numerical calculations, Bernstein determined, in the same paper, the following upper and lower bounds for fl: 0.278 < fl < 0.286. Now, the average of these bounds is 0.282, which, as Bernstein noted as a "curious coincidence," is very close to 1/(2~/~ -) = 0.2820947917 .... This observation has over the years become known as the Bemstein Conjecture: Is fl = 1/(2r We show here that the Bemstein conjecture is false. In addition, we determine rigorous upper and lower bounds for fl, and by means of the Richardson extrapolation procedure, estimate /~ to approximately 50 decimal places.
Summary. We investigate the location of the zeros of the normalized generalized Bessel polynomials and the normalized reversed generalized Bessel polynomials. Also, the rate at which these zeros approach certain well-defined curves is investigated. On the basis of numerical computations and graphs, four new conjectures are proposed.
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