Extraction of orthogonal polynomials from generating function for reciprocal of odd numbers

In the present paper the orthogonality relations, exhibited by both numerator and denominator polynomials of both even and odd order convergents of a regular C-fraction of a power series with coefficients as reciprocal of odd numbers are described. The two sequences of convergents are nothing but diagonal and upper diagonal Pade approximants for the power series. The two orthogonal polynomials extracted from denominators are shown to be classical orthogonal polynomials and two orthogonal polynomials extracted from numerators are shown to be non-classical orthogonal polynomials..     

The convergence of padé approximants and the size of the power series coefficients

Let f be a power series ∑a_izⁱ with complex coefficients. The (n. n) Pade approximant to f is a rational function P/Q where P and Q are polynomials, Q(z) ? 0, of degree ≦ n such that f(z)Q(z)-P(z) = Az²ⁿ⁺¹ + higher degree terms. It is proved that if the coefficients a_i satisfy a certain growth condition, then a corresponding subsequence of the sequence of (n, n) Pade approximants converges to f in the region where the power series f converges, except on an exceptional set E having a certain Hausdorff measure 0. It is also proved that the result is best possible in the sense that we may have divergence on E. In particular,there exists an entire function f such that the sequence of (ny n) Pade approximants diverges everywhere (except at 0)     

Eisenstein series associated with Γ<Subscript>0</Subscript>(2)

In this paper, we define the normalized Eisenstein series ℘, e, and associated with Γ₀(2), and derive three differential equations satisfied by them from some trigonometric identities. By using these three formulas, we define a differential equation depending on the weights of modular forms on Γ₀(2) and then construct its modular solutions by using orthogonal polynomials and Gaussian hypergeometric series. We also construct a certain class of infinite series connected with the triangular numbers. Finally, we derive a combinatorial identity from a formula involving the triangular numbers.      

The Hermite–Pade Approximations of Generalized Hypergeometric Series in Two Variables

We give examples of well-posed problems of joint Hermite–Pade approximations of series in two variables. We find Rodrigues formulas and integral representations for solutions. We also study the limit distribution of zeros of the corresponding polynomials. Constructions are based, on the one hand, on the classical Appel polynomials orthogonal in a triangle and, on the other hand, on various ways of proving Apery's theorem about irrationality of the number (3).     

Hall Numbers and the Composition Algebra of the Kronecker Algebra

We present formulas for the structure constants (Hall numbers) of the Hall algebra associated to the Kronecker algebra. The formulas which in some cases involve the classical Hall polynomials enable us to determine every Hall number. Using again these formulas we construct new PBW-bases with simple structure constants for the composition algebra , making possible the definition of the generic composition algebra via Hall polynomials.Presented by C. Ringel.     

Nonsymmetric interpolation macdonald polynomials and \mathfrak{g}\mathfrak{l}_n basic hypergeometric series

The Knop-Sahi interpolation Macdonald polynomials are inhomogeneous and nonsymmetric generalisations of the well-known Macdonald polynomials. In this paper we apply the interpolation Macdonald polynomials to study a new type of basic hypergeometric series of type . Our main results include a new q-binomial theorem, a new q-Gauss sum, and several transformation formulae for series. *Supported by the ANR project MARS (BLAN06-2 134516). **Supported by the NSF grant DMS-0401387. ***Supported by the Australian Research Council.     

Hermite polynomials on the plane

The space of bivariate generalised Hermite polynomials of degree n is invariant under rotations. We exploit this symmetry to construct an orthonormal basis for which consists of the rotations of a single polynomial through the angles , ℓ=0,...n. Thus we obtain an orthogonal expansion which retains as much of the symmetry of as is possible. Indeed we show that a continuous version of this orthogonal expansion exists.      

Two classes of special functions using Fourier transforms of some finite classes of classical orthogonal polynomials

Some orthogonal polynomial systems are mapped onto each other by the Fourier transform. The best-known example of this type is the Hermite functions, i.e., the Hermite polynomials multiplied by , which are eigenfunctions of the Fourier transform. In this paper, we introduce two new examples of finite systems of this type and obtain their orthogonality relations. We also estimate a complicated integral and propose a conjecture for a further example of finite orthogonal sequences.

     

Block diagonal polynomials

In this paper we introduce and study a certain class of polynomials in spaces with unconditional finite dimensional decomposition. Some applications to the existence of copies of in spaces of polynomials and to the stabilization of polynomial algebras are given.

     

Multiple orthogonal polynomials for classical weights

A new set of special functions, which has a wide range of applications from number theory to integrability of nonlinear dynamical systems, is described. We study multiple orthogonal polynomials with respect to 1$"> weights satisfying Pearson's equation. In particular, we give a classification of multiple orthogonal polynomials with respect to classical weights, which is based on properties of the corresponding Rodrigues operators. We show that the multiple orthogonal polynomials in our classification satisfy a linear differential equation of order . We also obtain explicit formulas and recurrence relations for these polynomials.

     

Biorthogonal exponential sequences with weight function on the real line and an orthogonal sequence of trigonometric functions

Some orthogonal functions can be mapped onto other orthogonal functions by the Fourier transform. In this paper, by using the Fourier transform of Stieltjes-Wigert polynomials, we derive a sequence of exponential functions that are biorthogonal with respect to a complex weight function like on . Then we restrict these introduced biorthogonal functions to a special case to obtain a sequence of trigonometric functions orthogonal with respect to the real weight function on .

