Meixner functions and polynomials related to Lie algebra representations

Groenevelt, Wolter; Koelink, Erik

doi:10.1088/0305-4470/35/1/306

Cited by 22 publications

(50 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Note that the strange series representations do not appear in the decomposition of theorem 2.4. The decomposition in theorem 2.4 looks similar to the decomposition of the tensor product of a positive and a negative discrete series representation of the Lie algebra su(1, 1), see [10]. However for the quantum algebra U q su(1, 1) the action of the Casimir in the tensor product is bounded, contrary to the Lie algebra case, where the action of the Casimir in the tensor product is unbounded.…”

Section: )mentioning

confidence: 85%

See 1 more Smart Citation

Bilinear Summation Formulas from Quantum Algebra Representations

Groenevelt

2004

Ramanujan J

Self Cite

View full text Add to dashboard Cite

Abstract. The tensor product of a positive and a negative discrete series representation of the quantum algebra Uq su(1, 1) decomposes as a direct integral over the principal unitary series representations. Discrete terms can appear, and these terms are a finite number of discrete series representations, or one complementary series representation. From the interpretation as overlap coefficients of little q-Jacobi functions and Al-Salam and Chihara polynomials in base q and base q −1 , two closely related bilinear summation formulas for the Al-Salam and Chihara polynomials are derived. The formulas involve Askey-Wilson polynomials, continuous dual q-Hahn polynomials and little q-Jacobi functions. The realization of the discrete series as q-difference operators on the spaces of holomorphic and anti-holomorphic functions, leads to a bilinear generating function for a certain type of 2ϕ1-series, which can be considered as a special case of the dual transmutation kernel for little q-Jacobi functions.

show abstract

Section: )mentioning

confidence: 85%

“…These Meixner functions can be considered as non-polynomial extensions of the Meixner polynomials. The goal of this paper is to find q-analogues of the results of [10] using representation theory of U q su(1, 1) . The method we use is different from the method applied in [10].…”

Section: Introductionmentioning

confidence: 99%

Bilinear Summation Formulas from Quantum Algebra Representations

Groenevelt

2004

Ramanujan J

Self Cite

View full text Add to dashboard Cite

show abstract

“…At the present work we again establish that the ISW satisfy the mathematical structure given by (14), for a particular forms of f ISW y h ISW from which it is possible to construct the constitutive elements of the basic algebraic anatomy of the KHQA.…”

Section: The Infinite Square Wellmentioning

confidence: 52%

“…Similarly to the previous systems, we have a probability density associated to the coherent state which is immediately extracted from the explicit form of the coherent state. We have established then, that the PTP also satisfy the algebraic structure given by (14) with characteristic functions of the form…”

Section: The Pöschl-teller Potentialsmentioning

confidence: 99%

“…Within the domain of condensed matter, we have realizations on the following quantum potentials: infinite square well, the Pöschl-Teller potentials [17] and Calogero-Sutherland model [16]. Other realizations from the su(1, 1) algebra arises from the mathematical physics at relation with the recursive properties of the special functions, namely Laguerre oscillators [18,19,20], Legendre and Chebyshev oscillators [19], Meixner Oscillators [14,21], and so on.The present paper is realized at the following way. In the section 2 we introduce KHQA in such way that the algebraic issues have been empathized and we explicit the hypercomputational characteristics of the g W−H algebra.…”

