Bivariate fibonacci like p-polynomials

In this article, we study the bivariate Fibonacci and Lucas p-polynomials (p ? 0 is integer) from which, specifying x, y and p, bivariate Fibonacci and Lucas polynomials, bivariate Pell and Pell-Lucas polynomials, Jacobsthal and Jacobsthal-Lucas polynomials, Fibonacci and Lucas p-polynomials, Fibonacci and Lucas p-numbers, Pell and Pell-Lucas p-numbers and Chebyshev polynomials of the first and second kind, are obtained. Afterwards, we obtain some properties of the bivariate Fibonacci and Lucas p-polynomials.     

On Operators on Polynomials Preserving Real-Rootedness and the Neggers-Stanley Conjecture

We refine a technique used in a paper by Schur on real-rooted polynomials. This amounts to an extension of a theorem of Wagner on Hadamard products of Pólya frequency sequences. We also apply our results to polynomials for which the Neggers-Stanley Conjecture is known to hold. More precisely, we settle interlacing properties for E-polynomials of series-parallel posets and column-strict labelled Ferrers posets.     

An extension of an inequality of I. Schur

Denote by π_n the set of all algebraic polynomials of degree at most n with complex coefficients. An inequality of I. Schur asserts that the first derivative of the transformed Tchebycheff polynomial has the greatest uniform norm in [?1, 1] among all f ∈ ??_n, where (1) Here we show that this extremal property of persists in the wider class of polynomials f ∈ π_n which vanish at ±1, and for which there exist n ? 1 points separating the zeros of and such that for j = 1, …, n ? 1. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Menke points on the real line and their connection to classical orthogonal polynomials

We investigate the properties of extremal point systems on the real line consisting of two interlaced sets of points solving a modified minimum energy problem. We show that these extremal points for the intervals [−1,1], [0,∞) and (−∞,∞), which are analogues of Menke points for a closed curve, are related to the zeros and extrema of classical orthogonal polynomials. Use of external fields in the form of suitable weight functions instead of constraints motivates the study of “weighted Menke points” on [0,∞) and (−∞,∞). We also discuss the asymptotic behavior of the Lebesgue constant for the Menke points on [−1,1].     

New ideas for decomposing nonlinearities in differential equations

In this paper we consider the decomposition for the nonlinearity in a differential equation for the solution by decomposition. By analyzing and transforming the Taylor expansion of the nonlinearity about the initial solution component, the decomposition of the nonlinearity is converted to the partitions of the solution sets for a class of Diophantine equations. This conversion simplifies the discussion and presents a new idea for decompositions. We enumerate five types of partitions and their corresponding decomposition polynomials. Each of the last four types contains infinitely many kinds of decomposition polynomials in the form of finite sums. In Types 2, 3 and 4, there is a parameter q and each value of q corresponds to a class of decomposition polynomials. In Type 5, each positive integer sequence {c_j} satisfying 1 = c₁ ? c₂ ? ? and j ? c_j for j = 2, 3, … corresponds to a class of decomposition polynomials. Four classes of the Adomian polynomials [R. Rach, A new definition of the Adomian polynomials, Kybernetes 37 (2008) 910-955] are derived as particular cases.     

Schur Q-polynomials, multiple hypergeometric series and enumeration of marked shifted tableaux

We study Schur Q-polynomials evaluated on a geometric progression, or equivalently q-enumeration of marked shifted tableaux, seeking explicit formulas that remain regular at q=1. We obtain several such expressions as multiple basic hypergeometric series, and as determinants and pfaffians of continuous q-ultraspherical or continuous q-Jacobi polynomials. As special cases, we obtain simple closed formulas for staircase-type partitions.     

Szeg? type polynomials and para-orthogonal polynomials

Szeg? type polynomials with respect to a linear functional M for which the moments M[tn]=μ−n are all complex, μ−n=μn and Dn≠0 for n?0, are considered. Here, Dn are the associated Toeplitz determinants. Para-orthogonal polynomials are also studied without relying on any integral representation. Relation between the Toeplitz determinants of two different types of moment functionals are given. Starting from the existence of polynomials similar to para-orthogonal polynomials, sufficient conditions for the existence of Szeg? type polynomials are also given. Examples are provided to justify the results.     

