arXiv subjects

E. Burlachenko

Publications and source records attributed to E. Burlachenko.

12 recordsLinked to original sources

arXiv2021

Generalized Riordan groups and zero generalized Pascal matrices

The generalized Riordan group consists of infinite lower triangular matrices that correspond to certain operators in the space of formal power series. Each such group contains the matrix (generalized Pascal matrix), elements of which are generalized binomial coefficients. Generalized Pascal matrices with non-negative elements form an infinite-dimensional vector space. The paper gives an idea of groups similar to the generalized Riordan groups, but associated with matrices, which in the space of generalized Pascal matrices correspond to the points at infinity; examples of such matrices are the matrix of $q$-binomial coefficients for $q=-1$ and the Pascal triangle modulo $2$. An analog of the Lagrange inversion theorem for these groups is given and the corresponding examples are considered.

math.NT↗

arXiv2020

$\alpha$, $\beta$-expansions of the Riordan matrices of the associated subgroup

We consider the group of the matrices $\left( 1,g\left( x \right) \right)$ isomorphic to the group of formal power series $g\left( x \right)=x+{{g}_{2}}{{x}^{2}}+...$ under composition: $\left( 1,{{g}_{2}}\left( x \right) \right)\left( 1,{{g}_{1}}\left( x \right) \right)=\left( 1,{{g}_{1}}\left( {{g}_{2}}\left( x \right) \right) \right)$. Denote $P_{k}^{\alpha }=\left( 1,x{{\left( 1-k\alpha {{x}^{k}} \right)}^{{-1}/{k}\;}} \right)$. Matrix $\left( 1,g\left( x \right) \right)$is decomposed into an infinite product of the matrices $P_{k}^{\alpha }$ with suitable exponents in two ways: to left-handed and right-handed products with respect to the matrix $P_{1}^{{{\alpha }_{1}}={{\beta }_{1}}}$: $\left( 1,g\left( x \right) \right)=...P_{k}^{{{\alpha }_{k}}}...P_{2}^{{{\alpha }_{2}}}P_{1}^{{{\alpha }_{1}}}=P_{1}^{{{\beta }_{1}}}P_{2}^{{{\beta }_{2}}}...P_{k}^{{{\beta }_{k}}}...$. We obtain two formulas expressing the coefficients of the series ${{\left( {g\left( x \right)}/{x}\; \right)}^{z}}$ in terms of the expansion coefficients ${{\alpha }_{i}}$, ${{\beta }_{i}}$ and introduce two one-parameter families of series $g_{\alpha }^{\left( t \right)}\left( x \right)$ and $g_{\beta }^{\left( t \right)}\left( x \right)$ associated with these expansions.

math.NT↗

arXiv2020

Numerator polynomials of the Riordan matrices

Riordan matrices are infinite lower triangular matrices corresponding to the certain operators in the space of formal power series. Generalized Euler polynomials ${{g}_{n}}\left( x \right)={{\left( 1-x \right)}^{n+1}}\sum\nolimits_{m=0}^{\infty }{{{p}_{n}}}\left( m \right){{x}^{m}}$, where ${{p}_{n}}\left( m \right)$ is the polynomial of degree $\le n$, are the numerator polynomials of the generating functions of diagonals of the ordinary Riordan matrices. Generalized Narayana polynomials ${{h}_{n}}\left( x \right)={{\left( 1-x \right)}^{2n+1}}\sum\nolimits_{m=0}^{\infty }{\left( m+1 \right)...\left( m+n \right){{p}_{n}}}\left( m \right){{x}^{m}}$ are the numerator polynomials of the generating functions of diagonals of the exponential Riordan matrices. In paper, the properties of these two types of numerator polynomials and the constructive relationships between them are considered. Separate attention is paid to the numerator polynomials of Riordan matrices associated with the family of series $_{\left( \beta \right)}a\left( x \right)=a\left( x{}_{\left( \beta \right)}{{a}^{\beta }}\left( x \right) \right)$.

