Associated Wilson polynomials

From a contiguous relation obtained by Wilson for terminating 2-balanced very well-poised₉ F ₈ hypergeometric functions of unit argument, we derive a pair of three term recurrence relations for very well-poised₇ F ₆'s. From these we obtain solutions to the recurrence relation for associated Wilson polynomials and spectral properties of the corresponding Jacobi matrix. A calculation of the basic weight function yields a generalization of Dougall's theorem.Communicated by Mourad Ismail.     

Connection and inversion coefficients for basic hypergeometric polynomials

In this paper, we give a closed-form expression of the inversion and the connection coefficients for general basic hypergeometric polynomial sets using some known inverse relations. We derive expansion formulas corresponding to all the families within the q-Askey scheme and we connect some d-orthogonal basic hypergeometric polynomials.     

Polynomial numerical hulls of matrix polynomials,II

In this article, we study some algebraic and geometrical properties of polynomial numerical hulls of matrix polynomials and joint polynomial numerical hulls of a finite family of matrices (possibly the coefficients of a matrix polynomial). Also, we study polynomial numerical hulls of basic A-factor block circulant matrices. These are block companion matrices of particular simple monic matrix polynomials. By studying the polynomial numerical hulls of the Kronecker product of two matrices, we characterize the polynomial numerical hulls of unitary basic A-factor block circulant matrices.     

On spectral analysis of non-monic matrix and operator polynomials,I. Reduction to monic polynomials

Three recent papers [1, 2, 3] developed the basic concepts of a spectral theory for matrix and operator monic polynomials. In this paper we continue the study, replacing the requirement of monicness by a weaker condition.     

Weight enumerators,intersection enumerators,and Jacobi polynomials

The intersection enumerator and the Jacobi polynomial in an arbitrary genus for a binary code are introduced. Adding the weight enumerator into our discussion, we give the explicit relations among them and give some of their basic properties.     

Special polynomials in free framed Lie algebra

A framed Lie algebra is an algebra with two operations which is a Lie algebra with respect to one of these operations. A basic example is a Lie algebra of vector fields on a manifold with connection where the covariant derivative serves as an additional operation. In a free framed Lie algebra, we distinguish a set of special polynomials that geometrically correspond to invariantly defined tensors. A necessary condition of being special is derived, and we presume that this condition is also sufficient. Translated from Algebra i Logika, Vol. 47, No. 5, pp. 571–583, September–October, 2008.     

Order and type of basic sets of polynomials associated with functions of algebraic semi block matrices

§ 1 is an introduction to basic sets of polynomials and to algebraic infinite matrices. In § 2 we investigate the order of basic sets associated with functions of algebraic semi block matrices whose elements are of a certain order of magnitude. In § 3 we investigate the type of such basic sets.     

Fueter polynomials in discrete Clifford analysis

Discrete Clifford analysis is a higher dimensional discrete function theory, based on skew Weyl relations. The basic notions are discrete monogenic functions, i.e. Clifford algebra valued functions in the kernel of a discrete Dirac operator. In this paper, we introduce the discrete Fueter polynomials, which form a basis of the space of discrete spherical monogenics, i.e. discrete monogenic, homogeneous polynomials. Their definition is based on a Cauchy–Kovalevskaya extension principle. We present the explicit construction for this discrete Fueter basis, in arbitrary dimension m and for arbitrary homogeneity degree k.     

Some properties of a class of sparse polynomials

We study an infinite class of sequences of sparse polynomials that have binomial coefficients both as exponents and as coefficients. This generalizes a sequence of sparse polynomials which arises in a natural way as graph theoretic polynomials. After deriving some basic identities, we obtain properties concerning monotonicity and log-concavity, as well as identities involving derivatives. We also prove upper and lower bounds on the moduli of the zeros of these polynomials.     

Schubert polynomials of types A--D

Schubert polynomials of type B, C, and D have been described first by S. Billey and M. Haiman [BH] using a combinatorial method. In this paper we give a unified algebraic treatment of Schubert polynomials of types A–D in the style of the Lascoux–Schützenberger theory in type A, i.e. Schubert polynomials are generated by the application of sequences of divided difference operators to “top polynomials”. The use of the creation operators for Q-Schur and P-Schur functions allows us to give: (1) simple and natural forms of the “top polynomials”, (2) formulas for the easy computation with all divided differences, (3) recursive structures, and (4) simplified derivations of basic properties. Received: 23 July 1998     

On the separation of basic semialgebraic sets by polynomials

In this paper we show that two disjoint basic closed semialgebraic sets, defined over a real closed field R, can be separated by a polynomial, if one of them has dimension 2. Counterexamples are given in all higher dimensions.     

