Open Access
Meixner-Pollaczek polynomials and the Heisenberg algebra
Tom H. Koornwinder
- pp 1-6
TLDR
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.read more
Citations
More filters
Journal Article
Convolutions for orthogonal polynomials from Lie and quantum algebra representations
H.T. Koelink,J. Van der Jeugt +1 more
TL;DR: In this article, the authors derived convolution identities for Al-Salam and Chihara polynomials by using the Clebsch-Gordan decomposition and the Racah coefficients.
Journal ArticleDOI
Shape Invariant Potentials in "Discrete Quantum Mechanics"
Satoru Odake,Ryu Sasaki +1 more
TL;DR: Shape invariance is an important ingredient of many exactly solvable quantum mechanics and several examples of shape invariant ''discrete quantum mechanical systems'' are introduced and discussed in some detail as mentioned in this paper.
Posted Content
Additions to the formula lists in \Hypergeometric orthogonal polynomials and their q-analogues" by Koekoek, Lesky and Swarttouw
TL;DR: In this article, the authors give a rather arbitrary choice of formulas for (q-)hypergeometric orthogonal polynomials which the author missed while consulting Chapters 9 and 14 in the book "Hypergeometric OOOP and their q-analogues" by Koekoek, Lesky and Swarttouw.
Journal ArticleDOI
On the Asymptotics of the Meixner—Pollaczek Polynomials and Their Zeros
X. Li,Roderick Wong +1 more
TL;DR: In this article, an infinite asymptotic expansion for the Meixner-Pollaczek polynomials was derived, which holds uniformly for -M≤α≤ M, where M can be any positive number.
Journal ArticleDOI
Solution of the Hausdorff moment problem by the use of Pollaczek polynomials
TL;DR: In this article, a new method for solving the Hausdorff moment problem using Pollaczek polynomials is presented, and it is shown that the expansion is asymptotically convergent in the L2-norm when the data (i.e., the moments) are perturbed by noise and finite in number.
References
More filters
Journal ArticleDOI
Form Sequences to Polynomials and Back, via Operator Orderings
TL;DR: In this article, Bender and Dunne showed that linear combinations of words $p$ and $q$ are subject to the relation $qp - pq = \imath, and that such linear combinations can be expressed as a polynomial in the symbol $z = \tfrac{1}{2}(qp+pq)
Linearization and connection problems for the Symmetric Meixner-Pollaczek polynomials
TL;DR: In this article, the authors considered the problem of finding orthogonal polynomials on the real line with respect to any positive real measure for failing to satisfy Favard's three term recurrence relation condition.
Journal ArticleDOI
A generating function and formulae defining the first-associated Meixner–Pollaczek polynomials
Khalid Ahbli,Zouhair Mouayn +1 more
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.
Proceedings ArticleDOI
Ideal interpolation: Mourrain's condition vs. $D$-invariance
TL;DR: In this article, it was shown that for more general polynomial spaces, D-invariance and being connected at 1 are unrelated, and that Mourrain's characterization need not hold when his condition is replaced by D-Invariance.
Journal ArticleDOI
The Weyl algebra, spherical harmonics, and Hahn polynomials
TL;DR: In this paper, the authors apply the duality technique of R. Howe to study the structure of the Weyl algebra and introduce a one-parameter family of ordering maps, where by an ordering map we understand a vector space isomorphism of the polynomial algebra on $\NR^{2d}$ with the weyl algebra generated by creation and annihilation operators.