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We study the distribution theory of some statistics related to records from a sequence of geometrically distributed random variables. The main novelty in this study is the introduction of q-calculus techniques. We obtain representations for the probability functions of some variables of interest, such as, for instance, record times, number of records, inter-record times, record indicators, etc., in terms of multiple q-integrals. It is remarkable to note that the expressions thus obtained are q-analogues of the corresponding well-known results in the absolutely continuous case. We also explore a duality between the results in the case of weak and ordinary records.

In this paper, the new version of the celebrated Montgomery identity is determined via quantum integral operators. By using it, certain quantum integral inequalities of Ostrowski type are established. Moreover, the relevant connection of the obtained results of this work with the derived results in previously published works is discussed.

We disprove and correct some recently obtained results regarding Montgomery identity for quantum integral operator and Ostrowski type inequalities involving convex functions., Comment: 5 pages

Using the Jackson integral, we obtain the q-integral analogue of Raabe's 1843 formula for Barnes multiple Hurwitz–Lerch zeta functions and Barnes and Vardi's multiple gamma functions. Our results generalize q-integral analogue of the Raabe type formulas for the Hurwitz zeta functions and log gamma functions in [9] . During the proof we also obtain a new formula on the relationship between the higher and lower orders Hurwitz zeta functions.

We provide an explicit expression for the first order q-difference system for the Jackson integral of symmetric Selberg type. The q-difference system gives a generalization of q-analog of contiguous relations for the Gauss hypergeometric function. As a basis of the system we use a set of the symmetric polynomials introduced by Matsuo in his study of the q-KZ equation. Our main result is an explicit expression for the coefficient matrix of the q-difference system in terms of its Gauss matrix decomposition. We introduce a class of symmetric polynomials called interpolation polynomials, which includes Matsuo's polynomials. By repeated use of three-term relations among the interpolation polynomials we compute the coefficient matrix.

In the current paper, we examine the $(p,q)$ -analogue of Kantorovich type Lupas–Schurer operators with the help of $(p,q)$ -Jackson integral. Then, we estimate the rate of convergence for the constructed operators by using the modulus of continuity in terms of a Lipschitz class function and by means of Peetre’s K-functionals based on Korovkin theorem. Moreover, we illustrate the approximation of the $(p,q)$ -Lupas–Schurer–Kantorovich operators to appointed functions by the help of Matlab algorithm and then show the comparison of the convergence of these operators with Lupas–Schurer operators based on $(p,q)$ -integers.

We investigate the connection problem for the Jackson integral of type $A_n$. Our connection formula implies a Slater type expansion of a bilateral multiple basic hypergeometric series as a linear combination of several specific multiple series. Introducing certain elliptic Lagrange interpolation functions, we determine the explicit form of the connection coefficients. We also use basic properties of the interpolation functions to establish an explicit determinant formula for a fundamental solution matrix of the associated system of $q$-difference equations.

A bosonization of the quantum affine superalgebra \(U_q(\widehat{sl}(M|N))\) is presented for an arbitrary level \(k \in \mathbf{C}\). The Wakimoto realization is given by using \(\xi -\eta \) system. The screening operators that commute with \(U_q(\widehat{sl}(M|N))\) are presented for the level \(k \ne -M+N\). New bosonization of the affine superalgebra \(\widehat{sl}(M|N)\) is obtained in the limit \(q \rightarrow 1\).

The purpose of this paper is to present a characterization of a certain Peetre K-functional in Lp[−1,1] norm, for 1 ≤ p ≤ 2 by means of a modulus of smoothness. This modulus is based on the classical one taken on a certain linear transform of the function., 2000 Mathematics Subject Classification: 41A25, 41A36.

