Basic Hypergeometric Functions Research Articles

On a q-Extension of the Leibniz Rule via Weyl Type of q-Derivative Operator

In the present paper we define a q-extension of the Leibniz rule for q-derivatives via Weyl type q-derivative operator. Expansions and summation formulae for the gener- alized basic hypergeometric functions of one and more variables are deduced as the appli- cations of the main result.

Exact solution of the discrete (1+1)-dimensional RSOS model in a slit with field and wall interactions

We present the solution of a linear restricted solid-on-solid (RSOS) model confined to a slit. We include a field-like energy, which equivalently weights the area under the interface, and also include independent interaction terms with both walls. This model can also be mapped to a lattice polymer model of Motzkin paths in a slit interacting with both walls including an osmotic pressure. This work generalizes the previous work on the RSOS model in the half-plane which has a solution that was shown recently to exhibit a novel mathematical structure involving basic hypergeometric functions 3ϕ2. Because of the mathematical relationship between the half-plane and slit this work hence effectively explores the underlying q-orthogonal polynomial structure to that solution. It also generalizes two other recent works: one on Dyck paths weighted with an osmotic pressure in a slit and another concerning Motzkin paths without an osmotic pressure term in a slit.

Exact solution of the discrete (1+1)-dimensional RSOS model with field and surface interactions

We present the solution of a linear restricted solid-on-solid (RSOS) model in a field. Aside from the origins of this model in the context of describing the phase boundary in a magnet, interest also comes from more recent work on the steady state of non-equilibrium models of molecular motors. While similar to a previously solved (non-restricted) SOS model in its physical behaviour, mathematically the solution is more complex. Involving basic hypergeometric functions 3ϕ2, it introduces a new form of solution to the lexicon of directed lattice path generating functions.

Sum of multiple $q$-zeta values

The generating function of the sums of multiple q-zeta values with fixed weights, depths and 1-heights, 2-heights,..., r-heights is represented in terms of specializations of basic hypergeometric functions.

Addition theorems via continued fractions

We show connections between a special type of addition formulas and a theorem of Stieltjes and Rogers. We use different techniques to derive the desirable addition formulas. We apply our approach to derive special addition theorems for Bessel functions and confluent hypergeometric functions. We also derive several additions theorems for basic hypergeometric functions. Applications to the evaluation of Hankel determinants are also given .

Basic-deformed quantum mechanics

Starting on the basis of $q$-symmetric oscillator algebra and on the associate $q$-calculus properties, we study a deformed quantum mechanics defined in the framework of the basic square-integrable wave functions space. In this context, we introduce a deformed Schroedinger equation, which satisfies the main quantum mechanics assumptions and admits, in the free case, plane wave functions that can be expressed in terms of the q-deformed exponential, originally introduced in the framework of the basic-hypergeometric functions.

Basic Hypergeometric Functions as Limits of Elliptic Hypergeometric Functions

We describe a uniform way of obtaining basic hypergeometric functions as limits of the elliptic beta integral. This description gives rise to the construction of a polytope with a different basic hypergeometric function attached to each face of this polytope. We can subsequently obtain various relations, such as transformations and three-term relations, of these functions by considering geometrical properties of this polytope. The most general functions we describe in this way are sums of two very-well-poised ${}_{10}\phi_9$'s and their Nassrallah-Rahman type integral representation.

Quantum mechanics in q-deformed calculus

Starting on the basis of q-deformed calculus and q-symmetric oscillator algebra, we introduce a generalized Schrödinger equation which admits factorized time-space solutions and the free plane wave functions can be expressed in terms of the so-called basic-hypergeometric functions. In this framework, q-deformed adjoint and q-hermitian operator properties occur i a natural way in order to satisfy the fundamental quantum mechanics asumptions.

Hypergeometric τ-Functions of the q-Painlevé System of Type E7(1)

We present the $\tau$-functions for the hypergeometric solutions to the $q$-Painlev\'e system of type $E_7^{(1)}$ in a determinant formula whose entries are given by the basic hypergeometric function ${}_8W_7$. By using the $W(D_5)$ symmetry of the function ${}_8W_7$, we construct a set of twelve solutions and describe the action of $\widetilde{W}(D_6^{(1)})$ on the set.

(q, ν)-deformation of generalized basic hypergeometric states

We provide with a (q, ν)-deformation of the generalized hypergeometric coherent states defined such that the normalization function is given by generalized basic hypergeometric functions. These states are eigenstates of suitably defined deformed lowering operators. We study the domain of convergence of the corresponding normalization function. On the basis of these states, we investigate generalized basic hypergeometric Husimi distributions and corresponding phase distributions as well as new analytic basis representations of arbitrary quantum states in Bargmann and Hardy spaces. The quantum statistical properties of the states, such as photon-counting statistics and quadrature squeezing are analytically and numerically discussed in the framework of conventional quantum optics.

