Self-reciprocal polynomials and coterm polynomials

We classify all self-reciprocal polynomials arising from reversed Dickson polynomials over \(\mathbb {Z}\) and \(\mathbb {F}_p\), where p is prime. As a consequence, we also obtain coterm polynomials arising from reversed Dickson polynomials.     

Addition theorems for the Appell polynomials and the associated classes of polynomial expansions

Various interesting and potentially useful properties and relationships involving the Bernoulli, Euler and Genocchi polynomials have been investigated in the literature rather extensively. Recently, the present authors (Srivastava and Pinter in Appl Math Lett 17:375–380, 2004) obtained addition theorems and other relationships involving the generalized Bernoulli polynomials ${B_n^{(\alpha)}(x)}$ and the generalized Euler polynomials ${E_n^{(\alpha)}(x)}$ of order α and degree n in x. The main purpose of this sequel to some of the aforecited investigations is to give several addition formulas for a general class of Appell sequences. The addition formulas, which are derived in this paper, involve not only the generalized Bernoulli polynomials ${B_n^{(\alpha)}(x)}$ and the generalized Euler polynomials ${E_n^{(\alpha)}(x)}$ , but also the generalized Genocchi polynomials ${G_n^{(\alpha)}(x)}$ , the Srivastava polynomials ${\mathcal{S}_{n}^{N}\left( x\right)}$ , several general families of hypergeometric polynomials and such orthogonal polynomials as the Jacobi, Laguerre and Hermite polynomials. Some umbral-calculus generalizations of the addition formulas are also investigated.     

The symmetric Meixner-Pollaczek polynomials with real parameter

The Symmetric Meixner-Pollaczek polynomials for λ>0 are well-studied polynomials. These are polynomials orthogonal on the real line with respect to a continuous, positive real measure. For λ?0, are also polynomials, however they are not orthogonal on the real line with respect to any real measure. This paper defines a non-standard inner product with respect to which the polynomials for λ?0, become orthogonal polynomials. It examines the major properties of the polynomials, for λ>0 which are also shared by the polynomials, for λ?0.     

An explicit formula for generalized potential polynomials and its applications

We define the generalized potential polynomials associated to an independent variable, and prove an explicit formula involving the generalized potential polynomials and the exponential Bell polynomials. We use this formula to describe closed type formulas for the higher order Bernoulli, Eulerian, Euler, Genocchi, Apostol-Bernoulli, Apostol-Euler polynomials and the polynomials involving the Stirling numbers of the second kind. As further applications, we derive several known identities involving the Bernoulli numbers and polynomials and Euler polynomials, and new relations for the higher order tangent numbers, the higher order Bernoulli numbers of the second kind, the numbers , the higher order Bernoulli numbers and polynomials and the higher order Euler polynomials and their coefficients.     

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.     

New methods for generating permutation polynomials over finite fields

We present two methods for generating linearized permutation polynomials over an extension of a finite field Fq. These polynomials are parameterized by an element of the extension field and are permutation polynomials for all nonzero values of the element. For the case of the extension degree being odd and the size of the ground field satisfying , these parameterized linearized permutation polynomials can be used to derive non-parameterized nonlinear permutation polynomials via a recent result of Ding et al.     

Several explicit formulae of Cauchy polynomials in terms of {r}-Stirling numbers

The integer values of Cauchy polynomials are expressed in terms of \({r}\)-Stirling numbers of the first kind. Several relations between the integral values of Bernoulli polynomials and those of Cauchy polynomials are obtained in terms of \({r}\)-Stirling numbers of both kinds. Also, we find a relation between the Cauchy polynomials and hyperharmonic numbers.     

Some new expansions for certain truncated <Emphasis Type="Italic">q</Emphasis>-series

We introduce the notion of \(\mathcal {R}_{\mu }\)-classical orthogonal polynomials, where \(\mathcal {R}_{\mu }\) is the degree raising shift operator for the sequence of Laguerre polynomials of parameter \(\mu \). Then we show that the Laguerre polynomials \(L^{(\mu )}_n(x), \ \mu \ne -m, \ m\ge 0\), are the only \(\mathcal {R}_{\mu }\)-classical orthogonal polynomials.     

Some polynomials associated with the $${\vavec{}}$$-Whitney numbes

In the present article, we study three families of polynomials associated with the r-Whitney numbers of the second kind. They are the r-Dowling polynomials, r-Whitney–Fubini polynomials and the r-Eulerian–Fubini polynomials. Then we derive several combinatorial results by using algebraic arguments (Rota’s method), combinatorial arguments (set partitions) and asymptotic methods.     

