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Combinatorial Proof of the Inversion Formula on the Kazhdan-Lusztig R-Polynomials
Kazhdan-Lusztig R-polynomial inversion formula Bruhat order
2014/6/3
In this paper, we present a combinatorial proof of the inversion formula on the Kazhdan-Lusztig R-polynomials. This problem was raised by Brenti. As a consequence, we obtain a combinatorial interpreta...
Branden's Conjectures on the Boros-Moll Polynomials
Boros-Moll Polynomials Real-rootedness Sturm sequence 3-log-concavity
2014/6/3
We prove two conjectures of Brändén on the real-rootedness of polynomials Qn(x) and Rn(x) which are related to the Boros-Moll polynomials Pn(x). In fact, we show that both Qn(x) and Rn(x) form St...
The Boros-Moll polynomials Pm(a) arise in the evaluation of a quartic integral. It has been conjectured by Boros and Moll that these polynomials are infinitely log-concave. In this paper, we show that...
Disposition Polynomials and Plane Trees
disposition disposition polynomial plane tree Prüfer correspondence
2014/6/3
We define the disposition polynomial Rm(x1, x2,..., xn) as ∏k=0m-1(x1 + x2 + ... + xn + k). When m=n-1, this polynomial becomes the generating function of plane trees with respect to the number of you...
Partially 2-Colored Permutations and the Boros-Moll Polynomials
partially 2-colored permutation Boros-Moll polynomial rising factorial logconcavity bijection
2014/6/3
We find a combinatorial setting for the coefficients of the Boros-Moll polynomials Pm(a) in terms of partially 2-colored permutations. Using this model, we give a combinatorial proof of a recurrence r...
The Interlacing Log-concavity of the Boros-Moll Polynomials
interlacing log-concavity log-concavity the Boros-Moll polynomials
2014/6/3
We introduce the notion of interlacing log-concavity of a polynomial sequence {Pm(x)}m≥0, where Pm(x) is a polynomial of degree m with positive coefficients. This sequence is said to be interlacingly ...
A further generalization of the Bernoulli polynomials and on the 2D-Bernoulli polynomials B_n^2(x,y)
Bernoulli numbers and Bernoulli polynomials Appell polynomials
2010/9/26
In this work we give some recurrence relations of the new generalized Bernoulli polynomials and numbers. Furthermore a relation is given between 2D-Bernoulli polynomials and 2-variable Hermite Kamp&ac...
Uniform approximation of a class of multivariable trigonometric interpolation polynomials in Euclidean space
triangle summation operator uniform convergence
2010/9/25
To improve the uniform convergence of the classical Lagrange operators of several variables, we construct a new operator with a class of summation factors. It is proved that the new operator converges...
On the structure and probabilistic interpretation of Askey-Wilson densities and polynomials with complex parameters
Askey-Wilson densities polynomials complex parameters
2010/11/11
We give equivalent forms of Askey-Wilson (AW) polynomials expressing them with a help of Al-Salam-Chihara polynomials. After restricting parameters of AW polynomials to complex conjugate pairs we giv...
Expansions of one density via polynomials orthogonal with respect to the other
density via polynomials orthogonal math
2010/11/11
We expand Chebyshev polynomials and some of its linear combination in linear combinations of q-Hermite, Rogers and Al Salam-Chihara polynomials and vice versa. We use these expansions to obtain expan...
Bollobas-Riordan and relative Tutte polynomials
Bollobas-Riordan relative Tutte polynomials
2010/11/8
We establish a relation between the Bollobas-Riordan polynomial of a ribbon graph with the relative Tutte polynomial of a plane graph obtained from the ribbon graph using its projection to the plane i...
Schur Positivity and the q-Log-convexity of the Narayana Polynomials
q-log-concavity q-log-convexity q-Narayana number Narayana polynomial lattice permutation Schur positivity Littlewood-Richardson rule
2014/6/3
Using Schur positivity and the principal specialization of Schur functions, we provide a proof of a recent conjecture of Liu and Wang on the q-log-convexity of the Narayana polynomials, and a proof of...
On the Combinatorics of the Boros-Moll Polynomials
Jacobi polynomials Boros-Moll polynomials reluctant function Meixner endofunction bi-colored permutation
2014/6/3
The Boros-Moll polynomials arise in the evaluation of a quartic integral. The original double summation formula does not imply the fact that the coefficients of these polynomials are positive. Boros a...
The q-log-convexity of the Narayana polynomials of type B
q-log-convexity Schur positivity Pieri's rule the Jacobi-Trudi identity principal specialization Narayana numbers of type B
2014/6/3
We prove a conjecture of Liu and Wang on the q-log-convexity of the Narayana polynomials of type B. By using Pieri's rule and the Jacobi-Trudi identity for Schur functions, we obtain an expansion of a...
Generalized determinantal identities involving Lucas polynomials
Fibonacci number Lucas number Fibonacci polynomial
2010/9/10
Determinants have played a significant part in various areas in mathematics. There are different perspectives on the study of determinants. Many problems on determinants of Fibonacci sequence and Luca...