Basic Hypergeometric Functions Research Articles

A study on unification of generalized hypergeometric function and Mittag-Leffler function with certain integral transforms of generalized basic hypergeometric function

This research article explores some new properties of generalized hypergeometric function and its q-analogue. The connections between ${}_{2}{{R}_{1}}^{\upsilon }(\mathfrak{z})$, the Wright function, and generalized Mittag-Leffler functions are explored. The authors introduce the q-analogue of generalized hypergeometric function denoted by ${}_{2}{{R}_{1}}^{\upsilon ,q}(\mathfrak{z})$ and discuss its properties and connections with q-Wright function and q-versions of generalized Mittag-Leffler functions. We get the q-integral transforms such as q-Mellin, q-Euler (beta), q-Laplace, q-sumudu, and q-natural transforms of Wright-type generalized q-hypergeometric function. This article contributes to the understanding of hypergeometric functions in q-calculus.

Certain Properties on Meromorphic Functions Defined by a New Linear Operator Involving the Mittag-Leffler Function

Our paper introduces a new linear operator using the convolution between a Mittag–Leffler Function and basic hypergeometric function. Use of the linear operator creates a new class of meromorphic functions defined in the punctured open unit disk. Consequently, the paper examines different aspects Apps and assets like, extreme points, coefficient inequality, growth and distortion. In conclusion, the work discusses modified Hadamard product and closure theorems.

ON TRANSFORMATION OF BASIC HYPERGEOMETRIC FUNCTIONS APPLYING FRACTIONAL q-DIFFERENTIAL OPERATOR

The object of this paper is to establish transformations in which a basic hypergeometric functions can be expressed as an infinite sum of functions of higher order by the application of fractional q-differential operator.Several well known hypergeometric functions and basic Kamp˙ e de F˙eriet function are expressed as an infinite sum of basic hypergeometric functions.Some special cases can be deduced as the application of the main results.

An Introductory Overview of Bessel Polynomials, the Generalized Bessel Polynomials and the q-Bessel Polynomials

Named essentially after their close relationship with the modified Bessel function Kν(z) 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 yn(x) and the generalized Bessel polynomials yn(x;α,β) 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 yn(x) and the generalized Bessel polynomials yn(x;α,β) involving the asymmetric parameters α and β. Each of these polynomial systems, as well as their reversed forms θn(x) and θn(x;α,β), 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 rFs 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 rΦs which also involves r symmetric numerator parameters and s symmetric denominator parameters.

Nonterminating transformations and summations associated with some q-Mellin–Barnes integrals

In many cases one may encounter an integral which is of q-Mellin–Barnes type. These integrals are easily evaluated using theorems which have a long history dating back to Slater, Askey, Gasper, Rahman and others. We derive some interesting q-Mellin–Barnes integrals and using them we derive transformation and summation formulas for nonterminating basic hypergeometric functions. The cases which we treat include ratios of theta functions, the Askey–Wilson moments, nonterminating well-poised ϕ23, nonterminating very-well-poised W45, W78, products of two nonterminating ϕ12's, square of a nonterminating well-poised ϕ12, a nonterminating W910, two nonterminating W1112's and several nonterminating summations which arise from the Askey–Roy and Gasper integrals.

