A conjecture of J.M. Carnicer, E. Mainar and J.M. Pe\~{ n}a states that the critical length of the space $P_{ n}\odot C_{ 1}$ generated by the functions $x^{ k}\sin x$ and $x^{ k}\cos x$ for $k=0,...n$ is equal to the first positive zero $j_{ n+\frac{ 1}{ 2},1}$ of the Bessel function $J_{ n+\frac{ 1}{ 2}}$ of the first kind. It is known that the conjecture implies the following statement (D3): the determinant of the Hankel matrix \begin{ equation} \left( \begin{ array} [c]{ ccc} f & f^{ \prime} & f^{ \prime\prime}\\ f^{ \prime} & f^{ \prime\prime} & f^{ \left( 3\right) }\\ f^{ \prime\prime} & f^{ \prime\prime\prime} & f^{ \left( 4\right) } \end{ array} \right) \label{ eqabstract} \end{ equation} does not have a zero in the interval $(0,j_{ n+\frac{ 1}{ 2},1})$ whenever $f=f_{ n}$ is given by $f_{ n}\left( x\right) =\sqrt{ \frac{ \pi}{ 2}} x^{ n+\frac{ 1}{ 2}}J_{ n+\frac{ 1}{ 2}}\left( x\right) .$ In this paper we shall prove (D3) and various generalizations.
Categories: math.CA cs.NA math.NA
