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Danil Krotkov

Publications and source records attributed to Danil Krotkov.

5 recordsLinked to original sources
arXiv2026

On a certain type of deformation of umbral orthogonal polynomials

In the present paper we derive complicated families of orthogonal polynomials in one variable from scratch using the known ones as building blocks. We recall the basics of operational formalism and introduce the notations we use throughout this paper. Then we establish through this formalism the known families of orthogonal polynomials in order to see the mechanism that lies at the heart of classical and hypergeometric $q=1$ orthogonal polynomials and their associated families, which arise form this viewpoint in a very natural manner.

math.NT↗
arXiv2026

On families of monic polynomials

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

math.NT↗
arXiv2019

Taking the logarithm of binomial type sequences: linear approach

In this paper we obtain the formal asymptotic expansion of the logarithms $\ln p_s(\alpha)$ of $p_s(\alpha)$, which are canonical continuations of polynomials of binomial type $p_n(\alpha)$. Our approach is based on linear methods which do not require the calculation of expansions $(p_s(\alpha)\alpha^{-s}-1)^k$, as opposed to the direct logarithmization.

math.NT↗
arXiv2019

On polynomials of binomial type, Ramanujan-Soldner constant and inverse logarithmic derivative operator

In this work we introduce interesting infinite series, related to Ramanujan-Soldner constant. Our method uses general properties of polynomials of binomial type and Lagrange inversion theorem. Also we study properties of the operator 1/dlog, acting on formal power series. In addition, several properties of polynomials of binomial type associated to elementary functions are discussed.

math.NT↗