Integrals involving products of Airy functions, their derivatives and Bessel functions

A new integral representation of the Hankel transform type is deduced for the function Fn(x,Z)=Zn−1Ai(x−Z)Ai(x+Z) with x∈R, Z>0 and n∈N. This formula involves the product of Airy functions, their derivatives and Bessel functions. The presence of the latter allows one to perform various transformations with respect to Z and obtain new integral formulae of the type of the Mellin transform, K-transform, Laplace and Fourier transform. Some integrals containing Airy functions, their derivatives and Chebyshev polynomials of the first and second kind are computed explicitly. A new representation is given for the function ²|Ai(z)| with z∈C.     

Type-B generalized triangulations and determinantal ideals

For n≥3, let Ω_n be the set of line segments between the vertices of a convex n-gon. For j≥2, a j-crossing is a set of j line segments pairwise intersecting in the relative interior of the n-gon. For k≥1, let Δ_n,k be the simplicial complex of (type-A) generalized triangulations, i.e. the simplicial complex of subsets of Ω_n not containing any (k+1)-crossing.The complex Δ_n,k has been the central object of many papers. Here we continue this work by considering the complex of type-B generalized triangulations. For this we identify line segments in Ω_2n which can be transformed into each other by a 180^°-rotation of the 2n-gon. Let F_n be the set Ω_2n after identification, then the complex D_n,k of type-B generalized triangulations is the simplicial complex of subsets of F_n not containing any (k+1)-crossing in the above sense. For k=1, we have that D_n,1 is the simplicial complex of type-B triangulations of the 2n-gon as defined in [R. Simion, A type-B associahedron, Adv. Appl. Math. 30 (2003) 2-25] and decomposes into a join of an (n−1)-simplex and the boundary of the n-dimensional cyclohedron. We demonstrate that D_n,k is a pure, k(n−k)−1+kn dimensional complex that decomposes into a kn−1-simplex and a k(n−k)−1 dimensional homology-sphere. For k=n−2 we show that this homology-sphere is in fact the boundary of a cyclic polytope. We provide a lower and an upper bound for the number of maximal faces of D_n,k.On the algebraical side we give a term order on the monomials in the variables X_ij,1≤i,j≤n, such that the corresponding initial ideal of the determinantal ideal generated by the (k+1) times (k+1) minors of the generic n×n matrix contains the Stanley-Reisner ideal of D_n,k. We show that the minors form a Gröbner-Basis whenever k∈{1,n−2,n−1} thereby proving the equality of both ideals and the unimodality of the h-vector of the determinantal ideal in these cases. We conjecture this result to be true for all values of k<n.     

Generalized triangulations and diagonal-free subsets of stack polyominoes

For n?3, let Ω_n be the set of line segments between vertices in a convex n-gon. For j?1, a j-crossing is a set of j distinct and mutually intersecting line segments from Ω_n such that all 2j endpoints are distinct. For k?1, let Δ_n,k be the simplicial complex of subsets of Ω_n not containing any (k+1)-crossing. For example, Δ_n,1 has one maximal set for each triangulation of the n-gon. Dress, Koolen, and Moulton were able to prove that all maximal sets in Δ_n,k have the same number k(2n-2k-1) of line segments. We demonstrate that the number of such maximal sets is counted by a k×k determinant of Catalan numbers. By the work of Desainte-Catherine and Viennot, this determinant is known to count quite a few other objects including fans of non-crossing Dyck paths. We generalize our result to a larger class of simplicial complexes including some of the complexes appearing in the work of Herzog and Trung on determinantal ideals.     

On discrete q-ultraspherical polynomials and their duals

A special case of the big q-Jacobi polynomials Pn(x;a,b,c;q), which corresponds to a=b=−c, is shown to satisfy a discrete orthogonality relation for imaginary values of the parameter a (outside of its commonly known domain 0<a<q⁻¹). Since Pn(x;qα,qα,−qα;q) tend to Gegenbauer (or ultraspherical) polynomials in the limit as q→1, this family represents another q-extension of these classical polynomials, different from the continuous q-ultraspherical polynomials of Rogers. For a dual family with respect to the polynomials Pn(x;a,a,−a;q) (i.e., for dual discrete q-ultraspherical polynomials) we also find new orthogonality relations with extremal measures.     

