A new result for hypergeometric polynomials

In some recent investigations involving differential operators for generalized Laguerre polynomials, Herman Bavinck (1996) encountered and proved a certain summation formula for the classical Laguerre polynomials. The main object of this sequel to Bavinck's work is to prove a generalization of this summation formula for a class of hypergeometric polynomials. The demonstration, which is presented here in the general case, differs markedly from the earlier proof given for the known special case. The general summation formula is also applied to derive the corresponding result for the classical Jacobi polynomials.

     

A new summation identity for the Srivastava-Singhal polynomials

In his recent investigations involving differential operators for some generalizations of the classical Laguerre polynomials, H. Bavinck [J. Phys. A Math. Gen. 29 (1996) L277-L279] encountered and proved a certain summation identity for the classical Laguerre polynomials. The main object of this sequel to Bavinck's work is to prove a generalization of this summation identity for the Srivastava-Singhal polynomials. The demonstration, which is presented here in the general case, differs markedly from the earlier proof given for the known special case. It is also indicated how the general summation identity can be applied to derive the corresponding result for one class of the Konhauser biorthogonal polynomials.     

Intersection Alexander polynomials

By considering a (not necessarily locally-flat) PL knot as the singular locus of a PL stratified pseudomanifold, we can use intersection homology theory to define intersection Alexander polynomials, a generalization of the classical Alexander polynomial invariants for smooth or PL locally-flat knots. We show that the intersection Alexander polynomials satisfy certain duality and normalization conditions analogous to those of ordinary Alexander polynomials, and we explore the relationships between the intersection Alexander polynomials and certain generalizations of the classical Alexander polynomials that are defined for non-locally-flat knots. We also investigate the relations between the intersection Alexander polynomials of a knot and the intersection and classical Alexander polynomials of the link knots around the singular strata. To facilitate some of these investigations, we introduce spectral sequences for the computation of the intersection homology of certain stratified bundles.     

A generalization of the remainder theorem and factor theorem

We propose a generalization of the classical Remainder Theorem for polynomials over commutative coefficient rings that allows calculating the remainder without using the long division method. As a consequence we obtain an extension of the classical Factor Theorem that provides a general divisibility criterion for polynomials. The arguments can be used in basic algebra courses and are suitable for building classroom/homework activities for college and high school students.     

Two-index Clifford-Hermite polynomials with applications in wavelet analysis

Clifford analysis may be regarded as a higher-dimensional analogue of the theory of holomorphic functions in the complex plane. It has proven to be an appropriate framework for higher-dimensional continuous wavelet transforms, based on specific types of multi-dimensional orthogonal polynomials, such as the Clifford-Hermite polynomials, which form the building blocks for so-called Clifford-Hermite wavelets, offering a refinement of the traditional Marr wavelets. In this paper, a generalization of the Clifford-Hermite polynomials to a two-parameter family is obtained by taking the double monogenic extension of a modulated Gaussian, i.e. the classical Morlet wavelet. The eventual goal being the construction of new Clifford wavelets refining the Morlet wavelet, we first investigate the properties of the underlying polynomials.     

An inequality for the multivariate normal distribution

Herman Chernoff used Hermite polynomials to prove an inequality for the normal distribution. This inequality is useful in solving a variation of the classical isoperimetric problem which, in turn, is relevant to data compression in the theory of element identification. As the inequality is of interest in itself, we prove a multivariate generalization of it using a different argument.     

Higher-order matching polynomials and d-orthogonality

We show combinatorially that the higher-order matching polynomials of several families of graphs are d-orthogonal polynomials. The matching polynomial of a graph is a generating function for coverings of a graph by disjoint edges; the higher-order matching polynomial corresponds to coverings by paths. Several families of classical orthogonal polynomials—the Chebyshev, Hermite, and Laguerre polynomials—can be interpreted as matching polynomials of paths, cycles, complete graphs, and complete bipartite graphs. The notion of d-orthogonality is a generalization of the usual idea of orthogonality for polynomials and we use sign-reversing involutions to show that the higher-order Chebyshev (first and second kinds), Hermite, and Laguerre polynomials are d-orthogonal. We also investigate the moments and find generating functions of those polynomials.     

