On families of monic polynomials

Abstract

In this paper we derive generalizations of different properties of monic polynomial families of binomial type, i.e. families of monic polynomials, for which the binomial theorem holds $$ p_n(\alpha+\beta)=\sum_{k=0}^n \left(\vphantom{\bigg|}\genfrac{}{}{0pt}{0}{n}{k}\right) p_k(\alpha)p_{n-k}(\beta) $$ Some trivial representations of general ''multiplication'' and ''derivative'' operators are derived. In addition we derive a formula for the logarithmic derivative of general monic polynomial $p_n(x)$ which reduces to the formula $$ \frac{1}{n}\frac{p_n'(x)}{p_n(x)} =\left(x+\frac{1}{\varphi'(y)}\left(\frac{d}{dy}-n\mathrm{L}\right)\right)^{-1}\cdot\left.\frac{\varphi(y)}{y\varphi'(y)}~\right|_{y=0} $$ derived by the author in binomial case, when the generating function of $p_n(x)$ equals to $e^{x\varphi(y)}$.