Searcharxiv⌕ Search

arXiv subjects

Oksana Bihun

Publications and source records attributed to Oksana Bihun.

At least 19 recordsLinked to original sources
arXiv2019

Orthogonal sequences constructed from quasi-orthogonal ultraspherical polynomials

Let $\displaystyle \{x_{k,n-1}\} _{k=1}^{n-1}$ and $\displaystyle \{x_{k,n}\} _{k=1}^{n},$ $n \in \mathbb{N}$, be two sets of real, distinct points satisfying the interlacing property $ x_{i,n} -3/2$, $λ\neq -1,0, (2k-1)/2, k=0,1,\ldots$. We plot and compare the zeros of $D_m^λ (x)$ and $C_m^λ (x)$ for several choices of $m \in \mathbb{N}$ and a range of values of the parameters $λ$ and $n$. For $-3/2 < λ< -1,$ the curves that the zeros of $D_m^λ (x)$ and $C_m^λ (x)$ approach are substantially different for large values of $m.$ When $-1 < λ< -1/2,$ the two curves have a similar shape while the curves are almost identical for $λ>-1/2.$

math.CA↗
arXiv2018

Solvable dynamical systems and isospectral matrices defined in terms of the zeros of orthogonal or otherwise special polynomials

Several recently discovered properties of multiple families of special polynomials (some orthogonal and some not) that satisfy certain differential, difference or q-difference equations are reviewed. A general method of construction of isospectral matrices defined in terms of the zeros of such polynomials is discussed. The method involves reduction of certain partial differential, differential difference and differential q-difference equations to systems of ordinary differential equations and their subsequent linearization about the zeros of polynomials in question. Via this process, solvable (in terms of algebraic operations) nonlinear first order systems of ordinary differential equations are constructed.

math-ph↗
arXiv2018

Time-dependent polynomials with one multiple root and new solvable dynamical systems

A time-dependent monic polynomial in the z variable with N distinct roots such that exactly one root has multiplicity m>=2 is considered. For k=1,2, the k-th derivatives of the N roots are expressed in terms of the derivatives of order j<= k of the first N coefficients of the polynomial and of the derivatives of order j<= k-1 of the roots themselves. These relations are utilized to construct new classes of algebraically solvable first order systems of ODEs as well as N-body problems. Multiple examples of solvable isochronous (all solutions are periodic with the same period) 2- and 3-body problems are provided.

math-ph↗
arXiv2018

Time-dependent polynomials with one double root, and related new solvable systems of nonlinear evolution equations

Recently new solvable systems of nonlinear evolution equations -- including ODEs, PDEs and systems with discrete time -- have been introduced. These findings are based on certain convenient formulas expressing the $k$-th time-derivative of a root of a time-dependent monic polynomial in terms of the $k$-th time-derivative of the coefficients of the same polynomial and of the roots of the same polynomial as well as their time-derivatives of order less than $k$. These findings were restricted to the case of generic polynomials without any multiple root. In this paper some of these findings -- those for $k=1$ and $k=2$ -- are extended to polynomials featuring one double root; and a few representative examples are reported of new solvable systems of nonlinear evolution equations.

math-ph↗
arXiv2018

Generalized Pseudospectral Method and Zeros of Orthogonal Polynomials

Via a generalization of the pseudospectral method for numerical solution of differential equations, a family of nonlinear algebraic identities satisfied by the zeros of a wide class of orthogonal polynomials is derived. The generalization is based on a modification of pseudospectral matrix representations of linear differential operators proposed in the paper, which allows these representations to depend on two, rather than one, sets of interpolation nodes. The identities hold for every polynomial family $\{p_ν(x)\}_{ν=0}^\infty$ orthogonal with respect to a measure supported on the real line that satisfies some standard assumptions, as long as the polynomials in the family satisfy differential equations $\mathcal{A} p_ν(x) =q_ν(x) p_ν(x)$, where $\mathcal{A}$ is a linear differential operator and each $q_ν(x)$ is a polynomial of degree at most $n_0 \in \mathbb{N}$; $n_0$ does not depend on $ν$. The proposed identities generalize known identities for classical and Krall orthogonal polynomials, to the case of the nonclassical orthogonal polynomials that belong to the class described above. The generalized pseudospectral representations of the differential operator $\mathcal{A}$ for the case of the Sonin-Markov orthogonal polynomials, also known as generalized Hermite polynomials, are presented. The general result is illustrated by new algebraic relations satisfied by the zeros of the Sonin-Markov polynomials.