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arXiv2019

Composition polynomials of RNA matrix and $B$-composition polynomials of Riordan pseudo-involution

Let $\left( g\left( x \right),xg\left( x \right) \right)$ be a Riordan matrix from the Bell subgroup. We denote ${{\left( g\left( x \right),xg\left( x \right) \right)}^{\varphi }}=\left( {{g}^{\left( \varphi \right)}}\left( x \right),x{{g}^{\left( \varphi \right)}}\left( x \right) \right)$, where a matrix power is defined in the standard way. The polynomials ${{c}_{n}}\left( x \right)$ such that ${{g}^{\left( \varphi \right)}}\left( x \right)=\sum\nolimits_{n=0}^{\infty }{{{c}_{n}}}\left( \varphi \right){{x}^{n}}$ will be called composition polynomials. We consider the composition polynomials of the RNA matrix. The construction associated with these polynomials allows the following generalization. If the matrix $\left( g\left( x \right),xg\left( x \right) \right)$ is a pseudo-involution, then there exists a numerical sequence ($B$-sequence) with the generating function $B\left( x \right)$ such that $g\left( x \right)=1+xg\left( x \right)B\left( {{x}^{2}}g\left( x \right) \right)$. The matrix whose $B$-sequence has the generating function $\varphi B\left( x \right)$ will be denoted by $\left( {{g}^{\left[ \varphi \right]}}\left( x \right),x{{g}^{\left[ \varphi \right]}}\left( x \right) \right)$. The polynomials ${{u}_{n}}\left( x \right)$ such that ${{g}^{\left[ \varphi \right]}}\left( x \right)=\sum\nolimits_{n=0}^{\infty }{{{u}_{n}}}\left( \varphi \right){{x}^{n}}$ will be called $B$-composition polynomials. Coefficients of these polynomials are expressed in terms of the $B$-sequence. We show that matrices whose rows correspond to the $B$-composition polynomials are connected with exponential Riordan matrices of the Lagrange subgroup in a certain way. The cases $B\left( x \right)={{\left( 1-x \right)}^{-1}}$ (RNA matrix), $B\left( x \right)=1+x$, $B\left( x \right)=C\left( x \right)$, where $C\left( x \right)$ is the Catalan series, are considered in detail.

math.NT↗

arXiv2019

Riordan arrays, Chebyshev polynomials, Fibonacci bases

Chebyshev polynomials and their modifications are attributes of various fields of mathematics. In particular, they are generating functions of the rows elements of certain Riordan matrices. In paper, we give a selection of some characteristic situations in which such matrices are involved. Using the columns and rows of these matrices, we will build the bases of the space of formal power series and the space of polynomials, the properties of which allow us to call them "Fibonacci bases".

math.NT↗

arXiv2018

Riordan-Dirichlet group

Riordan matrices are infinite lower triangular matrices that correspond to certain operators in the space of formal power series. In this paper, we introduce similar matrices for the space of formal Dirichlet series. We show that these matrices form a group similar to the Riordan group, and we derive an analog of the Lagrange inversion formula for this group. As an example of the application of these matrices, we obtain an analog of the Abel identities.

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arXiv2018

Exponential Riordan arrays and generalized Narayana polynomials

Generalized Euler polynomials ${{α}_{n}}\left( x \right)={{\left( 1-x \right)}^{n+1}}\sum\nolimits_{m=0}^{\infty }{{{p}_{n}}}\left( m \right){{x}^{m}}$, where ${{p}_{n}}\left( x \right)$ is the polynomial of degree $n$, are the numerator polynomials of the generating functions of diagonals of the ordinary Riordan arrays. Generalized Narayana polynomials ${{φ}_{n}}\left( x \right)={{\left( 1-x \right)}^{2n+1}}\sum\nolimits_{m=0}^{\infty }{\left( m+1 \right)...\left( m+n \right){{p}_{n}}}\left( m \right){{x}^{m}}$ are the numerator polynomials of the generating functions of diagonals of the exponential Riordan arrays. In present paper we consider the constructive relationship between these two types of numerator polynomials.

