Sequence Of Polynomials Research Articles

Integrable differential systems for deformed Laguerre–Hahn orthogonal polynomials

Our work studies sequences of orthogonal polynomials of the Laguerre–Hahn class, whose Stieltjes functions satisfy a Riccati type differential equation with polynomial coefficients, which are subject to a deformation parameter t. We derive systems of differential equations and give Lax pairs, yielding nonlinear differential equations in t for the recurrence relation coefficients and Lax matrices of the orthogonal polynomials. A specialisation to a non semi-classical case obtained via a Möbius transformation of a Stieltjes function related to a deformed Jacobi weight is studied in detail, showing this system is governed by a differential equation of the Painlevé type P. The particular case of P arising here has the same four parameters as the solution found by Magnus (1995 J. Comput. Appl. Math. 57 215–37) but differs in the boundary conditions.

Approximation of classes of Poisson integrals by Fejer means

The work is devoted to the investigation of problem of approximation of continuous periodic functions by trigonometric polynomials, which are generated by linear methods of summation of Fourier series. The simplest example of a linear approximation of periodic functions is the approximation of functions by partial sums of their Fourier series. However, the sequences of partial Fourier sums are not uniformly convergent over the class of continuous periodic functions. Therefore, a many studies is devoted to the research of the approximative properties of approximation methods, which are generated by transformations of the partial sums of Fourier series and allow us to construct sequences of trigonometrical polynomials that would be uniformly convergent for the whole class of continuous functions. Particularly, Fejer means have been widely studied in the last time. One of the important problems in this field is the study of asymptotic behavior of the upper bounds over a fixed classes of functions of deviations of the trigonometric polynomials. The aim of the work systematizes known results related to the approximation of classes of Poisson integrals of continuous functions by arithmetic means of Fourier sums, and presents new facts obtained for particular cases. The asymptotic behavior of the upper bounds on classes of Poisson integrals of periodic functions of the real variable of deviations of linear means of Fourier series, which are defined by applying the Fejer summation method is studied. The mentioned classes consist of analytic functions of a real variable, which are narrowing of bounded harmonic in unit disc functions of complex variable. In the work, asymptotic equalities for the upper bounds of deviations of Fejer means on classes of Poisson integrals were obtained.

A unified approach to generalized Pascal-like matrices: q-analysis

In this paper, we present a general method to construct q-analogues and other generalizations of Pascal-like matrices. Our matrices are obtained as functions of strictly lower triangular matrices and include several types of generalized Pascal-like matrices and matrices related with modified Hermite polynomials of two variables and other polynomial sequences. We find explicit expressions for products, powers, and inverses of the matrices and also some factorization formulas using this method.

Generalized Tribonacci Polynomials

In this paper, we investigate the generalized Tribonacci polynomials and we deal with, in detail, two special cases which we call them (r,s,t)-Tribonacci and (r,s,t)-Tribonacci-Lucas polynomials. We also introduce and investigate a new sequence and its two special cases namely the generalized co-Tribonacci, (r,s,t)-co-Tribonacci and (r,s,t)-co-Tribonacci-Lucas polynomials, respectively. We present Binet's formulas, generating functions, Simson formulas, and the summation formulas for these polynomial sequences. Moreover, we give some identities and matrices related to these polynomials. Furthermore, we evaluate the infinite sums of special cases of (r,s,t)-Tribonacci and (r,s,t)-Tribonacci-Lucas polynomials.

Linearization and connection coefficients of polynomial sequences: A matrix approach

For any sequence of polynomials {pk(t)} in one real or complex variable, where pk has degree k for k≥0, we find explicit expressions and recurrence relations for infinite matrices whose entries are the numbers d(n,m,k), called linearization coefficients, that satisfypn(t)pm(t)=∑k=0n+md(n,m,k)pk(t),n,m≥0. For any pair of polynomial sequences {uk(t)} and {pk(t)} we find infinite matrices whose entries are the numbers e(n,m,k) that satisfypn(t)pm(t)=∑k=0n+me(n,m,k)uk(t),n,m≥0. We also obtain recurrence relations and other properties of the linearization coefficients. Such results are obtained using only simple algebraic properties of infinite matrices. We apply the general results to general orthogonal polynomial sequences and to some simple families of orthogonal polynomials, such as the Chebyshev, Hermite, and Charlier families.

