Search Results (13)

In the geometric function theory of complex analysis, the investigation of the geometric properties of analytic functions using q-analogues of differential and integral operators is an important area of study, offering powerful tools for applications in numerical analysis and the solution of differential equations. Many topics, including complex analysis, hypergeometric series, and particle physics, have been generalized in q-calculus. In this study, first of all, we define the q-analogues of a differential operator ($D{R}_{\lambda ,q}^{m,n}$) by using the basic idea of q-calculus and the definition of convolution. Additionally, using the newly constructed operator ($D{R}_{\lambda ,q}^{m,n}$), we establish the q-analogues of two new integral operators ($F_{\lambda ,{\gamma }_{1,}{\gamma }_{2,\dots }{\gamma }_l}^{m,n,q}$ and $G_{\lambda ,{\gamma }_{1,}{\gamma }_{2,\dots }{\gamma }_l}^{m,n,q}$), and by employing these operators, new subclasses of the q-starlike and q-convex functions are defined. Sufficient conditions for the functions (f) that belong to the newly defined classes are investigated. Additionally, certain subordination findings for the differential operator ($D{R}_{\lambda ,q}^{m,n}$) and novel geometric characteristics of the q-analogues of the integral operators in these classes are also obtained. Our results are generalizations of results that were previously proven in the literature. Full article

By means of the q-derivative operator method, we review the q-beta integrals of Askey–Wilson and Nassrallah–Rahman. More integrals are evaluated by the author, making use of Bailey’s identity of well-poised bilateral ${}_6{\psi }_6$-series as well as the extended identity of Karlsson–Minton type for parameterized well-poised bilateral q-series. Full article

Named essentially after their close relationship with the modified Bessel function $K_{\nu }\left(z\right)$ of the second kind, which is known also as the Macdonald function (or, with a slightly different definition, the Basset function), the so-called Bessel polynomials $y_n\left(x\right)$ and the generalized Bessel polynomials $y_n(x;\alpha ,\beta )$ stemmed naturally in some systematic investigations of the classical wave equation in spherical polar coordinates. Our main purpose in this invited survey-cum-expository review article is to present an introductory overview of the Bessel polynomials $y_n\left(x\right)$ and the generalized Bessel polynomials $y_n(x;\alpha ,\beta )$ involving the asymmetric parameters $\alpha$ and $\beta$. Each of these polynomial systems, as well as their reversed forms ${\theta }_n\left(x\right)$ and ${\theta }_n(x;\alpha ,\beta )$, has been widely and extensively investigated and applied in the existing literature on the subject. We also briefly consider some recent developments based upon the basic (or quantum or q-) extensions of the Bessel polynomials. Several general families of hypergeometric polynomials, which are actually the truncated or terminating forms of the series representing the generalized hypergeometric function ${}_r{F}_s$ with r symmetric numerator parameters and s symmetric denominator parameters, are also investigated, together with the corresponding basic (or quantum or q-) hypergeometric functions and the basic (or quantum or q-) hypergeometric polynomials associated with ${}_r{\Phi }_s$ which also involves r symmetric numerator parameters and s symmetric denominator parameters. Full article

Since Gottlieb introduced and investigated the so-called Gottlieb polynomials in 1938, which are discrete orthogonal polynomials, many researchers have investigated these polynomials from diverse angles. In this paper, we aimed to investigate the q-extensions of these polynomials to provide certain q-generating functions for three sequences associated with a finite power series whose coefficients are products of the known q-extended multivariable and multiparameter Gottlieb polynomials and another non-vanishing multivariable function. Furthermore, numerous possible particular cases of our main identities are considered. Finally, we return to Khan and Asif’s q-Gottlieb polynomials to highlight certain connections with several other known q-polynomials, and provide its q-integral representation. Furthermore, we conclude this paper by disclosing our future investigation plan. Full article

This paper provides three classes of q-summation formulas in the form of general contiguous extensions of the first q-Kummer summation theorem. Their derivations are presented by using three methods, which are along the lines of the three types of well-known proofs of the q-Kummer summation theorem with a key role of the q-binomial theorem. In addition to the q-binomial theorem, the first proof makes use of Thomae’s q-integral representation and the second proof needs Heine’s transformation. Whereas the third proof utilizes only the q-binomial theorem. Subsequently, the applications of these summation formulas in obtaining the general contiguous extensions of the second and the third q-Kummer summation theorems are also presented. Furthermore, the investigated results are specialized to give many of the known as well as presumably new q-summation theorems, which are contiguous to the three q-Kummer summation theorems. This work is motivated by the observation that the basic (or q-) series and basic (or q-) polynomials, especially the basic (or q-) gamma and q-hypergeometric functions and basic (or q-) hypergeometric polynomials, are applicable particularly in several diverse areas including Number Theory, Theory of Partitions and Combinatorial Analysis as well as in the study of Combinatorial Generating Functions. Just as it is known in the theory of the Gauss, Kummer (or confluent), Clausen and the generalized hypergeometric functions, the parameters in the corresponding basic or quantum (or q-) hypergeometric functions are symmetric in the sense that they remain invariant when the order of the p numerator parameters or when the order of the q denominator parameters is arbitrarily changed. A case has therefore been made for the symmetry possessed not only by hypergeometric functions and basic or quantum (or q-) hypergeometric functions, which are studied in this paper, but also by the symmetric quantum calculus itself. Full article