     

Relative asymptotics for polynomials orthogonal with respect to a discrete Sobolev inner product

We investigate the asymptotic properties of orthogonal polynomials for a class of inner products including the discrete Sobolev inner products $\left\langle {h,{\text{ }}g} \right\rangle = \int h g d\mu + \sum {_{j = 1}^m } \sum {_{i = 0}^{N_j } M_{j,i} h^{(i)} (c_j )} g^{(i)} (c_j )$ , where μ is a certain type of complex measure on the real line, andc _j are complex numbers in the complement of supp(μ). The Sobolev orthogonal polynomials are compared with the orthogonal polynomials corresponding to the measure μ.     

A computation of minimal polynomials of special values of Siegel modular functions

Recently, Fukuda and Komatsu constructed units of a certain abelian extension of using special values of Siegel modular functions. In this paper, we determine the minimal polynomials of these units.

     

Strong Stieltjes distributions and orthogonal Laurent polynomials with applications to quadratures and Padé approximation

Starting from a strong Stieltjes distribution , general sequences of orthogonal Laurent polynomials are introduced and some of their most relevant algebraic properties are studied. From this perspective, the connection between certain quadrature formulas associated with the distribution and two-point Padé approximants to the Stieltjes transform of is revisited. Finally, illustrative numerical examples are discussed.

     

Orthogonal Laurent polynomials corresponding to certain strong Stieltjes distributions with applications to numerical quadratures

In this paper we shall be mainly concerned with sequences of orthogonal Laurent polynomials associated with a class of strong Stieltjes distributions introduced by A.S. Ranga. Algebraic properties of certain quadratures formulae exactly integrating Laurent polynomials along with an application to estimate weighted integrals on with nearby singularities are given. Finally, numerical examples involving interpolatory rules whose nodes are zeros of orthogonal Laurent polynomials are also presented.

     

Wilson Function Transforms Related to Racah Coefficients

The irreducible -representations of the Lie algebra consist of discrete series representations, principal unitary series and complementary series. We calculate Racah coefficients for tensor product representations that consist of at least two discrete series representations. We use the explicit expressions for the Clebsch–Gordan coefficients as hypergeometric functions to find explicit expressions for the Racah coefficients. The Racah coefficients are Wilson polynomials and Wilson functions. This leads to natural interpretations of the Wilson function transforms. As an application several sum and integral identities are obtained involving Wilson polynomials and Wilson functions. We also compute Racah coefficients for , which turn out to be Askey–Wilson functions and Askey–Wilson polynomials.This research was done during my stay at the Department of Mathematics at Chalmers University of Technology and Göteborg University in Sweden, supported by a NWO-TALENT stipendium of the Netherlands Organization for Scientific Research (NWO).     

Coefficients of multiplication formulas for classical orthogonal polynomials

In this paper using both analytic and algorithmic approaches, we derive the coefficients \(D_m(n,a)\) of the multiplication formula

$$\begin{aligned} p_n(ax)=\sum _{m=0}^nD_m(n,a)p_m(x) \end{aligned}$$

or the translation formula

$$\begin{aligned} p_n(x+a)=\sum _{m=0}^nD_m(n,a)p_m(x), \end{aligned}$$

where \(\{p_n\}_{n\ge 0}\) is an orthogonal polynomial set, including the classical continuous orthogonal polynomials, the classical discrete orthogonal polynomials, the \(q\)-classical orthogonal polynomials, as well as the classical orthogonal polynomials on a quadratic lattice and a \(q\)-quadratic lattice. We give a representation of the coefficients \(D_m(n,a)\) as a single, double or triple sum whereas in many cases we get simple representations.

     

-hypergeometric functions associated to character algebras

In this paper, we obtain certain discrete orthogonal polynomials expressed in terms of the -hypergeometric functions, from the eigenmatrices of character algebras.

     

Preserving geometric properties of the exponential matrix by block Krylov subspace methods

Given a large square real matrix A and a rectangular tall matrix Q, many application problems require the approximation of the operation . Under certain hypotheses on A, the matrix preserves the orthogonality characteristics of Q; this property is particularly attractive when the associated application problem requires some geometric constraints to be satisfied. For small size problems numerical methods have been devised to approximate while maintaining the structure properties. On the other hand, no algorithm for large A has been derived with similar preservation properties. In this paper we show that an appropriate use of the block Lanczos method allows one to obtain a structure preserving approximation to when A is skew-symmetric or skew-symmetric and Hamiltonian. Moreover, for A Hamiltonian we derive a new variant of the block Lanczos method that again preserves the geometric properties of the exact scheme. Numerical results are reported to support our theoretical findings, with particular attention to the numerical solution of linear dynamical systems by means of structure preserving integrators. AMS subject classification (2000) 65F10, 65F30, 65D30     

A noncommutative symmetric system over the Grossman-Larson Hopf algebra of labeled rooted trees

In this paper, we construct explicitly a noncommutative symmetric ( CS) system over the Grossman-Larson Hopf algebra of labeled rooted trees. By the universal property of the CS system formed by the generating functions of certain noncommutative symmetric functions, we obtain a specialization of noncommutative symmetric functions by labeled rooted trees. Taking the graded duals, we also get a graded Hopf algebra homomorphism from the Connes-Kreimer Hopf algebra of labeled rooted forests to the Hopf algebra of quasi-symmetric functions. A connection of the coefficients of the third generating function of the constructed CS system with the order polynomials of rooted trees is also given and proved.