A Transformation Connecting Products of Generalised Basic Hypergeometric Functions

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.

A three-complex-parameter class of orthogonal Laurent polynomials on the unit circle associated with basic hypergeometric or q q -hypergeometric functions is considered. To be precise, we consider the orthogonality properties of the sequence of polynomials { 2 Φ 1 ( q − n , q b + 1 ; q − c + b − n ; q , q − c + d − 1 z ) } n = 0 ∞ \{\,_2\Phi _1(q^{-n},q^{b+1};q^{-c+b-n}; q, q^{-c+d-1}z)\}_{n=0}^{\infty } , where 0 > q > 1 0 > q > 1 and the complex parameters b b , c c and d d are such that b ≠ − 1 , − 2 , … b \neq -1, -2, \ldots , c − b + 1 ≠ − 1 , − 2 , … c-b+1 \neq -1, -2, \ldots , R e ( d ) > 0 \mathcal {R}e(d) > 0 and R e ( c − d + 2 ) > 0 \mathcal {R}e(c-d+2) > 0 . Explicit expressions for the recurrence coefficients, moments, orthogonality and also asymptotic properties are given. By a special choice of the parameters, results regarding a class of Szegő polynomials are also derived.

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.

Representations of the quantum superalgebra Uq[osp(1/2)] and their relations to the basic hypergeometric functions are investigated. We first establish Clebsch–Gordan decomposition for the superalgebra Uq[osp(1/2)] in which the representations having no classical counterparts are incorporated. Formulae for these Clebsch–Gordan coefficients are derived, and is observed that they may be expressed in terms of the Q-Hahn polynomials. We next investigate representations of the quantum supergroup OSpq(1/2) which are not well defined in the classical limit. Employing the universal -matrix, the representation matrices are obtained explicitly, and found to be related to the little Q-Jacobi polynomials. Characteristically, the relation Q = −q is satisfied in all cases. Using the Clebsch–Gordan coefficients derived here, we construct new noncommutative spaces that are covariant under the coaction of the even-dimensional representations of the quantum supergroup OSpq(1/2).

We advocate the exploitation of symmetry (recurrence relation) techniques for the derivation of properties associated with families of basic hypergeometric functions, in analogy with the local Lie theory techniques for ordinary hypergeometric functions. Here these ideas are applied to the (continuous) Askey-Wilson polynomials, introduced by Askey and Wilson, to obtain a strikingly simple derivation of their orthogonality relations. 1. In t roduct ion . In [1] the authors and A.K. Agarwal introduced symmetry techniques for the study of families of basic hypergeometric functions, in analogy with the local Lie theory techniques for ordinary hypergeometric functions. The fundamental objects in this study are the recurrence relations obeyed by the families: generating functions and identities for each family are characterized in terms of the recurrence relations. The functions studied in [1] are of the form ^ / a i , . . . , a r _ y > (ai;q)k'(ar;q)k z k V 6 i , . . . , 6 8 ' ' / f^Q(bi;q)k'(b8;q)k{q;q)k and many-variable extension. Here , . _ J 1 if k = 0 ( f l ; g , ' ( l a ) ( l g a ) . . . ( l c * 1 o ) if fc = 1,2,... and \q . ( , , . ) ( . ( I ^ ) ( I ^ E.G. KALNINS AND W. MILLER, JR. 225 Here Eff(z) = f(qz). Relation (2.1A) follows from T(az;q)J-;q) = J ^ ( l q)(aqz;q)k.1 (^— ;q) \Z /k q' Z / k-l and (2.IB) follows from KaZ;q)k(±q)k = (1 a b q ^ ^ q ) ^ ^ q ) ^ The first relation was pointed out by Askey and Wilson [2]; we have not found (2. IB) in the literature. Consider relation (2.IB); it suggests that there should be an operator Aaq-^,bq-^,cq,dq) . Aa,b,c,d) T , , a A l± mapping Cn t o <f)n . Indeed we find l(cq ^,dq^iaq' ^ibq^)ji{aq-^ ,bq-^,cqAq) (2.3) ^(l-Ht-^) ,(aM

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.

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.

We introduce the $q$-Hahn PushTASEP—an integrable stochastic interacting particle system that is a three-parameter generalization of the PushTASEP, a well-known close relative of the TASEP (totally asymmetric simple exclusion process). The transition probabilities in the $q$-Hahn PushTASEP are expressed through the $_4\phi _3$ basic hypergeometric function. Under suitable limits, the $q$-Hahn PushTASEP degenerates to all known integrable (1+1)-dimensional stochastic systems with a pushing mechanism. One can thus view our new system as a pushing counterpart of the $q$-Hahn TASEP introduced by Povolotsky [37]. We establish Markov duality relations and contour integral formulas for the $q$-Hahn PushTASEP. In a $q\to 1$ limit of our process we arrive at a random recursion, which, in a special case, appears to be similar to the inverse-Beta polymer model. However, unlike in recursions for Beta polymer models, the weights (i.e., the coefficients of the recursion) in our model depend on the previous values of the partition function in a nontrivial manner.

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

We consider the generalized eigenvalue problem Aψ = λBψ for two operators A, B. Self-similar closure of this problem under a simplest Darboux transformation gives rise to two possible types of regular algebras of dimension 2 with generators A, B. Realization of the operators A, B by tri-diagonal operators leads to a theory of biorthogonal rational functions. We find the general solution of this problem in terms of the ordinary and basic hypergeometric functions. In special cases we obtain general Padé interpolation tables for the exponential and power function on uniform and exponential grids.

In this talk, we give a brief overview of several aspects of the theory of the chiral Potts model, including higher-genus solutions of the star–triangle and tetrahedron equations, cyclic representations of affine quantum groups, basic hypergeometric functions at root of unity, and possible applications.

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

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.