A generating function and formulae defining the first-associated Meixner–Pollaczek polynomials
18 Feb 2018-Integral Transforms and Special Functions (Informa UK Limited)-Vol. 29, Iss: 5, pp 352-366
TL;DR: In this article, the first associated Meixner-Pollaczek polynomials arising from nonlinear coherent states with anti-holomorphic coefficients were identified as orthogonal polynomial arising from coherent states.
Abstract: While considering nonlinear coherent states with anti-holomorphic coefficients z¯n/xn!, we identify as first-associated Meixner–Pollaczek polynomials the orthogonal polynomials arising from...
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01 Jan 1988
TL;DR: In this paper, an alternative proof is given for the connection between a system of continuous Hahn polynomials and identities for symmetric elements in the Heisenberg algebra, which was first observed by Bender, Mead, and Pinsky [Phys. Rev. Lett. 56, 2445 (1986); J. Math. 28, 509 (1987)].
Abstract: An alternative proof is given for the connection between a system of continuous Hahn polynomials and identities for symmetric elements in the Heisenberg algebra, which was first observed by Bender, Mead, and Pinsky [Phys. Rev. Lett. 56, 2445 (1986); J. Math. Phys. 28, 509 (1987)]. The continuous Hahn polynomials turn out to be Meixner–Pollaczek polynomials. Use is made of the connection between Laguerre polynomials and Meixner–Pollaczek polynomials, the Rodrigues formula for Laguerre polynomials, an operational formula involving Meixner–Pollaczek polynomials, and the Schrodinger model for the irreducible unitary representations of the three‐dimensional Heisenberg group.
34 citations
TL;DR: In this paper, a new proof of Wimp's formula for the associated Pollaczek polynomials Pnλz;a,b,c,d,e,f,c was presented.
Abstract: This paper is mainly devoted to generating functions of Pollaczek and other related polynomials. We first present a new proof of Wimp's formula for the associated Pollaczek polynomials Pnλz;a,b,c. ...
6 citations
TL;DR: In this article, a new form of a generating function for the c-associated Meixner-Pollaczek polynomials was derived, which is of particular importance since its several r...
Abstract: In the present paper we first derive a new form of a generating function for the c-associated Meixner-Pollaczek polynomials. This generating function is of particular importance since its several r...
2 citations
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TL;DR: In this article, a closed-form expression for the associated Meixner polynomials from which they derive closedform expressions for associated Charlier and Laguerre polynomial by a limit procedure is given.
Abstract: We give a closed-form expression for the associated Meixner polynomials from which we derive closed-form expressions for the associated Charlier and Laguerre polynomials by a limit procedure. These formulas are then used to derive generating functions of these polynomials. Also, we investigate the Mehler-Heine type asymptotic formulas for the associated Meixner and Charlier polynomials. Some applications and consequences of our results are also mentioned.
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TL;DR: In this paper, a falshe Verschiebung der Spektrallinie in der Richtung des helleren Teiles des Hintergrundes angibt.
