Notes on particular symmetric polynomials with applications

This paper presents some applications using several properties of three important symmetric polynomials: elementary symmetric polynomials, complete symmetric polynomials and the power sum symmetric polynomials. The applications includes a simple proof of El-Mikkawy conjecture in [M.E.A. El-Mikkawy, Appl. Math. Comput. 146 (2003) 759-769] and a very easy proof of the Newton-Girard formula. In addition, a generalization of Stirling numbers is obtained.     

Elementary symmetric polynomials in Shamir's scheme

The concept of k-admissible tracks in Shamir's secret sharing scheme over a finite field was introduced by Schinzel et al. (2009) [10]. Using some estimates for the elementary symmetric polynomials, we show that the track (1,…,n) over Fp is practically always k-admissible; i.e., the scheme allows to place the secret as an arbitrary coefficient of its generic polynomial even for relatively small p. Here k is the threshold and n the number of shareholders.     

On global optimizations with polynomials

We consider the problem of finding the (unconstrained) global minimum of a real valued polynomial . We study the problem of finding the bounds of global minimizers. It is shown that the unconstrained optimization reduces to some constrained optimizations which can be approximated by solving some convex linear matrix inequality (LMI) problems. This research was partly supported by the National Science Foundation of China under grant No. 10671145.     

Inverse functions of polynomials and orthogonal polynomials as operator monotone functions

We study the operator monotonicity of the inverse of every polynomial with a positive leading coefficient. Let be a sequence of orthonormal polynomials and the restriction of to , where is the maximum zero of . Then and the composite are operator monotone on . Furthermore, for every polynomial with a positive leading coefficient there is a real number so that the inverse function of defined on is semi-operator monotone, that is, for matrices , implies

     

Notes on the geometry of the space of polynomials

We show that the symmetric injective tensor product space is not complex strictly convex if E is a complex Banach space of dim E ≥ 2 and if n ≥ 2 holds. It is also reproved that ?_∞ is finitely represented in if E is infinite‐dimensional and if n ≥ 2 holds, which was proved in the other way in [3]. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A basic class of symmetric orthogonal polynomials using the extended Sturm-Liouville theorem for symmetric functions

In this research, by applying the extended Sturm-Liouville theorem for symmetric functions, a basic class of symmetric orthogonal polynomials (BCSOP) with four free parameters is introduced and all its standard properties, such as a generic second order differential equation along with its explicit polynomial solution, a generic orthogonality relation, a generic three term recurrence relation and so on, are presented. Then, it is shown that four main sequences of symmetric orthogonal polynomials can essentially be extracted from the introduced class. They are respectively the generalized ultraspherical polynomials, generalized Hermite polynomials and two other sequences of symmetric polynomials, which are finitely orthogonal on (−∞,∞) and can be expressed in terms of the mentioned class directly. In this way, two half-trigonometric sequences of orthogonal polynomials, as special sub-cases of BCSOP, are also introduced.     

A proof of a conjecture about a symmetric system of orthogonal polynomials

ABSTRACT

The purpose of this note is to give an affirmative answer to a conjecture appearing in Berg [Open problems. Integral Transforms Spec Funct. 2015;26(2):90–95].     

Extension of polynomials and John's theorem for symmetric tensor products

We show that for every infinite-dimensional normed space and every there are extendible -homogeneous polynomials which are not integral. As a consequence, we prove a symmetric version of a result of John.

     

Stationary point conditions for the FB merit function associated with symmetric cones

For the symmetric cone complementarity problem, we show that each stationary point of the unconstrained minimization reformulation based on the Fischer-Burmeister merit function is a solution to the problem, provided that the gradient operators of the mappings involved in the problem satisfy column monotonicity or have the Cartesian P₀-property. These results answer the open question proposed in the article that appeared in Journal of Mathematical Analysis and Applications 355 (2009) 195-215.     

A note on extrema of linear combinations of elementary symmetric functions

This note provides a new approach to a result of Foregger [T.H. Foregger, On the relative extrema of a linear combination of elementary symmetric functions, Linear Multilinear Algebra 20 (1987) pp. 377–385] and related earlier results by Keilson [J. Keilson, A theorem on optimum allocation for a class of symmetric multilinear return functions, J. Math. Anal. Appl. 15 (1966), pp. 269–272] and Eberlein [P.J. Eberlein, Remarks on the van der Waerden conjecture, II, Linear Algebra Appl. 2 (1969), pp. 311–320]. Using quite different techniques, we prove a more general result from which the others follow easily. Finally, we argue that the proof in [Foregger, 1987] is flawed.     