math.CA↗
arXiv2018

Polynomials Whose Coefficients Coincide with Their Zeros

In this paper we consider monic polynomials such that their coefficients coincide with their zeros. These polynomials were first introduced by S. Ulam. We combine methods of algebraic geometry and dynamical systems to prove several results. We obtain estimates on the number of Ulam polynomials of degree $N$. We provide additional methods to obtain algebraic identities satisfied by the zeros of Ulam polynomials, beyond the straightforward comparison of their zeros and coefficients. To address the question about existence of orthogonal Ulam polynomial sequences, we show that the only Ulam polynomial eigenfunctions of hypergeometric type differential operators are the trivial Ulam polynomials $\{x^N\}_{N=0}^\infty$. We propose a family of solvable $N$-body problems such that their stable equilibria are the zeros of certain Ulam polynomials.

math.CA↗
arXiv2017

The Chazy XII Equation and Schwarz Triangle Functions

Dubrovin [Lecture Notes in Math., Vol. 1620, Springer, Berlin, 1996, 120-348] showed that the Chazy XII equation $y'''- 2yy''+3y'^2 = K(6y'-y^2)^2$, $K \in \mathbb{C}$, is equivalent to a projective-invariant equation for an affine connection on a one-dimensional complex manifold with projective structure. By exploiting this geometric connection it is shown that the Chazy XII solution, for certain values of $K$, can be expressed as $y=a_1w_1+a_2w_2+a_3w_3$ where $w_i$ solve the generalized Darboux-Halphen system. This relationship holds only for certain values of the coefficients $(a_1,a_2,a_3)$ and the Darboux-Halphen parameters $(α, β, γ)$, which are enumerated in Table 2. Consequently, the Chazy XII solution $y(z)$ is parametrized by a particular class of Schwarz triangle functions $S(α, β, γ; z)$ which are used to represent the solutions $w_i$ of the Darboux-Halphen system. The paper only considers the case where $α+β+γ<1$. The associated triangle functions are related among themselves via rational maps that are derived from the classical algebraic transformations of hypergeometric functions. The Chazy XII equation is also shown to be equivalent to a Ramanujan-type differential system for a triple $(\hat{P}, \hat{Q},\hat{R})$.

math.CA↗
arXiv2017

New Properties of the Zeros of Krall Polynomials

We identify a class of remarkable algebraic relations satisfied by the zeros of the Krall orthogonal polynomials that are eigenfunctions of linear differential operators of order higher than two. Given an orthogonal polynomial family {p_n(x)}, we relate the zeros of the polynomial p_N with the zeros of p_m for each m <= N (the case m=N corresponding to the relations that involve the zeros of p_N only). These identities are obtained by exacting the similarity transformation that relates the spectral and the (interpolatory) pseudospectral matrix representations of linear differential operators, while using the zeros of the polynomial p_N as the interpolation nodes. The proposed framework generalizes known properties of classical orthogonal polynomials to the case of nonclassical polynomial families of Krall type. We illustrate the general result by proving new remarkable identities satisfied by the Krall-Legendre, the Krall-Laguerre and the Krall-Jacobi orthogonal polynomials.

math.CA↗
arXiv2016

Generations of solvable discrete-time dynamical systems

A technique is introduced which allows to generate -- starting from any solvable discrete-time dynamical system involving N time-dependent variables -- new, generally nonlinear, generations of discrete-time dynamical systems, also involving N time-dependent variables and being as well solvable by algebraic operations (essentially by finding the N zeros of explicitly known polynomials of degree N). The dynamical systems constructed using this technique may also feature large numbers of arbitrary constants, and they need not be autonomous. The solvable character of these models allows to identify special cases with remarkable time evolutions: for instance, isochronous or asymptotically isochronous discrete-time dynamical systems. The technique is illustrated by a few examples.

math-ph↗
arXiv2016

Novel solvable many-body problems

Novel classes of dynamical systems are introduced, including many-body problems characterized by nonlinear equations of motion of Newtonian type ("acceleration equals forces") which determine the motion of points in the complex plane. These models are solvable, namely their configuration at any time can be obtained from the initial data by algebraic operations, amounting to the determination of the zeros of a known time-dependent polynomial in the independent variable z. Some of these models are multiply periodic, isochronous or asymptotically isochronous; others display scattering phenomena.

math-ph↗
arXiv2015

Generations of monic polynomials such that the coefficients of the polynomials of the next generation coincide with the zeros of the polynomials of the current generation, and new solvable many-body problems

