Properties of Liron’s Polynomials

Abstract

For real nonzero k, put \[S_m (k) = \frac{1}{2}\sum_{n = 0}^\infty {\alpha _n^{ - 2m - 2} } ,\] where $\alpha _n $ runs through the nonzero roots of tan $\alpha = k\alpha $. Liron [5] found a generating function for $S_m (k)$ and showed that $S_m (k) = (k - 1)^{ - m - 1} P_{m + 1} (k)$, where $P_{m + 1} (k)$ is a polynomial in k of degree $m + 1$. Carlitz [1] showed that $P_{m + 1} (k)$ has coefficients that are closely related to tangent coefficients of higher order. In the present paper additional properties of $P_{m + 1} (k)$ are derived. In particular it is shown that $P_{m + 1} (k)$ can be expressed simply in terms of $P_m (k)$, $P'_m (k)$ and $P'_{m + 1} (k)$, and this result leads to a simple recurrence formula for the coefficients. Also formulas for $P'_{m + 1} (1)$ and $P''_{m + 1} (1)$ are found, and Carlitz’s result that $(3m + 3)!P_{m + 1} (k)/(m + 1)!$ has integral coefficients is sharpened.