Representation for the Lagrangian numerical differentiation formula involving elementary symmetric functions

By using elementary symmetric functions, this paper presents an explicit representation for the Lagrangian numerical differentiation formula as well as the error estimate for local approximation. And we also point out that the numerical differentiation formula constructed by Li [J.P. Li, General explicit difference formulas for numerical differentiation, J. Comput. Appl. Math. 183 (2005) 29-52] is actually a special case of the Lagrangian numerical differentiation formula to approximate the values of the derivatives at the nodes.     

Laplace and Segal-Bargmann transforms on Hermitian symmetric spaces and orthogonal polynomials

Let be a complex bounded symmetric domain of tube type in a complex Jordan algebra V and let be its real form in a formally real Euclidean Jordan algebra J⊂V; is a bounded realization of the symmetric cone in J. We consider representations of H that are gotten by the generalized Segal-Bargmann transform from a unitary G-space of holomorphic functions on to an L²-space on . We prove that in the unbounded realization the inverse of the unitary part of the restriction map is actually the Laplace transform. We find the extension to of the spherical functions on and find their expansion in terms of the L-spherical polynomials on , which are Jack symmetric polynomials. We prove that the coefficients are orthogonal polynomials in an L²-space, the measure being the Harish-Chandra Plancherel measure multiplied by the symbol of the Berezin transform. We prove the difference equation and recurrence relation for those polynomials by considering the action of the Lie algebra and the Cayley transform on the polynomials on . Finally, we use the Laplace transform to study generalized Laguerre functions on symmetric cones.     

On the extensions of homogeneous polynomials

We investigate the problem of the uniqueness of the extension of -homogeneous polynomials in Banach spaces. We show in particular that in a nonreflexive Banach space that admits contractive projection of finite rank of at least dimension 2, for every there exists an -homogeneous polynomial on that has infinitely many extensions to . We also prove that under some geometric conditions imposed on the norm of a complex Banach lattice , for instance when satisfies an upper -estimate with constant one for some , any -homogeneous polynomial on attaining its norm at with a finite rank band projection , has a unique extension to its bidual . We apply these results in a class of Orlicz sequence spaces.

     

On a family of symmetric polynomials

In this paper, we provide two simple approaches to the explicit expression of a family of symmetric polynomials introduced and studied in Milovanovi? and Cvetkovi? [J. Math. Anal. Appl. 311 (2005) 191], thereby improving on their observations.     

On a set of orthogonal polynomials associated with a quantized physical model

In this paper we investigate a set of orthogonal polynomials. We relate the polynomials to the Biconfluent Heun equation and present an explicit expression for the polynomials in terms of the classical Hermite polynomials. The orthogonality with a varying measure and the recurrence relation are also presented.     

LLT polynomials,chromatic quasisymmetric functions and graphs with cycles

We use a Dyck path model for unit-interval graphs to study the chromatic quasisymmetric functions introduced by Shareshian and Wachs, as well as unicellular LLT polynomials, revealing some parallel structure and phenomena regarding their $\mathrm{e}$-positivity.The Dyck path model is also extended to circular arc digraphs to obtain larger families of polynomials, giving a new extension of LLT polynomials. Carrying over a lot of the non-circular combinatorics, we prove several statements regarding the $\mathrm{e}$-coefficients of chromatic quasisymmetric functions and LLT polynomials, including a natural combinatorial interpretation for the $\mathrm{e}$-coefficients for the line graph and the cycle graph for both families. We believe that certain $\mathrm{e}$-positivity conjectures hold in all these families above.Furthermore, beyond the chromatic analogy, we study vertical-strip LLT polynomials, which are modified Hall–Littlewood polynomials.     

A combinatorial approach to the 2D-Hermite and 2D-Laguerre polynomials

The first author has recently proved a Kibble–Slepian type formula for the 2D-Hermite polynomials $\{H_{m,n}(z,\overline{z})\}$ which extends the Poisson kernel for these polynomials. We provide a combinatorial proof of a closely related formula. The combinatorial structures involved are the so-called m-involutionary ℓ-graphs. They are enumerated in two different manners: first globally, then as the exponential of their connected components. We also give a combinatorial model for the 2D-Laguerre polynomials and study their linearization coefficients.