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In an earlier work, a method was introduced for obtaining indefinite q-integrals of q-special functions from the second-order linear q-difference equations that define them. In this paper, we reformulate the method in terms of q-Riccati equations, which are nonlinear and first order. We derive q-integrals using fragments of these Riccati equations, and here only two specific fragment types are examined in detail. The results presented here are for the q-Airy function, the Ramanujan function, the discrete q-Hermite I and II polynomials, the q-hypergeometric functions, the q-Laguerre polynomials, the Stieltjes-Wigert polynomial, the little q-Legendre and the big q-Legendre polynomials. [ABSTRACT FROM AUTHOR]

AbstractThe fractional q-calculus has attracted the interest of a large number of academics over the last four decades or so, due mainly to a wide range of applications that cover natural sciences to social sciences. Many fractional q-calculus operators, particularly those involving various q-special functions, have been deeply studied and widely applied. In this paper, we aim to establish certain image formulas of Saigo fractional q-integral operators involving the product of generalized q-hypergeometric series and a general class of q-polynomials that are primarily expressed in terms of generalized q-hypergeometric series in a systematic manner. We demonstrate their use by studying q-Konhouser biorthogonal polynomials and q-Jacobi polynomials. Additionally, some fascinating special cases of our main findings are taken into consideration, and pertinent connections between some of the findings presented here and those from earlier studies are also made.

The fractional q-calculus has attracted the interest of a large number of academics over the last four decades or so, due mainly to a wide range of applications that cover natural sciences to social sciences. Many fractional q-calculus operators, particularly those involving various q-special functions, have been deeply studied and widely applied. In this paper, we aim to establish certain image formulas of Saigo fractional q-integral operators involving the product of generalized q-hypergeometric series and a general class of q-polynomials that are primarily expressed in terms of generalized q-hypergeometric series in a systematic manner. We demonstrate their use by studying q-Konhouser biorthogonal polynomials and q-Jacobi polynomials. Additionally, some fascinating special cases of our main findings are taken into consideration, and pertinent connections between some of the findings presented here and those from earlier studies are also made. [ABSTRACT FROM AUTHOR]

First we consider a new trapezium identity for twice differentiable functions in quantum integrals. This identity investigates our main results that consist some integral inequalities of trapezium type using the newly introduced q-derivatives for the m-convex functions over finite intervals of real numbers. Some special cases are discussed in details to support our theoretical results. [ABSTRACT FROM AUTHOR]

Abstract In this article, we introduce a new concept of quantum integrals which is called T q κ 2 ${}^{\kappa _{2}}T_{q}$ -integral. Then we prove several properties of this concept of quantum integrals. Moreover, we present several Hermite–Hadamard type inequalities for T q κ 2 ${}^{\kappa _{2}}T_{q}$ -integral by utilizing differentiable convex functions. The results presented in this article are unification and generalization of the comparable results in the literature.

This paper aims to find Hermite–Hadamard-type inequalities for a generalized notion of integrals called q−h-integrals. Inequalities for q-integrals can be deduced by taking h=0 and are connected with several known results of q-Hermite–Hadamard inequalities. In addition, we analyzed q−h-integrals, q-integrals, and the corresponding inequalities for symmetric functions.

In this article, we introduce a new concept of quantum integrals which is called T q κ 2 -integral. Then we prove several properties of this concept of quantum integrals. Moreover, we present several Hermite–Hadamard type inequalities for T q κ 2 -integral by utilizing differentiable convex functions. The results presented in this article are unification and generalization of the comparable results in the literature. [ABSTRACT FROM AUTHOR]

We have considered three independent variable generalization of the third order mock theta functions. We have given some identities for these functions. We have also given q-integral representation and multibasic expansion for these functions. [ABSTRACT FROM AUTHOR]

In this paper, we give some new definitions for quantum integrals of two-variable functions. We establish quantum Hermite–Hadamard-type inequalities for coordinated convex functions via newly defined quantum integrals. [ABSTRACT FROM AUTHOR]

This article is refer to the study of q analogue, which play significant role in number theory and combinatorics of annihilation. Here, we aim to presenting a new definition of (q,k)-Mittag Leffler function and discuss it various analytical properties. Certain special cases are derived, from the obtained results. And, we also point out their relevance with known results. [ABSTRACT FROM AUTHOR]

Given a q-integrable function f on [0,∞), we define s(x) = ∫0x f(t)dqt and σ(s(x)) = 1/x ∫0x s(t)dqt for x > 0. It is known that if limx→∞ s(x) exists and is equal to A, then limx→∞ σ(s(x)) = A. But the converse of this implication is not true in gen- eral. Our goal is to obtain Tauberian conditions imposed on the general control modulo of s(x) under which the converse implication holds. These conditions generalize some previously obtained Tauberian conditions. [ABSTRACT FROM AUTHOR]

This paper contains a brief review of orthogonal polynomials in two and several variables. It supplements the Koornwinder survey [40]. Several recently discovered systems of orthogonal polynomials have been treated in this work. We did not provide any proofs of the theorem presented here but references to the original sources are given for the benefit of the interested reader. It is hoped that collecting these scattered results in one place will make them accessible for the user.