Solution to Laplace’s Equation Using Quantum Calculus

The quantum calculus emerged as a new type of unconventional calculus relevant to both mathematics and physics. The study of quantum calculus or q-calculus has three hundred years of history of development since the era of Euler and Bernoulli, and was appeared as one of the most arduous techniques to use it in mathematics as well as physical science. At present, it is used in diverged mathematical areas like number theory, orthogonal polynomials, basic hypergeometric functions, etc. Furthermore, in order to get analytical approximate solutions to the ordinary as well as partial differential equations, q-reduced differential technique and quantum separation of variable technique are used in mathematics, Mechanics, and physics. In this paper, Laplace’s equation, a well-known equation in both Physical and Mathematical sciences, has been solved extensively based on the basics of calculus, transformation methods, and q-separation of variable method. In addition, solutions to the Laplace’s equation as obtained by using different boundary conditions are revisited and reviewed. Consequently, all the necessary basics of q-calculus are displayed one by one, and thereafter, the process of finding its solution in view of quantum calculus is described extensively. In order to find out the exact solutions the dimensionality of all the parameters related to the problem has been described. As an essential outcome, it is also found that, as q tends to 1, the solution takes the form as it is in general physics. Hence, this article presents a review and extension that describe the solution to Laplace’s equation in view of both Leibnitz and quantum calculus. Thus, it can add a pedagogical exercise for the students of both physical and mathematical sciences to understand the usefulness of quantum calculus.

On summation formulas in probability theory

We offer some summation formulas that appear to have great utility in probability theory. The proofs require some recent results from analysis that have thus far been applied to basic hypergeometric functions.

Hyperbolic and trigonometric hypergeometric solutions to the star-star equation

We construct the hyperbolic and trigonometric solutions to the star-star relation via the gauge/YBE correspondence by using the three-dimensional lens partition function and superconformal index for a certain mathcal N=2 supersymmetric gauge dual theories. This correspondence relates supersymmetric gauge theories to exactly solvable models of statistical mechanics. The equality of partition functions for the three-dimensional supersymmetric dual theories can be written as an integral identity for hyperbolic and basic hypergeometric functions.

A new identity for the sum of products of the generalized basic hypergeometric functions

We prove a duality relation for the generalized basic hypergeometric functions. It forms a q-extension of a recent result of the second- and the third-named authors and generalizes both a q-hypergeometric identity due to the third-named author (jointly with Feng and Yang) and a recent identity for the Heine’s \({}_2\phi _{1}\) function due to Suzuki. We further explore various consequences of our identity leading to several presumably new multi-term relations for both terminating and non-terminating generalized basic hypergeometric series. Moreover, we give confluent versions of our results and furnish a number of explicit examples.

Utility of integral representations for basic hypergeometric functions and orthogonal polynomials

We describe the utility of integral representations for sums of basic hypergeometric functions. In particular we use these to derive an infinite sequence of transformations for symmetrizations over certain variables which the functions possess. These integral representations were studied by Bailey, Slater, Askey, Roy, Gasper and Rahman and were also used to facilitate the computation of certain outstanding problems in the theory of basic hypergeometric orthogonal polynomials in the $q$-Askey scheme. We also generalize and give consequences and transformation formulas for some fundamental integrals connected to nonterminating basic hypergeometric series and the Askey--Wilson polynomials. We express a certain integral of a ratio of infinite $q$-shifted factorials as a symmetric sum of two basic hypergeometric series with argument $q$. The result is then expressed as a $q$-integral. Examples of integral representations applied to the derivation of generating functions for Askey--Wilson are given and as well the computation of a missing generating function for the continuous dual $q$-Hahn polynomials.

Extensions to generalizations of hypergeometric and basic hypergeometric functions of inequalities for quotients of hypergeometric functions that arose in research on financing efficiency of securities-based crowdfunding

Extensions to generalizations of hypergeometric and basic hypergeometric functions of inequalities for quotients of hypergeometric functions that arose in research on financing efficiency of securities-based crowdfunding

General Summation Formulas Contiguous to the q-Kummer Summation Theorems and Their Applications

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.

On a generalization of starlike functions

In this paper, we mainly study the order of q-starlikeness of the well-known basic hypergeometric function. In addition, we discuss the Bieberbach-type problem and the second order Hankel determinant for a generalized class of starlike functions.

A General Family of q-Hypergeometric Polynomials and Associated Generating Functions

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.

