相关论文: The Bailey lemma and Kostka polynomials
We show that explicit forms for certain polynomials~$\psi^{(a)}_m(n)$ with the property \[ \psi^{(a+1)}_m(n) = \sum_{\nu=1}^n \psi_m^{(a)}(\nu) \] can be found (here, $a,m,n\in\mathbb{N}_0$). We use these polynomials as a basis to express…
A combinatorial theory for type $R_I$ orthogonal polynomials is given. The ingredients include weighted generalized Motzkin paths, moments, continued fractions, determinants, and histories. Several explicit examples in the Askey scheme are…
Double Kostka polynomials are polynomials indexed by a pair of double partitions. As in the ordinary case, double Kostka polynomials are defined in terms of Schur functions and Hall-Littlewood functions associated to double partitions. In…
Elementary transformations of equations $A\psi=\lambda\psi$ are considered. The invertibility condition (Theorem 1) is established and similar transformations of Riccati equations in the case of second order differential operator $A$ are…
Recently, it has been shown by Ighachanea and Akkouchia \cite{0.1} that using binomial coefficients, one can derive some new refinements of Holder's inequalities. This inequalities then can be applied to a wide class of special functions…
We construct a filtration on integrable highest weight module of an affine Lie algebra whose adjoint graded quotient is a direct sum of global Weyl modules. We show that the graded multiplicity of each Weyl module there is given by a…
Associated to each random variable $Y$ having a finite moment generating function, we introduce a different generalization of the Stirling numbers of the second kind. Some characterizations and specific examples of such generalized numbers…
We study inverse factorial series and their relation to Stirling numbers of the first kind. We prove a special representation of the polylogarithm function in terms of series with such numbers. Using various identities for Stirling numbers…
In this paper we derive generalizations of different properties of monic polynomial families of binomial type, i.e. families of monic polynomials, for which the binomial theorem holds $$ p_n(\alpha+\beta)=\sum_{k=0}^n…
Determining the explicit forms and modularity for string functions and branching coefficients for Kac--Moody algebras after Kac, Peterson, and Wakimoto is a long-standing, yet wide-open, problem and recently a connection has been made…
The development of high-degree interpolation polynomials which use the values of the function and its subsequent derivatives is reformulated. Also, we present a variant of new formula in barycentric form.
A classification of ordinary differential equations and finite-difference equations in one variable having polynomial solutions (the generalized Bochner problem) is given. The method used is based on the spectral problem for a polynomial…
A bracket polynomial on the integers is a function formed using the operations of addition, multiplication and taking fractional parts. For a fairly large class of bracket polynomials we show that if p is a bracket polynomial of degree k-1…
In this article, we give an account of some recent irreducibility testing criteria for polynomials having integer coefficients over the field of rational numbers.
Departing from a class of infinite series with central binomial coefficients in the numerator and depending on a positive integer parameter, we first extend known identities to all complex parameters. Then we use various methods, including…
We prove a conjecture of Lecouvey, which proposes a closed, positive combinatorial formula for symplectic Kostka-Foulkes polynomials, in the case of rows of arbitrary weight. To show this, we construct a new algorithm for computing…
We prove an inverse relation and a family of convolution formulas involving partial Bell polynomials. Known and some presumably new combinatorial identities of convolution type are discussed. Our approach relies on an interesting…
We introduce iterated beta integrals, a new class of iterated integrals on the universal abelian covering of the punctured projective line that unifies hyperlogarithms and classical beta integrals while preserving their fundamental…
The generalized Kostka polynomials are the Poincare polynomials of isotypic components of certain graded GL(n)-modules. The former satisfy a monotonicity property arising from natural surjections of the corresponding modules. This…
We study a refined version of the Linnik problem on the asymptotic behavior of the number of representations of integer $m$ by an integral polynomial as $m$ tends to infinity. We assume that the polynomial arises from invariant theory, and…