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1.
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 qt-Kostka numbers and we show that they are polynomials in qt with integer coefficients by using the strong factorization property of Macdonald polynomials. We conjecture that multivariate qt-Kostka numbers are in fact polynomials in qt with nonnegative integer coefficients, which generalizes the celebrated Macdonald’s positivity conjecture.  相似文献   

2.
The 6j-symbols for representations of the q-deformed algebra of polynomials on \(\mathrm {SU}(2)\) are given by Jackson’s third q-Bessel functions. This interpretation leads to several summation identities for the q-Bessel functions. Multivariate q-Bessel functions are defined, which are shown to be limit cases of multivariate Askey–Wilson polynomials. The multivariate q-Bessel functions occur as 3nj-symbols.  相似文献   

3.
We give a new and elementary proof of Jackson’s terminating q-analogue of Dixon’s identity by using recurrences and induction.  相似文献   

4.
The aim of this paper is to prove an \(\mathcal {L}_q^1 \cap \mathcal {L}_q^2\) versions of Nash and Carlson’s inequalities for a class of q-integral operator \(\mathcal {T}_q\) with a bounded kernel. As applications, we give q-analogues of Nash and Carlson’s inequalities for the q-Fourier-cosine, q-Fourier-sine, q-Dunkl and q-Bessel Fourier transforms.  相似文献   

5.
In the space L p , 1 ≤ p < 2, on the half-line with power weight, Jackson’s inequality between the value of the best approximation of a function by even entire functions of exponential type and its modulus of continuity defined by means of a generalized shift operator is well known. The question of the sharpness of the inequality remained open. For the constant in Jackson’s inequality, we obtain a lower bound, which proves its sharpness.  相似文献   

6.
We investigate two kinds of q-analogues of generalized Stirling numbers. One is a q-analogue of Hsu-Shiue’s generalized Stirling numbers and the other is a q-analogue of Comtet’s numbers. In particular, we derive a q-analogue of an equality of Dobinski-type. Moreover, a determinant of the matrix consisting of the q-analogue of Comtet’s numbers is evaluated.  相似文献   

7.
We study super-braided Hopf algebras Λ primitively generated by finite-dimensional right crossed (or Drinfeld-Radford-Yetter) modules Λ1 over a Hopf algebra A which are quotients of the augmentation ideal A + as a crossed module by right multiplication and the adjoint coaction. Here super-bosonisation Open image in new window provides a bicovariant differential graded algebra on A. We introduce Λ m a x providing the maximal prolongation, while the canonical braided-exterior algebra Λ min = B ?1) provides the Woronowicz exterior calculus. In this context we introduce a Hodge star operator ? by super-braided Fourier transform on B ?1) and left and right interior products by braided partial derivatives. Our new approach to the Hodge star (a) differs from previous approaches in that it is canonically determined by the differential calculus and (b) differs on key examples, having order 3 in middle degree on k[S 3] with its 3D calculus and obeying the q-Hecke relation ?2 = 1 + (q ? q ?1)? in middle degree on k q [S L 2] with its 4D calculus. Our work also provides a Hodge map on quantum plane calculi and a new starting point for calculi on coquasitriangular Hopf algebras A whereby any subcoalgebra \(\mathcal {L}\subseteq A\) defines a sub-braided Lie algebra and \({\Lambda }^{1}\subseteq \mathcal {L}^{*}\) provides the required data A + → Λ1.  相似文献   

8.
We study the q-bracket operator of Bloch and Okounkov, recently examined by Zagier and other authors, when applied to functions defined by two classes of sums over the parts of an integer partition. We derive convolution identities for these functions and link both classes of q-brackets through divisor sums. As a result, we generalize Euler’s classic convolution identity for the partition function and obtain an analogous identity for the totient function. As corollaries, we generalize Stanley’s Theorem on the number of ones in all partitions of n, and provide several new combinatorial results.  相似文献   

9.
Both in Majid's double-bosonization theory and in Rosso's quantum shuffle theory, the rankinductive and type-crossing construction for U_q(g)'s is still a remaining open question. In this paper, working in Majid's framework, based on the generalized double-bosonization theorem we proved before, we further describe explicitly the type-crossing construction of U_q(g)'s for(BCD)_n series directly from type An-1via adding a pair of dual braided groups determined by a pair of(R, R′)-matrices of type A derived from the respective suitably chosen representations. Combining with our results of the first three papers of this series, this solves Majid's conjecture, i.e., any quantum group U_q(g) associated to a simple Lie algebra g can be grown out of U_q(sl_2)recursively by a series of suitably chosen double-bosonization procedures.  相似文献   

10.
We find new sufficient conditions for the existence of a 0’-limitwise monotonic function defining the order for a computable η-like linear order L, i.e., of a function G such that L q∈? G(q). Namely, we define the notions of left local maximal block and right local maximal block and prove that if the sizes of these blocks in a computable η-like linear order L are bounded then there is a 0’-limitwise monotonic function G with L = ∑ q∈? G(q).  相似文献   

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