The Stability of Graded Multiplicity in the Setting of the Kostant-Rallis Theorem

From a combinatorial point of view, we approach the problem of finding a graded generalization of the Kostant-Rallis theorem concerning the K-harmonic polynomials on p. Specifically, for each classical symmetric pair we obtain a stable range where the multiplicity of an irreducible K-representation in the degree d harmonic polynomials can be expressed in terms of Littlewood-Richardson coefficients.     

CONVERGENCE OF DERIVATIVES OF GENERALIZED BERNSTEIN OPERATORS

In the present paper, we obtain estimations of convergence rate derivatives of the q-Bernstein polynomials Bn (f, qn ; x) approximating to f ′ (x) asn →∞, which is a generalization of that relating the classical case qn = 1. On the other hand, we study the convergence properties of derivatives of the limit q-Bernstein operators B ∞ (f, q; x) as q → 1- .     

A fractional version of the peano-sard theorem

Sard's classical generalization of the Peano kernel theorem provides an extremely useful method for expressing and calculating sharp bounds for approximation errors. The error is expressed in terms of a derivative of the underlying function. However, we can apply the theorem only if the approximation is exact on a certain set of polynomials.

In this paper, we extend the Peano-Sard theorem to the case that the approximation is exact for a class of generalized polynomials (with non-integer exponents). As a result, we obtain an expression for the remainder in terms of a fractional derivative of the function under consideration. This expression permits us to give sharp error bounds as in the classical situation. An application of our results to the classical functional (vanishing on polynomials) gives error bounds of a new type involving weighted Sobolev-type spaces. In this way, we may state estimates for functions with weaker smoothness properties than usual.

The standard version of the Peano-Sard theory is contained in our results as a special case.     

Outcomes of the Abel Identity

We discuss some outcomes of an umbral generalization of the Abel identity. First we prove that a concise proof of the Lagrange inversion formula can be deduced from it. Second, we show that the whole class of Sheffer sequences, if manipulated to an umbral level, coincides with the subclass of Abel polynomials. Finally, we apply these techniques to obtain explicit formulae for some classical polynomial sequences, even in non Sheffer cases (Chebyshev and Gegenbauer polynomials).     

Non-autonomous Julia sets with escaping critical points

We consider non-autonomous iteration which is a generalization of standard polynomial iteration where we deal with Julia sets arising from composition sequences for arbitrarily chosen polynomials with uniformly bounded degrees and coefficients. In this paper, we look at examples where all the critical points escape to infinity. In the classical case, any example of this type must be hyperbolic and there can be only one Fatou component, namely the basin at infinity. This result remains true in the non-autonomous case if we also require that the dynamics on the Julia set be hyperbolic or semi-hyperbolic. However, in general it fails and we exhibit three counterexamples of sequences of quadratic polynomials all of whose critical points escape but which have bounded Fatou components.     

Second structure relation for semiclassical orthogonal polynomials

Classical orthogonal polynomials are characterized from their orthogonality and by a first or second structure relation. For the semiclassical orthogonal polynomials (a generalization of the classical ones), we find only the first structure relation in the literature. In this paper, we establish a second structure relation. In particular, we deduce it by means of a general finite-type relation between a semiclassical polynomial sequence and the sequence of its monic derivatives.     

Fiedler companion linearizations for rectangular matrix polynomials

The development of new classes of linearizations of square matrix polynomials that generalize the classical first and second Frobenius companion forms has attracted much attention in the last decade. Research in this area has two main goals: finding linearizations that retain whatever structure the original polynomial might possess, and improving properties that are essential for accurate numerical computation, such as eigenvalue condition numbers and backward errors. However, all recent progress on linearizations has been restricted to square matrix polynomials. Since rectangular polynomials arise in many applications, it is natural to investigate if the new classes of linearizations can be extended to rectangular polynomials. In this paper, the family of Fiedler linearizations is extended from square to rectangular matrix polynomials, and it is shown that minimal indices and bases of polynomials can be recovered from those of any linearization in this class via the same simple procedures developed previously for square polynomials. Fiedler linearizations are one of the most important classes of linearizations introduced in recent years, but their generalization to rectangular polynomials is nontrivial, and requires a completely different approach to the one used in the square case. To the best of our knowledge, this is the first class of new linearizations that has been generalized to rectangular polynomials.     

