On families of monic polynomials

TL;DR

This paper explores generalizations of properties of monic polynomial families of binomial type, deriving new formulas for their multiplication, derivatives, and logarithmic derivatives, extending known results to broader classes.

Contribution

It introduces generalized properties and formulas for monic polynomial families of binomial type, expanding the theoretical framework beyond classical cases.

Findings

Derived trivial representations of multiplication and derivative operators.

Established a general formula for the logarithmic derivative of monic polynomials.

Extended binomial type properties to broader polynomial families.

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 $e^{x\varphi(y)}$.