Zeros of the partial sums of \(\cos(z)\) and \(\sin(z)\). I (Q5934412)

The authors prove refinements of two theorems of \textit{M. Kappert} [Numer. Math. 74, No. 4, 397-417 (1996; Zbl 0865.30002)] on the asymptotic behaviour of the zeros of the polynomials \[ \sum^{n/2}_{k=0}{(-1)^k (nz)^{2k} \over(2k)!}, n\text{ even, resp. }\sum^{(n-1)/2}_{k=0}{(-1)^k (nz)^{2k+1} \over(2k-1)!}, n\text{ odd, for }n\to \infty. \] These results enable them to show that the theorems cited above are best possible.