This is the second part of the project "unified theory of classical orthogonal polynomials of a discrete variable derived from the eigenvalue problems of Hermitian matrices." In a previous paper, orthogonal polynomials having Jackson integral measures were not included, since such measures cannot be obtained from single infinite dimensional Hermitian matrices. Here we show that Jackson integral measures for the polynomials of the big q-Jacobi family are the consequence of the recovery of self-adjointness of the unbounded Jacobi matrices governing the difference equations of these polynomials. The recovery of self-adjointness is achieved in an extended '2 Hilbert space on which a direct sum of two unbounded Jacobi matrices acts as a Hamiltonian or a difference Schrodinger operator for an infinite dimensional eigenvalue problem. The polynomial appearing in the upper/lower end of the Jackson integral constitutes the eigenvector of each of the two-unbounded Jacobi matrix of the direct sum. We also point out that the orthogonal vectors involving the q-Meixner (q-Charlier) polynomials do not form a complete basis of the '2 Hilbert space, based on the fact that the dual q-Meixner polynomials introduced in a previous paper fail to satisfy the orthogonality relation. The complete set of eigenvectors involving the q-Meixner polynomials is obtained by constructing the duals of the dual q-Meixner polynomials which require the two-component Hamiltonian formulation. An alternative solution method based on the closure relation, the Heisenberg operator solution, is applied to the polynomials of the big q-Jacobi family and their duals and q-Meixner (q-Charlier) polynomials. Published by AIP Publishing., Article, JOURNAL OF MATHEMATICAL PHYSICS. 59(1):013504 (2018)

In this paper we extend our previous results concerning Jackson integral solutions of the boundary quantum Knizhnik-Zamolodchikov equations with diagonal K-operators to higher-spin representations of quantum affine $\mathfrak{sl}_2$. First we give a systematic exposition of known results on $R$-operators acting in the tensor product of evaluation representations in Verma modules over quantum $\mathfrak{sl}_2$. We develop the corresponding fusion of $K$-operators, which we use to construct diagonal $K$-operators in these representations. We construct Jackson integral solutions of the associated boundary quantum Knizhnik-Zamolodchikov equations and explain how in the finite-dimensional case they can be obtained from our previous results by the fusion procedure., 36 pages; some small additions and corrections concerning mero-uniform convergence (Defn. 6.1) and rectified some notation issues for the function \mathcal{Y} (p21 and onwards). appears in Annales Henri Poincar\'e, 2014

A Jackson integral of type BC n is a multisum generalization of the very-well-poised-balanced ψ 2 r 2 r basic hypergeometric series. We state an explicit product formula for the determinant of a matrix with entries given by the BC n type Jackson integrals. In order to show this, we treat the determinant as a solution of a holonomic q-difference equation. In particular we give the q-difference equation explicitly as a two-term recurrence relation, which the determinant satisfies, by introducing a set of new symmetric polynomials via the symplectic Schur functions.

We derive a $$q$$ -generalization of Euler’s decomposition formula for the $$q$$ MZVs recently introduced by Y. Ohno, J. Okuda, and W. Zudilin. This answers a question posed by these authors in J Number Theory 132, 144–155 (2012).

We study a q-difference equation of a BCn-type Jackson integral introduced by van Diejen. The BCn-type Jackson integral is a multiple q-series generalized from a q-analogue of Selberg's integral. The equation is characterized by some new symmetric polynomials defined via symplectic Schur functions. As an application, we give another proof of a product formula for the BCn-type Jackson integral, which is equivalent to the so-called q-Macdonald?Morris identity for the root system BCn first obtained by Gustafson.

For a multivariable q-series called Jackson integral associated with irreducible reduced root systems, a sufficient condition for convergence of it with respect to parameters is given. Its asymptotic behavior as a function of its parameters is studied. For its application, we give another proof of G2-type summation formula investigated by Gustafson in an appendix.

In this paper we study the q-analogue of the jα Bessel function (see (1)) which results after minor changes from the so-called Exton function studied by Koornwinder and Swarttow. Our objective is first to establish, using only the q -Jackson integral and the q-derivative, some properties of this function with proofs similar to the classical case; second to construct the associated q-Fourier analysis which will be used in a coming work to construct the q-analogue of the Bessel-hypergroup.