Bracket words: A generalisation of Sturmian words arising from generalised polynomials

Generalised polynomials are maps constructed by applying the floor function, addition, and multiplication to polynomials. Despite superficial similarity, generalised polynomials exhibit many phenomena which are impossible for polynomials. In particular, there exist generalised polynomial sequences which take only finitely many values without being periodic; examples of such sequences include the Sturmian words, as well as more complicated sequences like ⌊ 2 { π n 2 + 2 n ⌊ 3 n ⌋ } ⌋ \left \lfloor 2\left \{ \pi n^2 + \sqrt {2}n\left \lfloor \sqrt {3}n \right \rfloor \right \} \right \rfloor . The purpose of this paper is to investigate letter-to-letter codings of finitely-valued generalised polynomial sequences, which we call bracket words, from the point of view of combinatorics on words. We survey existing results on generalised polynomials and their corollaries in terms of bracket words, and also prove several new results. Our main contribution is a polynomial bound on the subword complexity of bracket words.

A unified approach to multivariate polynomial sequences with real stability

We give new sufficient conditions for a sequence of multivariate polynomials to be real stable. As applications, we obtain several known results, such as the real stability of multivariate Eulerian polynomials, multivariate Bell polynomials and multivariate polynomials over Stirling permutations, in a unified manner. And we also show some new results, such as the real stability of multivariate Lah polynomials and multivariate polynomials over Jacobi-Stirling permutations, and the proper position property of multivariate orthogonal polynomials and multivariate matching polynomials. Finally, we obtain a multivariate generalization of Fisk's result by presenting a 2×2 matrix, which preserves the proper position property.

Tame key polynomials

We introduce a new method of constructing complete sequences of key polynomials for simple extensions of tame fields. In our approach the key polynomials are taken to be the minimal polynomials over the base field of suitably constructed elements in its algebraic closure, with the extensions generated by them forming an increasing chain. In the case of algebraic extensions, we generalize the results to countably generated infinite tame extensions over henselian but not necessarily tame fields. In the case of transcendental extensions, we demonstrate the central role that is played by the implicit constant fields, which reveals the tight connection with the algebraic case.

Multiple orthogonal polynomials associated with branched continued fractions for ratios of hypergeometric series

The main objects of the investigation presented in this paper are branched-continued-fraction representations of ratios of contiguous hypergeometric series and type II multiple orthogonal polynomials on the step-line with respect to linear functionals or measures whose moments are ratios of products of Pochhammer symbols. This is an interesting case study of the recently found connection between multiple orthogonal polynomials and branched continued fractions that gives a clear example of how this connection leads to considerable advances on both topics.We start by obtaining new results about generating polynomials of lattice paths and total positivity of matrices and giving new contributions to the general theory of the connection between multiple orthogonal polynomials and branched continued fractions with emphasis on its application to the analysis of multiple orthogonal polynomials. Then, we construct new branched continued fractions for ratios of contiguous hypergeometric series. We give conditions for positivity of the coefficients of these branched continued fractions and we show that the ratios of products of Pochhammer symbols are generating polynomials of lattice paths for a special case of the branched continued fractions under study. Next, we introduce a family of type II multiple orthogonal polynomials on the step-line associated with those branched continued fractions. We present a formula as terminating hypergeometric series for these polynomials, we study their differential properties, and we find an explicit recurrence relation satisfied by them. Finally, we focus the analysis of the multiple orthogonal polynomials to the cases where the corresponding branched-continued-fraction coefficients are all positive. In those cases, the orthogonality conditions can be written using measures on the positive real line involving Meijer G-functions and we obtain results about the location of the zeros and the asymptotic behaviour of the polynomials.Specialisations of the multiple orthogonal polynomials studied here include the classical Laguerre, Jacobi, and Bessel orthogonal polynomials, multiple orthogonal polynomials with respect to Nikishin systems of two measures involving modified Bessel functions, confluent hypergeometric functions, and Gauss' hypergeometric function, and multiple orthogonal polynomials with respect to Meijer G-functions used to investigate the singular values of products of Ginibre random matrices as well as a r-orthogonal polynomial sequence with constant recurrence coefficients for any positive integer r and particular instances of the Jacobi-Piñeiro polynomials.