Polynomials and identities on real Banach spaces

We study the duality theory for real polynomials and functions on Banach spaces. Our approach leads to a unified treatment and generalization of some classical results on linear identities and polynomial characterizations due to Fréchet, Mazur, Orlicz, Reznick, Wilson, and others.     

Varying Jacobi–Krall orthogonal polynomials: local asymptotic behaviour and zeros

Krall orthogonal polynomials are well known and they constitute a generalization of classical orthogonal polynomials obtained by addition of positive masses located at some points on the real line. In this contribution we consider two families of Krall polynomials already known in the literature, but now the corresponding absolutely continuous measure is perturbed by a sequence of nonnegative masses located at the point 1 in the Jacobi case and at the end points of the interval of orthogonality in the Gegenbauer case. We analyze the asymptotic behaviour of these varying Krall orthogonal polynomials in the neighbourhood of the points where the perturbation has been done. To do this we use Mehler–Heine type asymptotic formulae. As a consequence we can establish limit relations between the zeros of these polynomials and the ones of the Bessel functions of the first kind (or linear combinations of them). We do some numerical experiments to illustrate the results.     

Formal orthogonal polynomials and Hankel/Toeplitz duality

For classical polynomials orthogonal with respect to a positive measure supported on the real line, the moment matrix is Hankel and positive definite. The polynomials satisfy a three term recurrence relation. When the measure is supported on the complex unit circle, the moment matrix is positive definite and Toeplitz. Then they satisfy a coupled Szeg recurrence relation but also a three term recurrence relation. In this paper we study the generalization for formal polynomials orthogonal with respect to an arbitrary moment matrix and consider arbitrary Hankel and Toeplitz matrices as special cases. The relation with Padé approximation and with Krylov subspace iterative methods is also outlined.This research was supported by the National Fund for Scientific Research (NFWO), project Lanczos, grant #2.0042.93.     

Parametric Poincaré-Perron theorem with applications

We prove a parametric generalization of the classical Poincaré-Perron theorem on stabilizing recurrence relations, where we assume that the varying coefficients of a recurrence depend on auxiliary parameters and converge uniformly in these parameters to their limiting values. As an application, we study convergence of the ratios of families of functions satisfying finite recurrence relations with varying functional coefficients. For example, we explicitly describe the asymptotic ratio for two classes of biorthogonal polynomials introduced by Ismail and Masson.     

On the Global Stability of Solutions of Moment Systems in Nonequilibrium Thermodynamics

In this paper, we study the linearization of the Cauchy problem and the mixed problem for the system of Grad--Hermite moments in nonequilibrium thermodynamics in the neighborhood of the equilibrium state. Stability conditions for solutions of the Cauchy problem are proved as a generalization of the classical Hermite--Biller theorem on stable polynomials. For the mixed problem, we prove an analog of the Vishik--Lyusternik theorem on small singular perturbations of general elliptic problems. The last observation allows us to introduce the Shapiro--Lopatinskii condition, which implies the well-posedness of the mixed problem.     

The stable computation of formal orthogonal polynomials

For many applications — such as the look-ahead variants of the Lanczos algorithm — a sequence of formal (block-)orthogonal polynomials is required. Usually, one generates such a sequence by taking suitable polynomial combinations of a pair of basis polynomials. These basis polynomials are determined by a look-ahead generalization of the classical three term recurrence, where the polynomial coefficients are obtained by solving a small system of linear equations. In finite precision arithmetic, the numerical orthogonality of the polynomials depends on a good choice of the size of the small systems; this size is usually controlled by a heuristic argument such as the condition number of the small matrix of coefficients. However, quite often it happens that orthogonality gets lost.We present a new variant of the Cabay-Meleshko algorithm for numerically computing pairs of basis polynomials, where the numerical orthogonality is explicitly monitored with the help of stability parameters. A corresponding error analysis is given. Our stability parameter is shown to reflect the condition number of the underlying Hankel matrix of moments. This enables us to prove the weak and strong stability of our method, provided that the corresponding Hankel matrix is well-conditioned.This work was partially supported by the HCM project ROLLS, under contract CHRX-CT93-0416.