math.NT↗

arXiv2017

Riordan arrays and generalized Euler polynomials

Generalization of the Euler polynomials ${{A}_{n}}\left( x \right)={{\left( 1-x \right)}^{n+1}}\sum\nolimits_{m=0}^{\infty }{{{m}^{n}}{{x}^{m}}}$ are the polynomials ${{α}_{n}}\left( x \right)={{\left( 1-x \right)}^{n+1}}\sum\nolimits_{m=0}^{\infty }{{{u}_{n}}}\left( m \right){{x}^{m}}$, where ${{u}_{n}}\left( x \right)$ is the polynomial of degree $n$. These polynomials appear in various fields of mathematics, which causes a variety of methods for their study. In present paper we will consider generalized Euler polynomials as an attribute of the theory of Riordan arrays. From this point of view, we will consider the transformations associated with them, with a participation of such objects as binomial sequences, Stirling numbers, multinomial coefficients, shift operator, and demonstrate a constructiveness of the chosen point of view.

math.NT↗

arXiv2017

$B$-expansion of pseudo-involution in the Riordan group

Each numerical sequence $\left( {{b}_{0}},{{b}_{1}},{{b}_{2}},... \right)$ with the generating function $B\left( x \right)$ defines the pseudo-involution in the Riordan group $\left( 1,xg\left( x \right) \right)$ such that $g\left( x \right)=1+xg\left( x \right)B\left( {{x}^{2}}g\left( x \right) \right)$. In the present paper we realize a simple idea: express the coefficients of the series ${{g}^{m}}\left( x \right)$ in terms of the coefficients of the series $B\left( x \right)$. Obtained expansion has a bright combinatorial character, sheds light on the connection of the pseudo-involution in the Riordan group with the generalized binomial series, and is also useful for finding the series $g\left( x \right)$ by the given series $B\left( x \right)$. We compare this expansion with the similar expansion for the sequence $\left( 1,{{a}_{1}},{{a}_{2}},... \right)$ with the generating function $A\left( x \right)$ such that $g\left( x \right)=A\left( xg\left( x \right) \right)$.

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arXiv2017

Algebra of formal power series, isomorphic to the algebra of formal Dirichlet series

Ordinary algebra of formal power series in one variable is convenient to study by means of the algebra of Riordan matrices and the Riordan group. In this paper we consider algebra of formal power series without constant term, isomorphic to the algebra of formal Dirichlet series. To study it, we introduce matrices, similar to the Riordan matrices. As a result, some analogies between two algebras becomes visible. For example, the Bell polynomials (polynomials of partitions of number $n$ into $m$ parts) play a certain role in the ordinary algebra. Similar polynomials (polynomials of decompositions of number $n$ into $m$ factors) play a similar role in the considered algebra. Analog of the Lagrange series for the considered algebra is also exists. In connection with this analogy, we introduce matrix group, similar to the Riordan group and called the Riordan-Dirichlet group. As an example, we consider analog of the Abel's identities for this group.

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arXiv2016

Generalized Riordan arrays and zero generalized Pascal matrices

Generalized Pascal matrix whose elements are generalized binomial coefficients is included in the group of generalized Riordan arrays. There is a special set of generalized Riordan arrays defined by parameter $q$. If $q=0$, they are ordinary Riordan arrays, if $q=1$, they are exponential Riordan arrays. In other cases, except $q=-1$, they are arrays associated with the $q$-binomial coefficients as well as the exponential Riordan arrays are associated with the ordinary binomial coefficients. Case $q=-1$ does not fit into the concept of generalized Riordan arrays, but it is necessary to expand for it. Introduced a special class of matrices, each of which is a limiting case of a certain set of generalized Pascal matrices. It is shown that every such matrix included in the matrix group similar to the generalized Riordan group.

math.CO↗

arXiv2016

Fractal generalized Pascal matrices

Set of generalized Pascal matrices whose elements are generalized binomial coefficients is considered as an integral object. The special system of generalized Pascal matrices, based on which we are building fractal generalized Pascal matrices, is introduced. Pascal matrix (Pascal triangle) is the Hadamard product of the fractal generalized Pascal matrices. The concept of zero generalized Pascal matrices, an example of which is the Pascal triangle modulo 2, arise in connection with the system of matrices introduced.

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