In our present investigation, we develop two new Bailey lattices. We describe a number of q-multisums new forms with multiple variables for the basic hypergeometric series which arise as consequences of these two new Bailey lattices. As applications, two new transformations for basic hypergeometric by using the unit Bailey pair are derived. Besides it, we use this Bailey lattice to get some kind of mock theta functions. Our results are shown to be connected with several earlier works related to the field of our present investigation. Full article

Basic (or q-) series and basic (or q-) polynomials, especially the basic (or q-) hypergeometric functions and the basic (or q-) hypergeometric polynomials are studied extensively and widely due mainly to their potential for applications in many areas of mathematical and physical sciences. Here, in this paper, we introduce a general family of q-hypergeometric polynomials and investigate several q-series identities such as an extended generating function and a Srivastava-Agarwal type bilinear generating function for this family of q-hypergeometric polynomials. We give a transformational identity involving generating functions for the generalized q-hypergeometric polynomials which we have introduced here. We also point out relevant connections of the various q-results, which we investigate here, with those in several related earlier works on this subject. We conclude this paper by remarking that it will be a rather trivial and inconsequential exercise to give the so-called $(p,q)$-variations of the q-results, which we have investigated here, because the additional parameter p is obviously redundant. Full article

Our investigation is motivated essentially by the demonstrated applications of the basic (or q-) series and basic (or q-) polynomials, especially the basic (or q-) hypergeometric functions and basic (or q-) hypergeometric polynomials, in many diverse areas. Here, in this paper, we use two q-operators $\mathbb{T}(a,b,c,d,e,y{D}_x)$ and $\mathbb{E}(a,b,c,d,e,y{\theta }_x)$ to derive two potentially useful generalizations of the q-binomial theorem, a set of two extensions of the q-Chu-Vandermonde summation formula and two new generalizations of the Andrews-Askey integral by means of the q-difference equations. We also briefly describe relevant connections of various special cases and consequences of our main results with a number of known results. Full article

This contribution deals with the sequence ${\left\{{\mathbb{U}}_n^{\left(a\right)}(x;q,j)\right\}}_{n\ge 0}$ of monic polynomials in x, orthogonal with respect to a Sobolev-type inner product related to the Al-Salam–Carlitz I orthogonal polynomials, and involving an arbitrary number j of q-derivatives on the two boundaries of the corresponding orthogonality interval, for some fixed real number $q\in (0,1)$. We provide several versions of the corresponding connection formulas, ladder operators, and several versions of the second order q-difference equations satisfied by polynomials in this sequence. As a novel contribution to the literature, we provide certain three term recurrence formula with rational coefficients satisfied by ${\mathbb{U}}_n^{\left(a\right)}(x;q,j)$, which paves the way to establish an appealing generalization of the so-called J-fractions to the framework of Sobolev-type orthogonality. Full article

We provide several new q-congruences for truncated basic hypergeometric series, mostly of arbitrary order. Our results include congruences modulo the square or the cube of a cyclotomic polynomial, and in some instances, parametric generalizations thereof. These are established by a variety of techniques including polynomial argument, creative microscoping (a method recently introduced by the first author in collaboration with Zudilin), Andrews’ multiseries generalization of the Watson transformation, and induction. We also give a number of related conjectures including congruences modulo the fourth power of a cyclotomic polynomial. Full article

Demonstrating the striking symmetry between calculus and q-calculus, we obtain q-analogues of the Bateman, Pasternack, Sylvester, and Cesàro polynomials. Using these, we also obtain q-analogues for some of their generating functions. Our q-generating functions are given in terms of the basic hypergeometric series ${}_4{\varphi }_5$ , ${}_5{\varphi }_5$ , ${}_4{\varphi }_3$ , ${}_3{\varphi }_2$ , ${}_2{\varphi }_1$ , and q-Pochhammer symbols. Starting with our q-generating functions, we are also able to find some new classical generating functions for the Pasternack and Bateman polynomials. Full article

By applying a classical method, already employed by Cauchy, to a terminating curious summation by one of the authors, a new curious bilateral q-series identity is derived. We also apply the same method to a quadratic summation by Gessel and Stanton, and to a cubic summation by Gasper, respectively, to derive a bilateral quadratic and a bilateral cubic summation formula. Full article