Pastro polynomials and Sobolev-type orthogonal polynomials on the unit circle based on a q-difference operator

ABSTRACT Applications of orthogonal polynomials on the unit circle have attracted the attention of many researchers in recent years. Pastro polynomials, which are basic hypergeometric polynomials, are known to be biorthogonal polynomials on the unit circle. However, with special choice of parameters they provide one of the nicest examples of orthogonal polynomials on the unit circle. Our objective here is to consider some properties of three sequences of polynomials which are related to these Pastro orthogonal polynomials on the unit circle by a q-difference operator. We also provide information regarding connection formulas, bounds for the connection coefficients as well as outer relative asymptotics associated with these sequences of polynomials.

Characterization of orthogonal polynomials on lattices

We consider two sequences of orthogonal polynomials and such that with , and are sequences of complex numbers, , is the identity operator, x defines a lattice, and . We show that under some natural conditions, both involved orthogonal polynomials sequences and are semiclassical whenever k = m. Some particular cases are studied closely where we characterize the continuous dual Hahn and Wilson polynomials for quadratic lattices.

Central factorial numbers associated with sequences of polynomials

Many important special numbers appear in the expansions of some polynomials in terms of central factorials and vice versa, for example, central factorial numbers, degenerate central factorial numbers, and central Lah numbers which are recently introduced. Here we generalize this to any sequence of polynomials such that deg . The aim of this paper is to study the central factorial numbers of the second kind associated with any sequence of polynomials and of the first kind associated with any sequence of polynomials, in a unified and systematic way with the help of umbral calculus technique. The central factorial numbers associated with any sequence of polynomials enjoy orthogonality and inverse relations. We illustrate our results with many examples and obtain interesting orthogonality and inverse relations by applying such relations for the central factorial numbers associated with any sequence of polynomials to each of our examples.

On the Singular Lowering Operator $$\textbf{D}_u$$ and Application to the Singular Laguerre–Hahn Polynomial Sequence with Class Zero of Hermite Type

On the Singular Lowering Operator $$\textbf{D}_u$$ and Application to the Singular Laguerre–Hahn Polynomial Sequence with Class Zero of Hermite Type

Quadratic Decomposition of Bivariate Orthogonal Polynomials

We describe the relation between the systems of bivariate orthogonal polynomial associated to a symmetric weight function and associated to some particular Christoffel modifications of the quadratic decomposition of the original weight. We analyze the construction of a symmetric bivariate orthogonal polynomial sequence from a given one, orthogonal to a weight function defined on the first quadrant of the plane. In this description, a sort of Bäcklund type matrix transformations for the involved three term matrix coefficients plays an important role. Finally, we take as a case study relations between the classical orthogonal polynomials defined on the ball and those on the simplex.

Complex Jacobi matrices generated by Darboux transformations

In this paper, we study complex Jacobi matrices obtained by the Christoffel and Geronimus transformations at a nonreal complex number, including the properties of the corresponding sequences of orthogonal polynomials. We also present some invariant and semi-invariant properties of Jacobi matrices under such transformations. For instance, we show that a Nevai class is invariant under the transformations in question, which is not true in general, and that the ratio asymptotic still holds outside the spectrum of the corresponding symmetric complex Jacobi matrix but the spectrum could include one extra point. In principal, these transformations can be iterated and, for example, we demonstrate how Geronimus transformations can lead to RII-recurrence relations, which in turn are related to orthogonal rational functions and pencils of Jacobi matrices.

Coherent pairs of moment functionals of the second kind and associated orthogonal polynomials and Sobolev orthogonal polynomials

Given a pair of quasi-definite moment functionals {v0,v1} we introduce the concept of coherence of the second kind in terms of an algebraic relation that the corresponding sequences of orthogonal polynomials satisfy. We characterize such moment functionals and give some illustrative examples taking into account they are semiclassical of class at most one. The relation between the corresponding monic Jacobi matrices is stated. For a pair of moment functionals satisfying the coherence property of the second kind, a Sobolev inner product is introduced. The connection formulas between the sequence of monic orthogonal polynomials associated with such a Sobolev inner product and the sequence of monic polynomials orthogonal with respect to the moment functional v0 are given.

