Basic hypergeometric functions and orthogonal Laurent polynomials

Abstract. In this paper, the expansion of a basic confluent hypergeometric function in terms of an incomplete q-gamma function is investigated and shown to be an asymptotic expansion for a basic confluent hypergeometric function and a q-analogue of the Hadamard expansion. The general expansion associated with a basic hypergeometric function is introduced. In special cases, this provides alternative derivations of the q-Hadamard expansion. Expansions and asymptotics for modified q-Bessel functions are derived as special cases from the general expansion.

This work is motivated essentially by the fact that the applications of basic (or q-) hypergeometric functions are frequently needed in the form of summations, transformations, expansions, reductions, and integral formulas. The objective of this research paper is to study the applications of the general summation formulas contiguous to q-Kummer theorems in deriving several presumably new q-integral evaluations involving the basic (or q-) hypergeometric functions of type $$_{3} \Phi_{3}$$ and $$_{3} \Phi_{2}$$ . The results are further particularized to give specific integral evaluations in the form of summations to show the importance and utilizations of our main findings. We also derive the summation theorems for certain basic (or q-) Appell double hypergeometric functions, basic (or q-) Lauricella multiple hypergeometric functions, and certain basic (or q-) Laplace transforms by applying general summation formulas contiguous to q-Kummer theorems.

This project aimed to gather together the latest developments in research concerning complex-valued functions from the perspective of geometric function theory. Scholars contributions weresought on topics including, but not limited to: new classes of univalent and bi-univalent functions; studies regarding coefficient estimates including the FeketeSzego functional, Hankel determinants, and Toeplitz matrices; applications of different types of operators in geometric function theory including differential, integral, fractional, or quantum calculus operators; differential subordination and superordination theories in their classical form and also concerning their recent extensionsstrong and fuzzy differential subordination and superordination theories; applications of different hypergeometric functions and orthogonal polynomials in geometric function theory. The presentation of new results obtained by using any other techniques which can be applied in the field of complex analysis were also welcomed. Hopefully, through this project, new lines of research associatedwithgeometric function theory have been highlighted and willserve toboost development inthis field.

Darling [2] in 1932 gave two types (equations 11 and 18) of transformations connecting generalised hyper geometric functions. The first was studied by Bailey [1] and extended by Sears [4] to a transformation connecting products of basic hyper geometric functions of the type r+1ϕr × r+1ϕr. In a number of papers [6, 7, 8] the author has extended these results to both unilateral and bilateral series with bases q and q1/2. The second type of transformation by Darling for a product 0F1 × 3F2 was extended by Bailey [1] to a transformation between 1F0 × r+1Fr. In the same paper Bailey mentioned the transformation of a 0ϕ1 × 3ϕ2 without proof.

Contiguous relations, basic hypergeometric functions, and orthogonal polynomials, I

As we have seen in Chapter 13, representations of Chevalley groups lead to q-orthogonal polynomials with positive integral q. Basic hypergeometric functions and q-orthogonal polynomials for arbitrary (including complex) values of q are connected with quantum algebras and groups. They are considered in the present Chapter.

The Strong Chebyshev Distribution and Orthogonal Laurent Polynomials

Orthogonal Laurent polynomials and strong moment theory: a survey

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.

Contiguous relations, basic hypergeometric functions, and orthogonal polynomials. II. Associated big q-Jacobi polynomials

Contiguous relations, basic hypergeometric functions, and orthogonal polynomials. III. Associated continuous dual q-Hahn polynomials

The location of the zeros of general hypergeometric polynomials are linked with those of the classical orthogonal polynomials in some cases, notably 2 F 1 and 1 F 1 hypergeometric polynomials which have been extensively studied. In the case of 3 F 2 polynomials, less is known about their properties, including the location of their zeros, because there is, in general, no direct link with orthogonal polynomials. We consider two classes of 3 F 2 hypergeometric polynomials, each of which has a representation in terms of 2 F 1 polynomials. Our main result proves that the class of polynomials 3 F 2(– n, a, b;a – 1, d; x), a, b, d ∈ R, n ∈ N is quasi-orthogonal of order 1 on an interval that varies with the real parameters b and d. We discuss the apparent role played by the parameter a with regard to the location of all the zeros of this class of polynomials. We also prove results on the location of the zeros of the class 3 F 2(– n, b,(b – n)/2; b – n, (b – n – 1)/2; x), b ∈ R, n ∈N by using the orthogonality and quasi-orthogonality of factors involved in its representation.

This paper continues the analysis of a class of orthogonal polynomials in two variables on a region bounded by two straight lines and a parabola touching these lines, which was introduced by the first author. An explicit series expansion for these polynomials is obtained, which generalizes Constantine’s expansion of hypergeometric functions of $(2 \times 2)$ matrix argument in terms of James’ zonal polynomials. In two special cases the orthogonal polynomials turn out to be Appell’s hypergeometric $F_4 $-functions and certain hypergeometric functions in two variables of order three, respectively.

The subject of q-analysis, which is popularly known as quantum analysis, has its roots in such important areas as (for example) Mathematical Physics, Analytic Number Theory, and the Theory of Partitions. Our present investigation is motivated essentially by the potential for applications of q-series and q-polynomials, especially the basic (or q-) and bibasic hypergeometric functions and the corresponding hypergeometric polynomials. The main object of this paper is to investigate the Bailey transform, the Bailey pair and the WP-Bailey pair of sequences and some of their applications in order to establish a number transformation formulas for q-hypergeometric series as well as bi-basic hypergeometric series, together with several other related q-series identities.

PC-fractions and orthogonal Laurent polynomials for log-normal distributions