Q-Derivatives of Multivariable q-Hypergeometric Function with Respect to Their Parameters

We consider the q-derivatives of the Srivastava and Daoust basic multivariable hypergeometric function with respect to the parameters. This function embodies a entire number of various q-hypergeometric series of one and several variables. Explicit equations are given for general case of summation indexes with positive real coefficients. As an example derivatives of q-analog of non-confluent Horn type hypergeometric function $${{H}_{3}}$$ is presented.

A Note on Generalized q-Difference Equations and Their Applications Involving q-Hypergeometric Functions

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 T(a,b,c,d,e,yDx) and E(a,b,c,d,e,yθx) 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.

On basic Horn hypergeometric functions mathbf{H}_{3} and mathbf{H}_{4}

The purpose of this work is to demonstrate several interesting contiguous function relations and q-differential formulas for basic Horn hypergeometric functions mathbf{H}_{3} and mathbf{H}_{4}. Some properties of our main results are also constructed.

Operators of Basic (or q-) Calculus and Fractional q-Calculus and Their Applications in Geometric Function Theory of Complex Analysis

Basic (or q-) series and basic (or q-) polynomials, especially the basic (or q-) hypergeometric functions and basic (or q-) hypergeometric polynomials, are known to have widespread applications, particularly in several areas of number theory and combinatorial analysis such as (for example) the theory of partitions. Our usages here, in this survey-cum-expository article, of the q-calculus and the fractional q-calculus in geometric function theory of complex analysis are believed to encourage and motivate significant further developments on these and other related topics. By applying a fractional q-calculus operator, we define the subclasses $${\mathcal{S}}_{n}^{\alpha }(\lambda ,\beta ,b,q)$$ and $${\mathcal{G}}_{n}^{\alpha }(\lambda ,\beta ,b,q)$$ of normalized analytic functions with complex order and negative coefficients. Among the results investigated for each of these function classes, we derive their associated coefficient estimates, radii of close-to-convexity, starlikeness and convexity, extreme points and growth and distortion theorems. Our investigation here is motivated essentially by the fact that basic (or q-) series and basic (or q-) polynomials, especially the basic (or q-) hypergeometric functions and basic (or q-) hypergeometric polynomials, are applicable particularly in several areas of number theory such as the theory of partitions. In fact, basic (or q-) hypergeometric functions are useful also in a wide variety of fields including, for example, combinatorial analysis, finite vector spaces, lie theory, particle physics, nonlinear electric circuit theory, mechanical engineering, theory of heat conduction, quantum mechanics, cosmology and statistics (see also (Srivastava and Karlsson in Multiple Gaussian hypergeometric series. pp 350–351, 1985) and the references cited thereon). In the last section on conclusion, we choose to point out the fact that the results for the q-analogues, which we consider in this article for $$0< q < 1$$, can easily (and possibly trivially) be translated into the corresponding results for the (p, q)-analogues (with $$0< q < p \leqq 1$$) by applying some obvious parametric and argument variations, the additional parameter p being redundant. Several other families of such extensively- and widely-investigated linear convolution operators as (for example) the Dziok–Srivastava, Srivastava–Wright and Srivastava–Attiya linear convolution operators, together with their extended and generalized versions, are also briefly considered.

The q-derivative and differential equation

The q-calculus appeared as a connection between mathematics and physics. It has several applications in different mathematical areas such as number theory, combinatorics, orthogonal polynomials, basic hypergeometric functions, quantum theory, and electronics. Recently, a great interest to its applications in differential transform methods, in order to get analytical approximate solutions to the ordinary as well as partial differential equations. In this paper, we present some of the interesting definitions of q-calculus and q-derivatives. By using q-calculus, solutions of some differential equations could be generated.

On certain q-hypergeometric matrix functions of several variables and their matrix q-difference equations

ABSTRACTWe introduce q-Lauricella matrix functions of three and n-variables. The convergence of these matrix functions is examined for both and cases. The systems of matrix q-difference equations satisfied by these q-Lauricella matrix functions as well as basic Srivastava's triple hypergeometric matrix functions have been determined. Certain integral representations satisfied by these matrix functions have also been established.