6,961 research outputs found
Polynomial sequences on quadratic curves
In this paper we generalize the study of Matiyasevich on integer points over
conics, introducing the more general concept of radical points. With this
generalization we are able to solve in positive integers some Diophantine
equations, relating these solutions by means of particular linear recurrence
sequences. We point out interesting relationships between these sequences and
known sequences in OEIS. We finally show connections between these sequences
and Chebyshev and Morgan-Voyce polynomials, finding new identities
Note on Ward-Horadam H(x) - binomials' recurrences and related interpretations, II
We deliver here second new recurrence formula,
were array is appointed by sequence of
functions which in predominantly considered cases where chosen to be
polynomials . Secondly, we supply a review of selected related combinatorial
interpretations of generalized binomial coefficients. We then propose also a
kind of transfer of interpretation of coefficients onto
coefficients interpretations thus bringing us back to
and Donald Ervin Knuth relevant investigation decades
ago.Comment: 57 pages, 8 figure
Relative asymptotics for orthogonal matrix polynomials
In this paper we study sequences of matrix polynomials that satisfy a
non-symmetric recurrence relation. To study this kind of sequences we use a
vector interpretation of the matrix orthogonality. In the context of these
sequences of matrix polynomials we introduce the concept of the generalized
matrix Nevai class and we give the ratio asymptotics between two consecutive
polynomials belonging to this class. We study the generalized matrix Chebyshev
polynomials and we deduce its explicit expression as well as we show some
illustrative examples. The concept of a Dirac delta functional is introduced.
We show how the vector model that includes a Dirac delta functional is a
representation of a discrete Sobolev inner product. It also allows to
reinterpret such perturbations in the usual matrix Nevai class. Finally, the
relative asymptotics between a polynomial in the generalized matrix Nevai class
and a polynomial that is orthogonal to a modification of the corresponding
matrix measure by the addition of a Dirac delta functional is deduced
A family of linearizable recurrences with the Laurent property
We consider a family of non-linear recurrences with the Laurent property. Although these recurrences are not generated by mutations in a cluster algebra, they fit within the broader framework of Laurent phenomenon algebras, as introduced recently by Lam and Pylyavskyy. Furthermore, each member of this family is shown to be linearizable in two different ways, in the sense that its iterates satisfy both a linear relation with constant coefficients and a linear relation with periodic coefficients. Associated monodromy matrices and first integrals are constructed, and the connection with the dressing chain for Schrödinger operators is also explained
A unified approach to polynomial sequences with only real zeros
We give new sufficient conditions for a sequence of polynomials to have only
real zeros based on the method of interlacing zeros. As applications we derive
several well-known facts, including the reality of zeros of orthogonal
polynomials, matching polynomials, Narayana polynomials and Eulerian
polynomials. We also settle certain conjectures of Stahl on genus polynomials
by proving them for certain classes of graphs, while showing that they are
false in general.Comment: 19 pages, Advances in Applied Mathematics, in pres
The Kontorovich-Lebedev transform as a map between -orthogonal polynomials
A slight modification of the Kontorovich-Lebedev transform is an automorphism
on the vector space of polynomials. The action of this -transform
over certain polynomial sequences will be under discussion, and a special
attention will be given the d-orthogonal ones. For instance, the Continuous
Dual Hahn polynomials appear as the -transform of a 2-orthogonal
sequence of Laguerre type. Finally, all the orthogonal polynomial sequences
whose -transform is a -orthogonal sequence will be
characterized: they are essencially semiclassical polynomials fulfilling
particular conditions and is even. The Hermite and Laguerre polynomials are
the classical solutions to this problem.Comment: 27 page
- …