# Alexander DyachenkoUniversity College London | UCL · Department of Mathematics

Alexander Dyachenko

Doctor of Philosophy

## About

28

Publications

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48

Citations

## Publications

Publications (28)

We find the spectrum and eigenvectors of an arbitrary irreducible complex tridiagonal matrix with two-periodic main diagonal. This is expressed in terms of the spectrum and eigenvectors of the matrix with the same sub- and superdiagonals and zero main diagonal. Our result generalises some recent results where the latter matrix stemmed from certain...

We consider the problem of the reconstruction of a Schwarz matrix from exactly one given eigenvalue. This inverse eigenvalue problem leads to the Jacobi orthogonal polynomials~$\{P_k^{(-n,n)}\}_{k=0}^{n-1}$ that can be treated as a discrete finite analogue of Bessel polynomials.

Known already to the ancient Greeks, today trigonometric identities come in a large variety of tastes and flavours. In this large family there is a subfamily of interpolation-like identities discovered by Hermite and revived rather recently in two independent papers, one by Wenchang Chu and the other by Warren Johnson exploring various forms and ge...

Known already to the ancient Greeks, today trigonometric identities come in a large variety of tastes and flavours. In this large family there is a subfamily of interpolation-like identities discovered by Hermite and revived rather recently in two independent papers, one by Wenchang Chu and the other by Warren Johnson exploring various forms and ge...

Measures generating classical orthogonal polynomials are determined by Pearson’s equation, whose parameters usually provide the positivity of the measures. The case of general complex parameters (nonstandard) is also of interest; the non-Hermitian orthogonality with respect to (now complex-valued) measures is considered on curves in C. Some applica...

Given real parameters a,b,c and integer shifts n1,n2,m, we consider the ratio R(z)=2F1(a+n1,b+n2;c+m;z)/2F1(a,b;c;z) of the Gauss hypergeometric functions. We find a formula for ImR(x±i0) with x>1 in terms of real hypergeometric polynomial P, beta density and the absolute value of the Gauss hypergeometric function. This allows us to construct expli...

A method of generating differential operators is used to solve the spectral problem for a generalisation of the Sylvester-Kac matrix. As a by-product, we find a linear differential operator with polynomial coefficients of the first order that has a finite sequence of polynomial eigenfunctions generalising the operator considered by M. Kac. In addit...

We find the spectrum of an arbitrary irreducible complex tridiagonal matrix with two-periodic main diagonal provided that the spectrum of the matrix with the same sub- and superdiagonals and zero main diagonal is known. Our result substantially generalises some of the recent results on the Sylvester-Kac matrix and its certain main principal submatr...

We consider the ratio of two Gauss hypergeometric functions, in which the parameters of the numerator function differ from the respective parameters of the denominator function by integers. We derive explicit integral representations for this ratio based on a formula for its imaginary part. This work extends our recent results by lifting certain re...

We exhibit a lower-triangular matrix of polynomials T(a,c,d,e,f,g) in~six indeterminates that appears empirically to be coefficientwise totally positive, and which includes as a special case the Eulerian triangle. We prove the coefficientwise total positivity of T(a,c,0,e,0,0), which includes the reversed Stirling subset triangle.

A method of generating differential operators is used to solve the spectral problem for a generalisation of the Sylvester-Kac matrix. As a by-product, we find a linear differential operator with polynomial coefficients of the first order that has a finite sequence of polynomial eigenfunctions generalising the operator considered by M. Kac.

We consider the ratio of two Gauss hypergeometric functions with real parameters shifted by arbitrary integers. We find a formula for the jump of this ratio over the branch cut in terms of a real hypergeometric polynomial, the beta density and the absolute value of the Gauss hypergeometric function. This allows us to construct explicit integral rep...

We exhibit a lower-triangular matrix of polynomials $T(a,c,d,e,f,g)$ in six indeterminates that appears empirically to be coefficientwise totally positive, and which includes as a special case the Eulerian triangle. We prove the coefficientwise total positivity of $T(a,c,0,e,0,0)$, which includes the reversed Stirling subset triangle.