The notion of generations of monic polynomials such that the coefficients of the polynomials of the next generation coincide with the zeros of the polynomials of the current generation is introduced, and its relevance to the identification of endless sequences of new solvable many-body problems of "goldfish type" is demonstrated.

math-ph↗
arXiv2015

Properties of the zeros of generalized hypergeometric polynomials

We define the generalized hypergeometric polynomial of degree N in terms of the generalized hypergeometric function that depends on p parameters a_1, ..., a_p and q parameters b_1, ..., b_q. The parameters are "generic", possibly complex, numbers. In this paper we obtain a set of N nonlinear algebraic equations satisfied by the N zeros z_n of this polynomial. We moreover manufacture an NxN matrix L in terms of the 1+p+q parameters N, a_j, b_k characterizing this polynomial, and of its N zeros z_n. We show that the matrix L features N eigenvalues that depend only on the q parameters b_k, implying that this matrix is isospectral for the variations of the p parameters a_j. These eigenvalues are integer (or rational) numbers if the q parameters b_k are themselves integer (or rational) numbers: a nontrivial diophantine property.

math-ph↗
arXiv2015

Diophantine properties of the zeros of (monic) polynomials the coefficients of which are the zeros of Hermite polynomials

We introduce a monic polynomial p_N(z) of degree N whose coefficients are the zeros of the N-th degree Hermite polynomial. Note that there are N! such different polynomials p_N(z), depending on the ordering assignment of the N zeros of the Hermite polynomial of order N. We construct two NxN matrices M_1 and M_2 defined in terms of the N zeros of the polynomial p_N(z). We prove that the eigenvalues of M_1 and M_2 are the first N integers respectively the first N squared-integers, a remarkable isospectral and Diophantine property. The technique whereby these findings are demonstrated can be extended to other named polynomials.

math-ph↗
arXiv2015

Properties of the zeros of generalized basic hypergeometric polynomials

We define the generalized basic hypergeometric polynomial of degree $N \geq 1$ in terms of the generalized basic hypergeometric function, which depends on (arbitrary, generic, possibly complex) parameters $q \neq 1$, the $r \geq 0$ parameters $α_{j}$ and the $s \geq 0$ parameters $β_{k}$. In this paper we obtain a set of $N$ nonlinear algebraic equations satisfied by the $N$ zeros $ζ_{n}\equiv ζ_{n}\left( \underline{α},\underline{β};q;N\right) $ of this polynomial. We moreover identify an $\left( N\times N\right) $-matrix $\underline{M}\equiv \underline{M}\left( \underline{α},\underline{β};\underline{ζ};q;N\right) $ featuring the $N$ eigenvalues $μ_{n}=-q^{\left( s-r\right) \left( N-n\right) }\left(q^{-n}-1\right) ~\prod\limits_{j=1}^{r}\left( α_{j}~q^{N-n}-1\right)$, where $n=1,2,...,N.$ These $N$ eigenvalues depend only on the $r$ parameters $α_{j}$ (besides $q$ and $N$), implying that the $\left( N\times N\right) $-matrix $\underline{M}$ is isospectral for variations of the $s$ parameters $β_{k}$; and they clearly are rational numbers if $q$ and the $r$ parameters $α_{j}$ are themselves rational numbers: a nontrivial Diophantine property.

math-ph↗
arXiv2014

Properties of the zeros of the polynomials belonging to the q-Askey scheme

In this paper we provide properties -- which are, to the best of our knowledge, new -- of the zeros of the polynomials belonging to the q-Askey scheme. These findings include Diophantine relations satisfied by these zeros when the parameters characterizing these polynomials are appropriately restricted.

math-ph↗
arXiv2014

Properties of the zeros of the polynomials belonging to the Askey scheme

In this paper we provide properties---which are, to the best of our knowledge, new---of the zeros of the polynomials belonging to the Askey scheme. These findings include Diophantine relations satisfied by these zeros when the parameters characterizing these polynomials are appropriately restricted.

math.CA↗
arXiv2014

Solvable and/or integrable many-body models on a circle

Various many-body models are treated, which describe $N$ points confined to move on a plane circle. Their Newtonian equations of motion ("accelerations equal forces") are integrable, i. e. they allow the explicit exhibition of $N$ constants of motion in terms of the dependent variables and their time-derivatives. Some of these models are moreover solvable by purely algebraic operations, by (explicitly performable) quadratures and, finally, by functional inversions. The techniques to manufacture these models are not new; some of these models are themselves new; others are reinterpretations of known models.

math-ph↗