Explicit factorizations of cyclotomic polynomials over finite fields

Let q be a prime power and let \({\mathbb {F}}_q\) be a finite field with q elements. This paper discusses the explicit factorizations of cyclotomic polynomials over \(\mathbb {F}_q\). Previously, it has been shown that to obtain the factorizations of the \(2^{n}r\)th cyclotomic polynomials, one only need to solve the factorizations of a finite number of cyclotomic polynomials. This paper shows that with an additional condition that \(q\equiv 1 \pmod p\), the result can be generalized to the \(p^{n}r\)th cyclotomic polynomials, where p is an arbitrary odd prime. Applying this result we discuss the factorization of cyclotomic polynomials over finite fields. As examples we give the explicit factorizations of the \(3^{n}\)th, \(3^{n}5\)th and \(3^{n}7\)th cyclotomic polynomials.     

On shifted Eisenstein polynomials

We study polynomials with integer coefficients which become Eisenstein polynomials after the additive shift of a variable. We call such polynomials shifted Eisenstein polynomials. We determine an upper bound on the maximum shift that is needed given a shifted Eisenstein polynomial and also provide a lower bound on the density of shifted Eisenstein polynomials, which is strictly greater than the density of classical Eisenstein polynomials. We also show that the number of irreducible degree \(n\) polynomials that are not shifted Eisenstein polynomials is infinite. We conclude with some numerical results on the densities of shifted Eisenstein polynomials.     

Connection coefficients for Laguerre-Sobolev orthogonal polynomials

Laguerre-Sobolev polynomials are orthogonal with respect to an inner product of the form , where α>−1, λ?0, and , the linear space of polynomials with real coefficients. If dμ(x)=xαe−xdx, the corresponding sequence of monic orthogonal polynomials {Q_n^(α,λ)(x)} has been studied by Marcellán et al. (J. Comput. Appl. Math. 71 (1996) 245-265), while if dμ(x)=δ(x)dx the sequence of monic orthogonal polynomials {L_n^(α)(x;λ)} was introduced by Koekoek and Meijer (SIAM J. Math. Anal. 24 (1993) 768-782). For each of these two families of Laguerre-Sobolev polynomials, here we give the explicit expression of the connection coefficients in their expansion as a series of standard Laguerre polynomials. The inverse connection problem of expanding Laguerre polynomials in series of Laguerre-Sobolev polynomials, and the connection problem relating two families of Laguerre-Sobolev polynomials with different parameters, are also considered.     

Semigroups associated to generalized polynomials and some classical formulas

We study operator semigroups associated with a family of generalized orthogonal polynomials with Hermitian matrix entries. For this we consider a Markov generator sequence, and therefore a Markov semigroup, for the family of orthogonal polynomials on related to the generalized polynomials. We give an expression of the infinitesimal generator of this semigroup and under the hypothesis of diffusion we prove that this semigroup is also Markov. We also give expressions for the kernel of this semigroup in terms of the one-dimensional kernels and obtain some classical formulas for the generalized orthogonal polynomials from the correspondent formulas for orthogonal polynomials on .     

The Fourth Order Difference Equation for the Laguerre-Hahn Polynomials Orthogonal on Special Non-uniform Lattices

We firstly establish the fourth order difference equation satisfied by the Laguerre-Hahn polynomials orthogonal on special non-uniform lattices in general case, secondly give it explicitly for the cases of polynomials r-associated to the classical polynomials orthogonal on linear, q-linear and q-nonlinear (Askey-Wilson) lattices, and thirdly give it semi-explicitly for the class one Laguerre-Hahn polynomials orthogonal on linear lattice.     

Strong factorization property of Macdonald polynomials and higher-order Macdonald’s positivity conjecture

We prove a strong factorization property of interpolation Macdonald polynomials when q tends to 1. As a consequence, we show that Macdonald polynomials have a strong factorization property when q tends to 1, which was posed as an open question in our previous paper with Féray. Furthermore, we introduce multivariate q, t-Kostka numbers and we show that they are polynomials in q, t with integer coefficients by using the strong factorization property of Macdonald polynomials. We conjecture that multivariate q, t-Kostka numbers are in fact polynomials in q, t with nonnegative integer coefficients, which generalizes the celebrated Macdonald’s positivity conjecture.     

On generalized Fibonacci and Lucas polynomials

Let h(x) be a polynomial with real coefficients. We introduce h(x)-Fibonacci polynomials that generalize both Catalan’s Fibonacci polynomials and Byrd’s Fibonacci polynomials and also the k-Fibonacci numbers, and we provide properties for these h(x)-Fibonacci polynomials. We also introduce h(x)-Lucas polynomials that generalize the Lucas polynomials and present properties of these polynomials. In the last section we introduce the matrix Q_h(x) that generalizes the Q-matrix whose powers generate the Fibonacci numbers.     

Central polynomials and growth functions

The growth of central polynomials for the algebra of n × n matrices in characterstic zero was studied by Regev in [13]. Here we study the growth of central polynomials for any finite-dimensional algebra over a field of characteristic zero. For such an algebra A we prove the existence of two limits called the central exponent and the proper central exponent of A. They give a measure of the exponential growth of the central polynomials and the proper central polynomials of A. We study the range of such limits and we compare them with the PI-exponent of the algebra.