Recently, Kim and Oh expressed the Selberg integral in terms of the number of Young books which are a generalization of standard Young tableaux of shifted staircase shape. In this paper the generating function for Young books according to major index statistic is considered. It is shown that this generating function can be written as a Jackson integral which gives a new $q$-Selberg integral. It is also shown that the new $q$-Selberg integral has an expression in terms of Schur functions., 14 pages, 3 figures

The Ramanujan $_{1} \psi_{1}$ summation theorem is studied from the perspective of Jackson integrals, $q$-difference equations and connection formulae. This is an approach which has previously been shown to yield Bailey’s very-well-poised $_{6} \psi_{6}$ summation. Bilateral Jackson integral generalizations of the Dixon–Anderson and Selberg integrals relating to the type $A$ root system are identified as natural candidates for multidimensional generalizations of the Ramanujan $_{1} \psi_{1}$ summation theorem. New results of this type are announced, and furthermore they are put into context by reviewing from previous literature explicit product formulae for Jackson integrals relating to other roots systems obtained from the same perspective.

We state certain product formulae for Jackson integrals associated with irreducible reduced root systems. The Jackson integral is defined here as a sum over any full-rank sublattice of the coweight lattice for the root system. In particular, a Weyl group symmetry classification of the Jackson integrals is done when they have an expression of a product of the Jacobi elliptic theta functions. Most of the product formulae investigated by Aomoto, Macdonald and Gustafson appear in the list of classifications. A new product formula for an F4 root system is included in it.

We exhibit the double q-shuffle structure for the qMZVs recently introduced by Y. Ohno, J. Okuda and W. Zudilin., Comment: Submitted. Added paragraphs (5.2 and 5.3) on handling non-modified qMZVs. Red internal comment removed

In a new functional integral approach proposed for the model, we find the regime with a deformed integration measure in which the standard integral is replaced with the Jackson integral. We indicate the relation to a p-adic functional integral. For the magnetic and electronic subsystems in the effective functional that results from the operator formulation of the Hubbard model, we find the two-parametric quantum derivative resulting in the appearance of the quantum SU_{rq}(2) group. We establish the relation to the one-parametric quantum derivative and the standard derivative., 18 pages

In this note, we define a $q$-analogue of the Mahler measures by using the Jackson integral which we call the Jackson $q$-Mahler measures. Especially we study their classical limit for polynomials of one variable.

We introduce the (global) q-Whittaker function as the limit at t=0 of the q,t-spherical function extending the symmetric Macdonald polynomials to arbitrary eigenvalues. The construction heavily depends on the technique of the q-Gaussians developed by the author (and Stokman in the non-reduced case). In this approach, the q-Whittaker function is given by a series convergent everywhere, a kind of generating function for multi-dimensional q-Hermite polynomials (closely related to the level 1 Demazure characters). One of the applications is a q-version of the Shintani- Casselman- Shalika formula, which appeared directly connected with q-Mehta- Macdonald identities in terms of the Jackson integrals. This formula generalizes that of type A due to Gerasimov et al. to arbitrary reduced root systems. At the end of the paper, we obtain a q,t-counterpart of the Harish-Chandra asymptotic formula for the spherical functions, including the Whittaker limit., Comment: V2: a discussion of the one-dimensional case was added. V3: Jackson integration and growth estimates were added. V4: a q-variant of the Harish-Chandra asymptotic formula for spherical functions was added. V5: editing, some improvements, adding references. V6: General editing

We present an explicit expression for the $q$-difference system, which the $BC_1$-type Jackson integral ($q$-series) satisfies, as first order simultaneous $q$-difference equations with a concrete basis. As an application, we give a simple proof for the hypergeometric summation formula introduced by Gustafson and the product formula of the $q$-integral introduced by Nassrallah-Rahman and Gustafson.

We review the construction of a q-analogue of the Gaussian measure. We apply that construction to obtain a q-analogue of Feynman integrals and to compute explicitly an example of such integrals., Comment: 5 figures, 10 pages

Fix a complex numberqwith|q|< 1. LetT1…,Tnbe n-commutingq-difference operators defined byfor a functionf(x), x=(x1,…,xn)ε (C*)n. Consider a system of linearq-difference equations in several variables for a matrix valued functionon(C*)nas follows