Q-Calculus as operational algebra

This second paper on operational calculus is a continuation of Ernst, T. q-Analogues of some operational formulas. Algebras Groups Geom., 2006, 23(4), 354-374. We find multiple q-analogues of formulas in Carlitz, L. A note on the Laguerre polynomials. Michigan Math. J., 1960, 7, 219-223, for the Cigler q-Laguerre polynomials (Ernst, T. A method for q-calculus. J. Nonlinear Math. Phys., 2003, 10(4), 487-525). The q-Jacobi polynomials (Jacobi, C. G. J. Werke 6. Berlin, 1891) are treated in the same way, we generalize further to q-analogues of Manocha, H. L. and Sharma, B. L. (Some formulae for Jacobi polynomials. Proc. Cambridge Philos. Soc., 1966, 62, 459-462) and Singh, R. P. (Operational formulae for Jacobi and other polynomials. Rend. Sem. Mat. Univ. Padova, 1965, 35, 237-244). A field of fractions for Cigler's multiplication operator (Cigler, J. Operatormethoden fur q-Identitaten II, q-Laguerre-Polynome. Monatsh. Math., 1981, 91, 105-117) is used in the computations. The formulas for q-Jacobi polynomials are mostly formal. We find q-orthogonality relations for q-Laguerre, q-Jacobi, and q-Legendre polynomials using q-integration by parts. This q-Legendre polynomial is given here for the first time, we also find its q-difference equations. An inequality for a q-exponential function is proved. The q-difference equation for (p)phi(p-1) (a(1),...,a(p); b(1),...,b(p-1)vertical bar q, z) is given improving on Smith, E. R. Zur Theorie der Heineschen Reihe und ihrer Verallgemeinerung. Diss. Univ. Munchen 1911, p. 11, by using e(k) = elementary symmetric polynomial. Partial q-difference equations for the q-Appell and q-Lauricella functions are found, improving on Jackson, F. H. On basic double hypergeometric functions. Quart. J. Math., Oxford Ser., 1942, 13, 69-82, and Gasper, G. and Rahman, M. Basic hypergeometric series. Second edition. Cambridge, 2004, p. 299, where q-difference equations for q-Appell functions were given with different notation. The q-difference equation for Phi(1) can also be written in canonical form, a q-analogue of [p. 146] Mellin, H. J. Uber den Zusammenhang zwischen den linearen Differential- und Differenzengleichunge, Acta Math., 1901, 25, 139-164.

Basic hypergeometric functions and covariant spaces for even-dimensional representations of Uq[osp(1/2)

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).

A 𝑞-sampling theorem and product formula for continuous 𝑞-Jacobi functions

In this paper we derive a q-analogue of the sampling theorem for Jacobi functions. We also establish a product formula for the nonterminating version of the q-Jacobi polynomials. The proof uses recent results in the theory of q-orthogonal polynomials and basic hypergeometric functions.

The correlation functions of vertex operators and Macdonald polynomials

The n-point correlation functions introduced by Bloch and Okounkov have already found several geometric connections and algebraic generalizations. In this note we formulate a q,t-deformation of this n-point function. The key operator used in our formulation arises from the theory of Macdonald polynomials and affords a vertex operator interpretation. We obtain closed formulas for the n-point functions when n = 1,2 in terms of the basic hypergeometric functions. We further generalize the q,t-deformed n-point function to more general vertex operators.

Disturbing the Dyson conjecture, in a generally GOOD way

Dyson's celebrated constant term conjecture [F.J. Dyson, Statistical theory of the energy levels of complex systems I, J. Math. Phys. 3 (1962) 140–156] states that the constant term in the expansion of ∏ 1 ≦ i ≠ j ≦ n ( 1 − x i / x j ) a j is the multinomial coefficient ( a 1 + a 2 + ⋯ + a n ) ! / ( a 1 ! a 2 ! ⋯ a n ! ) . The definitive proof was given by I.J. Good [I.J. Good, Short proof of a conjecture of Dyson, J. Math. Phys. 11 (1970) 1884]. Later, Andrews extended Dyson's conjecture to a q-analog [G.E. Andrews, Problems and prospects for basic hypergeometric functions, in: R. Askey (Ed.), The Theory and Application of Special Functions, Academic Press, New York, 1975, pp. 191–224]. In this paper, closed form expressions are given for the coefficients of several other terms in the Dyson product, and are proved using an extension of Good's idea. Also, conjectures for the corresponding q-analogs are supplied. Finally, perturbed versions of the q-Dixon summation formula are presented.

Reconstruction of universal Drinfeld twists from representations

Universal Drinfeld twists are inner automorphisms which relate the coproduct of a quantum enveloping algebra to the coproduct of the undeformed enveloping algebra. Even though they govern the deformation theory of classical symmetries and have appeared in numerous applications, no twist for a semisimple quantum enveloping algebra has ever been computed. It is argued that universal twists can be reconstructed from their well-known representations. A method to reconstruct an arbitrary element of the enveloping algebra from its irreducible representations is developed. For the twist this yields an algebra valued generating function to all orders in the deformation parameter, expressed by a combination of basic and ordinary hypergeometric functions. It is shown how the generating function can be expanded to the formal power series of the twist. An explicit expression for the universal twist of su(2) is given up to third order.

Regular algebras of dimension 2, the generalized eigenvalue problem and Padé interpolation

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.

Metaplectic Representation of ospq(1/2) Algebra and Basic Hypergeometric Functions

In this paper we use the bosonic realization of ospq(1/2) algebra to obtain its metaplectic representation. The group element for this algebra is shown to be described in terms of the basic hypergeometric function.

A 𝑞-analogue of the Whittaker-Shannon-Kotel’nikov sampling theorem

The Whittaker-Shannon-Kotel’nikov (WSK) sampling theorem plays an important role not only in harmonic analysis and approximation theory, but also in communication engineering since it enables engineers to reconstruct analog signals from their samples at a discrete set of data points. The main aim of this paper is to derive a q q -analogue of the Whittaker-Shannon-Kotel’nikov sampling theorem. The proof uses recent results in the theory of q q -orthogonal polynomials and basic hypergeometric functions, in particular, new results on the addition theorems for q q -exponential functions.

G-continued fractions for basic hypergeometric functions II

In this paper we apply a modification of a generalized Pringsheim's theorem to obtain a G-continued fraction expansion for the quotient of two contiguous basic hypergeometric functions in arbitrarily many variables. As an application we obtain a G-continued fraction extension of the Rogers–Ramanujan continued fraction.