mentioning

confidence: 99%

See 1 more Smart Citation

Quantum hypercomputation based on the dynamical algebra

Sicard¹,

Ospina²,

Vélez³

2006

J. Phys. A: Math. Gen.

View full text Add to dashboard Cite

Abstract. An adaptation of Kieu's hypercomputational quantum algorithm (KHQA) is presented. The method that was used was to replace the Weyl-Heisenberg algebra by other dynamical algebra of low dimension that admits infinite-dimensional irreducible representations with naturally defined generalized coherent states. We have selected the Lie algebra su(1, 1), due to that this algebra posses the necessary characteristics for to realize the hypercomputation and also due to that such algebra has been identified as the dynamical algebra associated to many relatively simple quantum systems. In addition to an algebraic adaptation of KHQA over the algebra su(1, 1), we presented an adaptations of KHQA over some concrete physical referents: the infinite square well, the infinite cylindrical well, the perturbed infinite cylindrical well, the Pöschl-Teller potentials, the Holstein-Primakoff system, and the Laguerre oscillator. We conclude that it is possible to have many physical systems within condensed matter and quantum optics on which it is possible to consider an implementation of KHQA. Quantum hypercomputation based on the dynamical algebra su(1, 1) 2 IntroductionThe hypercomputers compute functions or numbers, or more generally solve problems or carry out tasks, that cannot be computed or solved by a Turing machine (TM) [1,2]. Starting from that seems to be the first published model of hypercomputation, which is called the Turing's oracle machines [3]; the formulations of models and algorithms of hypercomputation have applied a wide spectrum of underlying theories [1,4,5]. It is precisely due to the existence of Turing's oracle machines that J. Copeland and D. Proudfoot introduced the term 'hypercomputation' by 1999 [6] for to replace the wrong expressions such as 'super-Turing computation', 'computing beyond Turing's limit', and 'breaking the Turing barrier', and similar.Recently Tien D. Kieu has proposed an quantum algorithm to solve the TM incomputable ‡ problem named Hilbert's tenth problem, using as physical referent the well known simple harmonic oscillator (SHO), which by effect of the second quantization has as associated dynamical algebra the Weyl-Heisenberg algebra denoted g W−H [8,9,10,11,12,13]. From the algebraic analysis of Kieu's hypercomputational quantum algorithm (KHQA), we have identified the underlying properties of the g W−H algebra which are necessary (but not sufficient) to guaranty KHQA works. Such properties are that the dynamical algebra admits infinite-dimensional irreducible representations with naturally associated coherent states.The importance of KHQA inside the field of hypercomputation, at the same tenor that the importance of hypercomputation within the domain of computer science, can not be sub-estimated. This algorithm is a plausible candidate for a practical implementation of the hypercomputation, maybe within the scope of the quantum optics. The adaptation of KHQA to a new physical referents different to the harmonic oscillator, opens the possibility of analyze news viable alterna...

show abstract

Continuous Hahn Functions as Clebsch-Gordan Coefficients

Groenevelt

Koelink

Rosengren

Theory and Applications of Special Functions

Self Cite

View full text Add to dashboard Cite

An explicit bilinear generating function for Meixner-Pollaczek polynomials is proved. This formula involves continuous dual Hahn polynomials, Meixner-Pollaczek functions, and nonpolynomial 3F2-hypergeometric functions that we consider as continuous Hahn functions. An integral transform pair with continuous Hahn functions as kernels is also proved. These results have an interpretation for the tensor product decomposition of a positive and a negative discrete series representation of su(1, 1) with respect to hyperbolic bases, where the Clebsch-Gordan coefficients are continuous Hahn functions.3 F 2 -hypergeometric functions, which we call continuous Hahn functions. We show that the continuous Hahn functions are the kernel in an integral transform, that generalizes the Jacobi function transform. We emphasize that the main part (sections 3 and 5) of this paper is analytic in nature, and that the Lie algebraic interpretation is mainly restricted to section 4.The Lie algebra su(1, 1) is generated by the three elements H, B and C. There are four classes of irreducible unitary representations for su(1, 1): discrete series, i.e. the positive and the negative discrete series representations, and continuous series, i.e. the principal unitary series and the complementary series representations. There are three kinds of basis elements on which the various representations can act: the elliptic, the parabolic and the hyperbolic basis elements. These three elements are related to conjugacy classes of the group SU (1, 1). We consider the tensor product of a positive and a negative discrete series representation, which decomposes into a direct integral over the principal unitary series representations. Under certain condition discrete terms can appear. The Clebsch-Gordan coefficients for the standard (elliptic) basis vectors are continuous dual Hahn polynomials. We compute the Clebsch-Gordan coefficients for the hyperbolic basis vectors, which

show abstract

Meixner functions and polynomials related to Lie algebra representations

Cited by 22 publications

References 14 publications

Bilinear Summation Formulas from Quantum Algebra Representations

Bilinear Summation Formulas from Quantum Algebra Representations

Quantum hypercomputation based on the dynamical algebra

Continuous Hahn Functions as Clebsch-Gordan Coefficients

Contact Info

Product

Resources

About