Various Nyquist polynomials and their spectral factorizations to obtain suitable finite impulse response (FIR) vestigial sideband (VSB) filters are summarized here. The filters of interest have transfer functions that are polynomials with complex coefficients which are symmetric, due to Lawton, and polynomials whose coefficients are conjugate symmetric or conjugate antisymmetric, which are also...

Let D be any integral domain of any characteristic. A polynomial p(x) ∈ D[x] is D-nice if p(x) and its derivative p′(x) split in D[x]. We give a complete description of all D-nice symmetric polynomials with four roots over integral domains D of any characteristic not equal to 2 by giving an explicit formula for constructing these polynomials and by counting equivalence classes of such D-nice po...

In this paper two families of rational solutions and associated special polynomials for the equations in the symmetric fourth Painlevé hierarchy are studied. The structure of the roots of these polynomials is shown to be highly regular in the complex plane. Further representations are given of the associated special polynomials in terms of Schur functions. The properties of these polynomials ar...

In this paper we define a family of polynomials closely related to the modified R-polynomials of the symmetric group and begin work toward a classification of the polynomials by using a combinatorial interpretation involving subwords of the maximal element in the Bruhat order. The problem of determining the precise conditions which make one of these polynomials zero motivates our work. We state...

The ring of symmetric functions Λ, with natural basis given by the Schur functions, arise in many different areas of mathematics. For example, as the cohomology ring of the grassmanian, and as the representation ring of the symmetric group. One may define a coproduct on Λ by the plethystic addition on alphabets. In this way the ring of symmetric functions becomes a Hopf algebra. The Littlewood–...

Using the action of the Yang–Baxter elements of the Hecke algebra on polynomials, we define two bases of polynomials in n variables. The Hall–Littlewood polynomials are a subfamily of one of them. For q = 0, these bases specialize to the two families of classical Key polynomials (i.e., Demazure characters for type A). We give a scalar product for which the two bases are adjoint to each other.

D. Grigoriev-G. Koshevoy recently proved that tropical Schur polynomials have (at worst) polynomial tropical semiring complexity. They also conjectured tropical skew Schur polynomials have at least exponential complexity; we establish a polynomial complexity upper bound. Our proof uses results about (stable) Schubert polynomials, due to R. P. Stanley and S. Billey-W. Jockusch-R. P. Stanley, tog...

This paper studies the elementary symmetric polynomials Sk(x) for x ∈ Rn. We show that if |Sk(x)|, |Sk+1(x)| are small for some k > 0 then |Sl(x)| is also small for all l > k. We use this to prove probability tail bounds for the symmetric polynomials when the inputs are only t-wise independent, that may be useful in the context of derandomization. We also provide examples of t-wise independent ...

This article is an attempt to understand the ubiquity of tableaux and of Pieri and Cauchy formulae for combinatorially defined families of symmetric functions. We show that such formulae are to be expected from symmetric functions arising from representations of Heisenberg algebras. The resulting framework that we describe is a generalization of the classical Boson-Fermion correspondence, from ...