Lidstone interpolation III. Several variables

A polynomial in a single variable is uniquely determined by its derivatives of even order at 0 and 1. More precisely, such an univariate polynomial can be written and a finite sum of f ( 2 n ) ( 0 ) Λ n ( 1 − z ) f^{(2n)}(0) \Lambda _n(1-z) and f ( 2 n ) ( 1 ) Λ n ( z ) f^{(2n)}(1) \Lambda _n(z) , ( n ≥ 0 n\ge 0 ), where the Λ n ( z ) \Lambda _n(z) are the Lidstone polynomials defined by the conditions ( d d z ) 2 k Λ n ( 0 ) = 0 and ( d d z ) 2 k Λ n ( 1 ) = δ k , n , k ≥ 0 , n ≥ 0. \begin{equation*} \left (\frac {\mathrm {d}}{\mathrm {d}z}\right )^{2k} \Lambda _n(0)=0\text { and } \left (\frac {\mathrm {d}}{\mathrm {d}z}\right )^{2k} \Lambda _n(1)=\delta _{k,n},\quad k\ge 0, \; n\ge 0. \end{equation*} We generalize this theory to n n variables, replacing the two points 0 0 , 1 1 in C \mathbb {C} with n + 1 n+1 points e _ 0 , e _ 1 , … , e _ n {\underline {e}}_0,{\underline {e}}_1,\dots ,{\underline {e}}_n in C n \mathbb {C}^n , where e _ 0 {\underline {e}}_0 is the origin of C n \mathbb {C}^n and e _ 1 , … , e _ n {\underline {e}}_1,\dots ,{\underline {e}}_n the canonical basis of C n \mathbb {C}^n . By selecting a suitable subset of even order derivatives at these n + 1 n+1 points, we show that any polynomial in n n variables has a unique expansion. We obtain generating series for these sequences of polynomials and we deduce an expansion for entire functions in C n \mathbb {C}^n of exponential type > π >\pi . We extend to several variables results due to Lidstone [Proceedings Edinburgh Math. Soc., II. Ser. (2)2, 16-19 (1930)], Poritsky [Trans. Amer. Math. Soc. 34 (1932), pp. 274–331], Whittaker [Proc. London Math. Soc. (2) 36 (1934), pp. 451–469], Schoenberg [Bull. Amer. Math. Soc. 42 (1936), pp. 284–288], Buck [Proc. Amer. Math. Soc. 6 (1955), pp. 793–796]. We also show that our results are, to a certain extent, best possible.

Sarnak's Conjecture for nilsequences on arbitrary number fields and applications

We formulate the generalized Sarnak's Möbius disjointness conjecture for an arbitrary number field K, and prove a quantitative disjointness result between polynomial nilsequences (Φ(g(n)Γ))n∈ZD and aperiodic multiplicative functions on OK, the ring of integers of K. Here D=[K:Q], X=G/Γ is a nilmanifold, g:ZD→G is a polynomial sequence, and Φ:X→C is a Lipschitz function. This result, being a generalization of a previous theorem of the author in [44], requires a significantly different approach, which involves with multi-dimensional higher order Fourier analysis, multi-linear analysis, orbit properties on nilmanifold, and an orthogonality criterion of Kátai in OK.We also use variations of this result to derive applications in number theory and combinatorics: (1) we prove a structure theorem for multiplicative functions on K, saying that every bounded multiplicative function can be decomposed into the sum of an almost periodic function (the structural part) and a function with small Gowers uniformity norm of any degree (the uniform part); (2) we give a necessary and sufficient condition for the Gowers norms of a bounded multiplicative function in OK to be zero; (3) we provide partition regularity results over K for a large class of homogeneous equations in three variables. For example, for a,b∈Z﹨{0}, we show that for every partition of OK into finitely many cells, where K=Q(a,b,a+b), there exist distinct and non-zero x,y belonging to the same cell and z∈OK such that ax2+by2=z2.

Stirling Numbers Associated with Sequences of Polynomials

Stirling Numbers Associated with Sequences of Polynomials

Sturm's Theorem for Min matrices

<abstract><p>In the present paper, we study Min matrix $ \mathcal{A}_{min} = \left[a_{min\left(i, j\right)}\right]_{i, j = 1}^n $, where $ a_s $'s are the elements of a real sequence $ \left\lbrace a_s\right\rbrace $. We first obtain a recurrence relation for the characteristic polynomial for matrix $ \mathcal{A}_{min} $, and some relations between the coefficients of its characteristic polynomial. Next, we show that the sequence of the characteristic polynomials of the $ i \times i \left(i \leq n\right) $ Min matrices satisfies the Sturm sequence properties according to different required conditions of the sequence $ \left\lbrace a_s\right\rbrace $. Using Sturm's Theorem, we get some results about the eigenvalues, such as the number of eigenvalues in an interval. Thus, we obtain the number of positive and negative eigenvalues of Min matrix $ \mathcal{A}_{min} $. Finally, we give an example to illustrate our results.</p></abstract>