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1.
The n-point correlation functions introduced by Bloch and Okounkov have already found several geometric connections and algebraic generalizations. In this note we formulate a q,t-deformation of this n-point function. The key operator used in our formulation arises from the theory of Macdonald polynomials and affords a vertex operator interpretation. We obtain closed formulas for the n-point functions when n = 1,2 in terms of the basic hypergeometric functions. We further generalize the q,t-deformed n-point function to more general vertex operators.  相似文献   

2.
For any homomorphism V on the space of symmetric functions, we introduce an operation that creates a q-analog of V. By giving several examples we demonstrate that this quantization occurs naturally within the theory of symmetric functions. In particular, we show that the Hall-Littlewood symmetric functions are formed by taking this q-analog of the Schur symmetric functions and the Macdonald symmetric functions appear by taking the q-analog of the Hall-Littlewood symmetric functions in the parameter t. This relation is then used to derive recurrences on the Macdonald q,t-Kostka coefficients.RésuméPour un homomorphisme V sur l'espace des fonctions symétriques, nous présentons une opération qui crée un q-analogue de V. En donnant plusieurs exemples nous démontrons que cette quantization se produit naturellement dans la théorie de fonctions symétriques. En particulier, nous prouvons que les fonctions symétriques de Hall-Littlewood sont constituées en prenant ce q-analogue des fonctions symétriques de Schur et les fonctions symétriques de Macdonald apparaissent en prenant le q-analogue des fonctions symétriques de Hall-Littlewood dans le paramètre t. Cette relation est alors employée pour dériver des récurrence sur les coefficients Macdonald q,t-Kostka.  相似文献   

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We introduce deformations of Kazhdan-Lusztig elements and specialised non-symmetric Macdonald polynomials, both of which form a distinguished basis of the polynomial representation of a maximal parabolic subalgebra of the Hecke algebra. We give explicit integral formula for these polynomials, and explicitly describe the transition matrices between classes of polynomials. We further develop a combinatorial interpretation of homogeneous evaluations using an expansion in terms of Schubert polynomials in the deformation parameters.  相似文献   

5.
The theory of congruences for the partition function p(n) depends heavily on the properties of half-integral weight Hecke operators. The subject has been complicated by the absence of closed formulas for the Hecke images P(z)|T(?2), where P(z) is the relevant modular generating function. We obtain such formulas using Euler?s Pentagonal Number Theorem and the denominator formula for the Monster Lie algebra. As a corollary, we obtain congruences for certain powers of Ramanujan?s Delta-function.  相似文献   

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We present a generic operator J defined on the vectorial space of polynomial functions and we address the problem of finding the polynomial sequences that coincide with the (normalized) J-image of themselves. The technique developed assembles different types of operators and initiates with a transposition of the problem to the dual space. We provide examples for a J limited to three terms.  相似文献   

8.
We study integral refinable operators of integral type exact on polynomials of even degree constructed by using refinable B-bases of GP type. We prove a general theorem of existence and uniqueness. Then we study the LpLp-norm of these operators and we give error bounds in approximating functions and their derivatives belonging to suitable classes. Numerical results and comparisons with other quasi-interpolatory operators having the same order of exactness on polynomial reproduction are presented.  相似文献   

9.
We consider the polynomials orthogonal with respect to the Sobolev type inner product

where and is a nonnegative integer. It is the purpose of this paper to show that these polynomials are eigenfunctions of a class of linear differential operators containing one that is of finite order if is a nonnegative integer and

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We find the raising and lowering operators for orthogonal polynomials on the unit circle introduced by Szego and for their four parameter generalization to biorthogonal rational functions on the unit circle.

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13.
As an example of a problem involving two finite contiguous domains, the boundary element method is used to compute the eddy current density in a conductor subjected to a transverse magnetic field.  相似文献   

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For extra-large Coxeter systems (m(s,r)>3), we construct a natural and explicit set of Soergel bimodules D={Dw}wW such that each Dw contains as a direct summand (or is equal to) the indecomposable Soergel bimodule Bw. When decategorified, we prove that D gives rise to a set {dw}wW that is actually a basis of the Hecke algebra. This basis is close to the Kazhdan–Lusztig basis and satisfies a positivity condition.  相似文献   

16.
We develop a factorization method for q-Racah polynomials. It is inspired by the approach to q-Hahn polynomials based on the q-Johnson scheme, but we do not use association scheme theory nor Gel'fand pairs but only manipulation of q-difference operators.  相似文献   

17.
An infinite summation formula of Hall-Littlewood polynomials due to Kawanaka is generalized to a finite summation formula, which implies, in particular, twelve more multiple q-identities of Rogers-Ramanujan type than those previously found by Stembridge and the last two authors.  相似文献   

18.
Consider the space of Drinfeld modular forms of fixed weight and type for Γ0(n)⊂GL2(Fq[T]). It has a linear form bn, given by the coefficient of tm+n(q−1) in the power series expansion of a type m modular form at the cusp infinity, with respect to the uniformizer t. It also has an action of a Hecke algebra. Our aim is to study the Hecke module spanned by b1. We give elements in the Hecke annihilator of b1. Some of them are expected to be nontrivial and such a phenomenon does not occur for classical modular forms. Moreover, we show that the Hecke module considered is spanned by coefficients bn, where n runs through an infinite set of integers. As a consequence, for any Drinfeld Hecke eigenform, we can compute explicitly certain coefficients in terms of the eigenvalues. We give an application to coefficients of the Drinfeld Hecke eigenform h.  相似文献   

19.
This paper is an attempt to stress the usefulness of the operational methods in the theory of special functions. Using operational methods, we derive summation formulae and generating relations involving various forms of Hermite-base polynomials and functions.  相似文献   

20.
In the present work, we introduce and study essentially a class of multi-dimensional modified fractional calculus operators involving a general class of polynomials in the kernel. These operators are considered in the space of functionsM γ (R + n ). Some mapping properties and fractional differential formulas are obtained. Also images of some elementary and special functions are established.  相似文献   

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