Relatively prime polynomials and nonsingular Hankel matrices over finite fields

The probability for two monic polynomials of a positive degree n with coefficients in the finite field Fq to be relatively prime turns out to be identical with the probability for an n×n Hankel matrix over Fq to be nonsingular. Motivated by this, we give an explicit map from pairs of coprime polynomials to nonsingular Hankel matrices that explains this connection. A basic tool used here is the classical notion of Bezoutian of two polynomials. Moreover, we give simpler and direct proofs of the general formulae for the number of m-tuples of relatively prime polynomials over Fq of given degrees and for the number of n×n Hankel matrices over Fq of a given rank.     

Sperner labellings: A combinatorial approach

In 2002, De Loera, Peterson and Su proved the following conjecture of Atanassov: let T be a triangulation of a d-dimensional polytope P with n vertices v₁,v₂,…,vn; label the vertices of T by 1,2,…,n in such a way that a vertex of T belonging to the interior of a face F of P can only be labelled by j if vj is on F; then there are at least n−d simplices labelled with d+1 different labels. We prove a generalisation of this theorem which refines this lower bound and which is valid for a larger class of objects.     

Intersective polynomials and the primes

Intersective polynomials are polynomials in Z[x] having roots every modulus. For example, P₁(n)=n² and P₂(n)=n²−1 are intersective polynomials, but P₃(n)=n²+1 is not. The purpose of this note is to deduce, using results of Green and Tao (2006) [8] and Lucier (2006) [16], that for any intersective polynomial h, inside any subset of positive relative density of the primes, we can find distinct primes p₁,p₂ such that p₁−p₂=h(n) for some integer n. Such a conclusion also holds in the Chen primes (where by a Chen prime we mean a prime number p such that p+2 is the product of at most 2 primes).     

Representability theorems for presheaves of spectra

Suppose that M is a simplicial model category and that F is a contravariant simplicial functor defined on M which takes values in pointed simplicial sets. This note displays conditions on the simplicial model category M and the functor F such that F is representable up to weak equivalence. The conditions on F are homotopy coherent versions of the classical conditions for Brown representability, while M should have the fundamental properties of the stable model structure for presheaves of spectra on a Grothendieck site.     

Galois groups of polynomials arising from circulant matrices

Circulant matrices are used to construct polynomials, associated with Chebyshev polynomials of the first kind, whose roots are real and made explicit. Then the Galois groups of the polynomials are computed, giving rise to new examples of polynomials with cyclic Galois groups and Galois groups of order p(p−1) that are generated by a cycle of length p and a cycle of length p−1.     

Spectra of certain types of polynomials and tiling of integers with translates of finite sets

We investigate Fuglede's spectral set conjecture in the special case when the set in question is a union of finitely many unit intervals in dimension 1. In this case, the conjecture can be reformulated as a statement about multiplicative properties of roots of associated with the set polynomials with (0,1) coefficients. Let be an N-term polynomial. We say that {θ₁,θ₂,…,θ_N−1} is an N-spectrum for A(x) if the θ_j are all distinct and

     

On cluster algebras arising from unpunctured surfaces II

We study cluster algebras with principal and arbitrary coefficient systems that are associated to unpunctured surfaces. We give a direct formula for the Laurent polynomial expansion of cluster variables in these cluster algebras in terms of certain paths on a triangulation of the surface. As an immediate consequence, we prove the positivity conjecture of Fomin and Zelevinsky for these cluster algebras.Furthermore, we obtain direct formulas for F-polynomials and g-vectors and show that F-polynomials have constant term equal to 1. As an application, we compute the Euler-Poincaré characteristic of quiver Grassmannians in Dynkin type A and affine Dynkin type .     

Quasisymmetric Schur functions

We introduce a new basis for quasisymmetric functions, which arise from a specialization of nonsymmetric Macdonald polynomials to standard bases, also known as Demazure atoms. Our new basis is called the basis of quasisymmetric Schur functions, since the basis elements refine Schur functions in a natural way. We derive expansions for quasisymmetric Schur functions in terms of monomial and fundamental quasisymmetric functions, which give rise to quasisymmetric refinements of Kostka numbers and standard (reverse) tableaux. From here we derive a Pieri rule for quasisymmetric Schur functions that naturally refines the Pieri rule for Schur functions. After surveying combinatorial formulas for Macdonald polynomials, including an expansion of Macdonald polynomials into fundamental quasisymmetric functions, we show how some of our results can be extended to include the t parameter from Hall-Littlewood theory.