In this paper we associate a class of Hurwitz matrix polynomials with Stieltjes positive definite matrix sequences. This connection leads to an extension of two classical criteria of Hurwitz stability for real polynomials to matrix polynomials: tests for Hurwitz stability via positive definiteness of block-Hankel matrices built from matricial Marko...

This paper aims at finding conditions on a Hamburger or Stieltjes moment sequence, under which the change of at most a finite number of its entries produces another sequence of the same type. It turns out that a moment sequence allows all small enough variations of this kind precisely when it is indeterminate. We also show that a determinate moment...

In this paper, we associate a class of Hurwitz matrix polynomials with Stieltjes positive definite matrix sequences. This connection leads to an extension of two classical criteria of Hurwitz stability for real polynomials to matrix polynomials: tests for Hurwitz stability via positive definiteness of block-Hankel matrices built from matricial Mark...

We introduce a new class of polynomials of multiple orthogonality with respect to the product of $r$ classical discrete weights on integer lattices with noninteger shifts. We give explicit representations in the form of the Rodrigues formulas. The case of two weights is described in detail.

This paper aims at finding conditions on a Hamburger or Stieltjes moment sequence, under which the change of at most a finite number of its entries produces another sequence of the same type. It turns out that a moment sequence allows all small enough variations of this kind precisely when it is indeterminate. We also show that a determinate moment...

In this work we study the solutions of the equation $z^pR(z^k)=\alpha$ with
nonzero complex $\alpha$, integer $p,k$ and $R(z)$ generating a (possibly
doubly infinite) totally positive sequence. It is shown that the zeros of
$z^pR(z^k)-\alpha$ are simple (or at most double in the case of real
$\alpha^k$) and split evenly among the sectors $\{\frac j...

This paper show that two doubly infinite series generate a totally nonnegative Hurwitz-type matrix if and only if their ratio represents an S-functions of a certain kind. The doubly infinite case needs a specific approach, since the ratios have no correspondent Stieltjes continued fraction. Another forthcoming publication (see Dyachenko, arXiv:1608...

This paper aims at extending the criterion that the quasi-stability of a polynomial is equivalent to the total nonnegativity of its Hurwitz matrix. We give a complete description of functions generating doubly infinite series with totally nonnegative Hurwitz and Hurwitz-type matrices (in a Hurwitz-type matrix odd and even rows come from two distinc...

Under the Hermite-Biehler method we understand the approach to problems of stability, which exploits a deep relation between Hurwitz-stable functions and mappings of the upper half of the complex plane into itself (i.e. R-functions). This method dates back to works by Hermite, Biehler, Hurwitz; in the first half of the XXth century it was extended...

In the present note we give an elementary proof of the necessary and
sufficient condition for a univariate function to belong the class $\mathcal
N_\varkappa^+$. Although this class was introduced mainly to deal with the
indefinite version of the Stieltjes moment problem (and corresponding
$\pi$-Hermitian operators), it can be useful far beyond the...

In the present note we obtain new results on two conjectures by Csordas et
al. regarding the interlacing property of zeros of special polynomials. These
polynomials came from the Jacobi tau methods for the Sturm-Liouville eigenvalue
problem. Their coefficients are the successive even derivatives of the Jacobi
polynomials $P_n(x;\alpha,\beta)$ evalu...

In this paper we determine a class of entire functions using conditions on
their odd and even parts. Further it is shown that the zeros of members of this
class are localized in a very special way. This result allows us to treat a
particular case of a conjecture by A. Sokal.

In this paper we fully describe functions generating the infinite totally
nonnegative Hurwitz matrices. In particular, we generalize the well-known
result by Asner and Kemperman on the total nonnegativity of the Hurwitz
matrices of real stable polynomials. An alternative criterion for entire
functions to generate a P\'olya frequency sequence is als...