Let F[∂;σ ,δ] be a ring of Ore polynomials over a field. We give a new deterministic algorithm for computing the Popov form P of a non-singular matrix A ∈ F[∂;σ ,δ]n×n . Our main focus is to ensure controlled growth in the size of coefficients from F in the case F = k(z), and even k = Q. Our algorithms are based on constructing from A a linear system over F and performing a structured fraction-...

An orthogonal Ore algebra is an abstraction of common properties of linear partial differential, shift and q-shift operators. Using orthogonal Ore algebras, we present an algorithm for finding hyperexponential solutions of a system of linear differential, shift and q-shift operators, or any mixture thereof, whose solution space is finite-dimensional. The algorithm is applicable to factoring mod...

We say that the polynomial sequence (Q (λ) n ) is a semiclassical Sobolev polynomial sequence when it is orthogonal with respect to the inner product 〈p, r〉S = 〈u, p r〉+ λ 〈u,DpDr〉 , where u is a semiclassical linear functional, D is the differential, the difference or the q–difference operator, and λ is a positive constant. In this paper we get algebraic and differential/difference properties ...

In the paper, the authors view some ordinary differential equations and their solutions from the angle of (the generalized) derivative polynomials and simplify some known identities for the Bernoulli numbers and polynomials, the Frobenius-Euler polynomials, the Euler numbers and polynomials, in terms of the Stirling numbers of the first and second kinds.

We present a framework for the study of q-differential equations satisfied by q-semi-classical orthogonal systems. As an example, we identify the q-differential equation satisfied by a deformed version of the little q-Jacobi polynomials as a guage transformation of a special case of the associated linear problem for q-PV I . We obtain a parametrization of the associated linear problem in terms ...

We establish a connection between a specialization of the nonsymmetric Macdonald polynomials and the Demazure characters of the corresponding affine Kac-Moody algebra. This allows us to obtain a representation-theoretical interpretation of the coefficients of the expansion of the specialized symmetric Macdonald polynomials in the basis formed by the irreducible characters of the associated fini...

The properties of matrix valued polynomials generated by the scalartype Rodrigues’ formulas are analyzed. A general representation of these polynomials is found in terms of products of simple differential operators. The recurrence relations, leading coefficients, completeness are established, as well as, in the commutative case, the second order equations for which these polynomials are eigenfu...

We present various results on the properties of the four infinite sets of the exceptional X` polynomials discovered recently by Odake and Sasaki [Phys. Lett. B 679 (2009), 414–417; Phys. Lett. B 684 (2010), 173–176]. These X` polynomials are global solutions of second order Fuchsian differential equations with ` + 3 regular singularities and their confluent limits. We derive equivalent but much...

We consider the class of polynomial differential equation x = 2 ( , ) ( , ) ( , ) n n m n m P x y P x y P x y + + + + , 2 ( , ) ( , ) ( , ) n n m n m y Q x y Q x y Q x y + + = + + . For , 1 m n ≥ where i P and i Q are homogeneous polynomials of degree i. Inside this class of polynomial differential equation we consider a subclass of Darboux integrable systems. Moreover, under additional conditi...

This paper presents a method for extracting the coupling matrix and the unloaded Q from the measured (or electromagnetic simulated) S-parameters of a narrow band cross-coupled resonator bandpass filter with losses. The Cauchy method is applied to determine the characteristic polynomials of the S-parameters of a filter in the normalized low-pass frequency domain. A five-parameter optimization me...