Abstract: v o n d e r Achse nur einen Betrag yon 0 \" , i 8 erreicht und dem EINSTEINeffekte entgegengesetzt ist, so dab die Resultate yon CAMPBELL nicht dutch denselben beeintr~chtigt werden. Ich muI3 nur hinzuffigen, dab der EinfluB der verschiedehen Tubusla.nge anf die Verzeiehnung hierbei nicht in Betracht gezogen wurde. Aber auch be~ der Ausmessung der Plat ten unter Einstellung mit einem Fadenkreuze kann das Vorhandensein des Koronarlichtes auf der einen Platte eine Fehlerquelle bedeuten. Bei der Ausmessung yon R6ntgenspektrographischen Platten hat BKCKLIN im Inst i tute yon SIEGBAttN den Effekt des unsymmetrisehen Kontrastes beobachtet und beschrieben. Die Folge desselben ist, dab das Messungsresultat eine falsehe Verschiebung der Spektrallinie in der Richtung des helleren Teiles des Hintergrundes angibt. Dieser Kontrasteffekt mug also be~drken, dab das Koronarlicht eine falsche Verschiebung des gemessenen Sternbildes in der Richtung yon der Sonne weg zur Folge hat, w~hrend wahrscheinlich richtig auf das Vergleiehsbi-ld eingestellt werden kann, da dasselbe nicht in der geschw/irzten Schieht eingebettet Iiegt. Ob aber diese Verschiebung yon einer solchen Gr613enordnung ist, dab sie in Betracht kommt, l~f3t sich nicht vorhersagen. AuI alle F~tlle erfordert eine m6glichst differenfielle Messung, dab die zu vergleichenden Photographien mit Expositionszeiten, die m6glichst gleiche Sternbilder geben, auf einer und derselben Platte aufgenommen werden, welche zwischen den Aufnahmen ein wenig um die Achse des Objektives in bezug auf das Feld gedreht wird. yon der Mikroz u r M a k r o m e c h a n i k . [ Die Naturlwis~enschaften
1,267 citations
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01 Apr 1985
1,175 citations
TL;DR: In this paper, the authors introduced a new class of nonlinear coherent states (NCSs) related to the photon-added coherent states and proposed a scheme to generate a class of even and odd NCSs in the centre-of-mass motion of a trapped, laser-cooled, two-level ion.
Abstract: Nonclassical states of light are of fundamental importance in quantum optics. The properties of these states are due to the quantal nature of the electromagnetic field. In recent years there have been many experimental demonstrations of the realizability of nonclassical states in various physical schemes such as resonance fluorescence, four-wave mixing, centre-of-mass motion of a trapped ion and manipulation of field in a cavity. It is of interest to introduce new classes of nonclassical states and investigate their properties. In this context, generalization of the concept of coherent states (CSs) has played a major role. New concepts such as the nonlinear coherent states (NCSs) and interference in phase space have emerged from such studies. In this tutorial review we are concerned with the construction of new classes of nonclassical states by generalizing the notion of CSs. Our investigations on the photon-added coherent states (PACSs) indicate that these states can be interpreted as NCSs. Also, we introduce a new class of nonclassical states related to the PACSs. Having introduced a realizable example of NCSs, we extend the notion of even and odd CSs to the case of NCSs by introducing even and odd NCSs. With this new definition we interrelate some of the well known states of light. We suggest a scheme to generate a class of even and odd NCSs in the centre-of-mass motion of a trapped, laser-cooled, two-level ion.
112 citations
TL;DR: In this article, the Hahn and Meixner polynomials of a discrete variable are analytically continued in the complex plane both in variable and parameter, leading to the origination of two systems of real polynomial systems orthogonal with respect to a continuous measure.
Abstract: The Hahn and Meixner polynomials belonging to the classical orthogonal polynomials of a discrete variable are analytically continued in the complex plane both in variable and parameter. This leads to the origination of two systems of real polynomials orthogonal with respect to a continuous measure. The Meixner polynomials of an imaginary argument obtained in this manner turned out to be known in the literature as the Pollaczek polynomials. The orthogonality relation for the Hahn polynomials with respect to a continuous measure is apparently new. A close connection between the Hahn polynomials of an imaginary argument and representations of the Lorentz group SO(3,1) is considered.
100 citations
TL;DR: In this paper, the authors construct an appropriate oscillator algebra such that the polynomials make up the eigenfunctions system of the oscillator hamiltonian and obtain the explicit form of the hamiltonians, the energy levels and the explicit forms of the impulse operators.
Abstract: For any orthogonal polynomials system on real line we construct an appropriate oscillator algebra such that the polynomials make up the eigenfunctions system of the oscillator hamiltonian. The general scheme is divided into two types: a symmetric scheme and a non-symmetric scheme. the general approach is illustrated by the examples of the classical orthogonal poly-nominals: Hermite, Jacobi and Laguerre Polynomials. For these polynomials we obtain the explicit form of the hamiltonians, the energy levels and the explicit form of the impulse operators
80 citations