On Legendre numbers (Q1065051)

The Legendre polynomials \(P_ n(x)\) are defined by \((1-2xt+t^ 2)^{- 1/2}=\sum^{\infty}_{n=0}P_ n(x) t^ n\), \(| t| <1\). For integers \(m\geq 0\), \(n\geq 0\), the Legendre number \(P^ m_ n\) is \(P_ n^{(m)}(0)\), the value of the m-th derivative of \(P_ n(x)\) at \(x=0\). In this paper, the author makes a systematic study of the numbers \(P^ m_ n\). He gives a general formula and a short table for \(P^ m_ n\) and evaluates some series involving \(P^ m_ n\). Further he expresses the higher derivatives of \(P_ n(x)\) at \(x=1\) in terms of \(P^ m_ n\).