In this paper, we study the existence of the solutions of some q-difference equations. The solutions of some nonlinear of q-difference equations of order four are obtained. Also, we find the solutions of some non homogeneous linear q-difference equations with constant and variables coefficiensis. [ABSTRACT FROM AUTHOR]

We construct q-B-splines using a new form of truncated power functions. We give basic properties to show that q-B-splines form a basis for quantum spline spaces. On the other hand, we derive algorithmic formulas for 1/q-integration and 1/q-differentiation for q-spline functions. Moreover, we show a way to find the polynomial pieces on each interval of a q-spline function.

A q-integral is a definite integral of a function of q having an expansion in non-negative powers of q for | q | < 1 {|q|<1} (q-series). In his book on hypergeometric series, N. J. Fine [N. J. Fine, Basic Hypergeometric Series and Applications, Math. Surveys Monogr. 27, American Mathematical Society, Providence, 1988] explicitly evaluated three q-integrals. For example, he showed that ∫ 0 e - π ∏ n = 1 ∞ ( 1 - q 2 ⁢ n ) 20 ( 1 - q n ) 16 ⁢ d ⁢ q = 1 16. \int_{0}^{e^{-\pi}}\prod_{n=1}^{\infty}\frac{(1-q^{2n})^{20}}{(1-q^{n})^{16}}% dq=\frac{1}{16}. In this paper, we prove a general theorem which allows us to determine a wide class of integrals of this type. This class includes the three q-integrals evaluated by Fine as well as some of those evaluated by L.-C. Zhang [L.-C. Zhang, Some q-integrals associated with modular forms, J. Math Anal. Appl. 150 1990, 264–273]. It also includes many new evaluations of q-integrals. [ABSTRACT FROM AUTHOR]

We introduce two q-analogues of the 2D-Hermite polynomials which are functions of two complex variables. We derive explicit formulas, orthogonality relations, raising and lowering operator relations, generating functions, and Rodrigues formulas for both families. We also introduce a q-2D analogue of the disk polynomials (Zernike polynomials) and derive similar formulas for them as well, including evaluating certain connection coefficients. Some of the generating functions may be related to Rogers-Ramanujan type identities. [ABSTRACT FROM AUTHOR]

This paper contains a brief review of orthogonal polynomials in two and several variables. It supplements the Koornwinder survey [40]. Several recently discovered systems of orthogonal polynomials have been treated in this work. We did not provide any proofs of the theorem presented here but references to the original sources are given for the benefit of the interested reader. It is hoped that collecting these scattered results in one place will make them accessible for the user. [ABSTRACT FROM AUTHOR]

Ramanujan's reciprocity theorem may be considered as a three-variable extension of Jacobi's triple product identity. Using the method of q-partial differential equations, we extend Ramanujan's reciprocity theorem to a seven-variable reciprocity formula. Using the same method, the Andrews-Askey integral formula is extended to a q-integral formula which has seven parameters with base q. [ABSTRACT FROM AUTHOR]

Abstract: In the present paper we propose the q analogue of the modified Beta operators. We apply q-derivatives to obtain the central moments of the discrete q-Beta operators. A direct result in terms of modulus of continuity for the q operators is also established. We have also used the properties of q integral to establish the recurrence formula for the moments of q analogue of the modified Beta operators. We also establish an asymptotic formula. In the end we have also present the modification of such q operators so as to have better estimate. [Copyright &y& Elsevier]

Abstract: A q-analogue of Wignersʼ 9-j symbols was found by the author in a recent paper where their orthogonality was also established. In this work we introduce a continuous version of these functions and prove their orthogonality with respect to a 2-dimensional extension of the Askey–Wilson weight. [Copyright &y& Elsevier]

Abstract: Recently we discovered several new Erdélyi type integrals. In the present paper, it is shown how the q-extensions of all those integrals involving and representing certain q-hypergeometric functions can be developed. The well-known special cases and applications of these q-integrals are also pointed out. [Copyright &y& Elsevier]

In an attempt to find a q-analogue of Weber and Schafheitlin's integral ∫∞ 0 x−ρ Jμ ( ax) Jν ( bx) dx which is discontinuous on the diagonal a = b the integral ∫∞ 0 x−ρ J(2) ν ( a(1 − q) x; q) J(1) μ ( b(1 − q) x; q) dx is evaluated where J(1) μ ( x; q) and J(2) μ ( x; q) are two of Jackson's three q-Bessel functions. It is found that the question of discontinuity becomes irrelevant in this case. Evaluations of this integral are also made in some interesting special cases. A biorthogonality formula is found as well as a Neumann series expansion for xρ in terms of J(2) ν+1+2 n ((1 − q) x; q). Finally, a q-Lommel function is introduced. [ABSTRACT FROM AUTHOR]

A q-integral representation of Rogers' q-ultraspherical polynomials C (x;β∥q) is obtained by using Sears' summation formula for balanced non-terminating φ series. It is then used to give a simple derivation of the Gasper-Rahman formula for the Poisson kernel of C (x;β∥q). As another application it is shown how this representation can be directly used to give an asymptotic expansion of the q-ultraspherical polynomials. [ABSTRACT FROM AUTHOR]

We construct q-B-splines using a new form of truncated power functions. We give basic properties to show that q-B-splines form a basis for quantum spline spaces. On the other hand, we derive algorithmic formulas for 1 / q -integration and 1 / q -differentiation for q-spline functions. Moreover, we show a way to find the polynomial pieces on each interval of a q-spline function. [ABSTRACT FROM AUTHOR]