Dickson polynomials over finite fields

In this paper we introduce the notion of Dickson polynomials of the $(k+1)$-th kind over finite fields ${\mathbb{F}}_{p^m}$ and study basic properties of this family of polynomials. In particular, we study the factorization and the permutation behavior of Dickson polynomials of the third kind.     

A canonical form and the distribution of values of generalized polynomials

Generalized polynomials are functions obtained from conventional polynomials by applying the operations of taking the integer part, addition, and multiplication. We construct a system of “basic” generalized polynomials with the property that any bounded generalized polynomial is representable as a piecewise polynomial function of these basic ones. Such a representation is unique up to a function vanishing almost everywhere, which solves the problem of determining whether two generalized polynomials are equal a.e. The basic generalized polynomials are jointly equidistributed, thus we also obtain an effective algorithm of finding the limiting distribution of values of one or several generalized polynomials.     

Generalized Lucas polynomials and relationships between the Fibonacci polynomials and Lucas polynomials

In this article, we find elements of the Lucas polynomials by using two matrices. We extend the study to the n-step Lucas polynomials. Then the Lucas polynomials and their relationship are generalized in the paper. Furthermore, we give relationships between the Fibonacci polynomials and the Lucas polynomials.     

On the Schur–Szegö composition of polynomials

The Schur–Szegö composition of two polynomials of degree ?n introduces an interesting semigroup structure on polynomial spaces and is one of the basic tools in the analytic theory of polynomials. In the present Note we show how it interacts with the stratification of polynomials according to the multiplicities of their zeros and we present the induced semigroup structure on the set of all ordered partitions of n. To cite this article: V. Kostov, B. Shapiro, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Generating functions for generalized Hermite polynomials associated with parabolic cylinder functions

ABSTRACT

In this article, we first give some basic properties of generalized Hermite polynomials associated with parabolic cylinder functions. We next use Weisner? group theoretic method and operational rules method to establish new generating functions for these generalized Hermite polynomials. The operational methods we use allow us to obtain unilateral, bilinear and bilateral generating functions by using the same procedure. Applications of generating functions obtained by Weisner? group theoretic method are discussed.     

Extremal polynomials for obtaining bounds for spherical codes and designs

We investigate two extremal problems for polynomials giving upper bounds for spherical codes and for polynomials giving lower bounds for spherical designs, respectively. We consider two basic properties of the solutions of these problems. Namely, we estimate from below the number of double zeros and find zero Gegenbauer coefficients of extremal polynomials. Our results allow us to search effectively for such solutions using a computer. The best polynomials we have obtained give substantial improvements in some cases on the previously known bounds for spherical codes and designs. Some examples are given in Section 6. This research was partially supported by the Bulgarian NSF under Contract I-35/1994.     

Basic r-symmetric tropical polynomials

We consider tropical polynomials in nr variables, divided into n blocks of r variables, and especially r-symmetric tropical polynomials, which are invariant under the action of the symmetric group $S_n$ on the blocks. We define a set of basic r-symmetric tropical polynomials and show that the basic 2-symmetric tropical polynomials give coordinates on ${\mathbb{R}}^{2n}/S_n$ more efficiently than known polynomials. Moreover, we present special cases for $r\ge 3$ where the basic polynomials separate orbits.     

Self-reciprocal polynomials and coterm polynomials

We classify all self-reciprocal polynomials arising from reversed Dickson polynomials over \(\mathbb {Z}\) and \(\mathbb {F}_p\), where p is prime. As a consequence, we also obtain coterm polynomials arising from reversed Dickson polynomials.     

Geometric relations between the zeros of polynomials

We consider the classical theorem of Grace, which gives a condition for a geometric relation between two arbitrary algebraic polynomials of the same degree. This theorem is one of the basic instruments in the geometry of polynomials. In some applications of the Grace theorem, one of the two polynomials is fixed. In this case, the condition in the Grace theorem may be changed. We explore this opportunity and introduce a new notion of locus of a polynomial. Using the loci of polynomials, we may improve some theorems in the geometry of polynomials. In general, the loci of a polynomial are not easy to describe. We prove some statements concerning the properties of a point set on the extended complex plane that is a locus of a polynomial.