R. Askey and S. K. Suslov :
“The \( q \) -harmonic oscillator and an analogue of the Charlier polynomials ,”
J. Phys. A
26 : 15
(1993 ),
pp. L693–L698 .
MR
1239353
Zbl
0859.33021
ArXiv
math/9307206
article
Abstract
People
BibTeX
A model of a \( q \) -harmonic oscillator based on \( q \) -Charlier polynomials of Al-Salam and Carlitz [1965] is discussed. A simple explicit realization of \( q \) -creation and \( q \) -annihilation operators, \( q \) -coherent states and an analogue of the Fourier transformation are found. A connection of the kernel of this transform with biorthogonal rational functions is observed.
@article {key1239353m,
AUTHOR = {Askey, R. and Suslov, S. K.},
TITLE = {The \$q\$-harmonic oscillator and an analogue
of the {C}harlier polynomials},
JOURNAL = {J. Phys. A},
FJOURNAL = {Journal of Physics A. Mathematical and
General},
VOLUME = {26},
NUMBER = {15},
YEAR = {1993},
PAGES = {L693--L698},
DOI = {10.1088/0305-4470/26/15/014},
NOTE = {ArXiv:math/9307206. MR:1239353. Zbl:0859.33021.},
ISSN = {0305-4470},
}
R. Askey and S. K. Suslov :
“The \( q \) -harmonic oscillator and the Al-Salam and Carlitz polynomials ,”
Lett. Math. Phys.
29 : 2
(1993 ),
pp. 123–132 .
MR
1245186
Zbl
0919.33010
ArXiv
math/9307207
article
Abstract
People
BibTeX
One more model of a \( q \) -harmonic oscillator based on the \( q \) -orthogonal polynomials of Al-Salam and Carlitz is discussed. The explicit form of \( q \) -creation and \( q \) -annihilation operators, \( q \) -coherent states and an analog of the Fourier transformation are established. A connection of the kernel of this transform with a family of self-dual biorthogonal rational functions is observed.
@article {key1245186m,
AUTHOR = {Askey, R. and Suslov, S. K.},
TITLE = {The \$q\$-harmonic oscillator and the
{A}l-{S}alam and {C}arlitz polynomials},
JOURNAL = {Lett. Math. Phys.},
FJOURNAL = {Letters in Mathematical Physics},
VOLUME = {29},
NUMBER = {2},
YEAR = {1993},
PAGES = {123--132},
DOI = {10.1007/BF00749728},
NOTE = {ArXiv:math/9307207. MR:1245186. Zbl:0919.33010.},
ISSN = {0377-9017},
}
R. Askey, N. M. Atakishiyev, and S. K. Suslov :
“An analog of the Fourier transformation for a \( q \) -harmonic oscillator ,”
pp. 57–63
in
Symmetries in science, VI: From the rotation group to quantum algebras
(Bregenz, Austria, 2–7 August 1992 ).
Edited by B. Gruber .
Plenum (New York ),
1993 .
MR
1258093
Zbl
0863.33020
ArXiv
math/9307205
incollection
Abstract
People
BibTeX
@incollection {key1258093m,
AUTHOR = {Askey, R. and Atakishiyev, N. M. and
Suslov, S. K.},
TITLE = {An analog of the {F}ourier transformation
for a \$q\$-harmonic oscillator},
BOOKTITLE = {Symmetries in science, {VI}: {F}rom
the rotation group to quantum algebras},
EDITOR = {Gruber, Bruno},
PUBLISHER = {Plenum},
ADDRESS = {New York},
YEAR = {1993},
PAGES = {57--63},
DOI = {10.1007/978-1-4899-1219-0_5},
NOTE = {(Bregenz, Austria, 2--7 August 1992).
ArXiv:math/9307205. MR:1258093. Zbl:0863.33020.},
ISBN = {9780306445842},
}
D. Gasper and M. Rakhman :
Bazisnye gipergeometricheskie ryady
[Basic hypergeometric series ].
Mir (Moscow ),
1993 .
With a foreword by Richard Askey. Translated from the English and with a preface by N. M. Atakishiev and S. K. Suslov.
Russian translation of 1990 original .
MR
1286735
Zbl
0848.33011
book
People
BibTeX
@book {key1286735m,
AUTHOR = {Gasper, Dzh. and Rakhman, M.},
TITLE = {Bazisnye gipergeometricheskie ryady
[Basic hypergeometric series]},
PUBLISHER = {Mir},
ADDRESS = {Moscow},
YEAR = {1993},
PAGES = {349},
NOTE = {With a foreword by Richard Askey. Translated
from the English and with a preface
by N.~M. Atakishiev and S.~K. Suslov.
Russian translation of 1990 original.
MR:1286735. Zbl:0848.33011.},
}
R. Askey, N. M. Atakishiyev, and S. K. Suslov :
“Fourier transformations for difference analogs of the harmonic oscillator ,”
pp. 140–144
in
Problems on workshop on high energy physics and field theory
(Protvino, Russia, 6–10 July 1992 ).
Edited by A. P. Samokhin and G. L. Rcheulishvili .
IHEP (Protvino, Russia ),
1995 .
incollection
Abstract
People
BibTeX
@incollection {key75421263,
AUTHOR = {Askey, Richard and Atakishiyev, N. M.
and Suslov, S. K.},
TITLE = {Fourier transformations for difference
analogs of the harmonic oscillator},
BOOKTITLE = {Problems on workshop on high energy
physics and field theory},
EDITOR = {Samokhin, A. P. and Rcheulishvili, G.
L.},
PUBLISHER = {IHEP},
ADDRESS = {Protvino, Russia},
YEAR = {1995},
PAGES = {140--144},
NOTE = {(Protvino, Russia, 6--10 July 1992).},
ISBN = {9785851650659},
}
R. A. Askey, M. Rahman, and S. K. Suslov :
“On a general \( q \) -Fourier transformation with nonsymmetric kernels ,”
J. Comput. Appl. Math.
68 : 1–2
(April 1996 ),
pp. 25–55 .
MR
1418749
Zbl
0871.33008
ArXiv
math/9411229
article
Abstract
People
BibTeX
Wiener used the Poisson kernel for the Hermite polynomials to deal with the classical Fourier transform. Askey, Atakishiyev and Suslov used this approach to obtain a \( q \) -Fourier transform by using the continuous \( q \) -Hermite polynomials. Rahman and Suslov extended this result by taking the Askey–Wilson polynomials, considered to be the most general continuous classical orthogonal polynomials. The theory of \( q \) -Fourier transformation is further extended here by considering a nonsymmetric version of the Poisson kernel with Askey–Wilson polynomials. This approach enables us to obtain some new results, for example, the complex and real orthogonalities of these kernels.
@article {key1418749m,
AUTHOR = {Askey, Richard A. and Rahman, Mizan
and Suslov, Serge\u{\i} K.},
TITLE = {On a general \$q\$-{F}ourier transformation
with nonsymmetric kernels},
JOURNAL = {J. Comput. Appl. Math.},
FJOURNAL = {Journal of Computational and Applied
Mathematics},
VOLUME = {68},
NUMBER = {1--2},
MONTH = {April},
YEAR = {1996},
PAGES = {25--55},
DOI = {10.1016/0377-0427(95)00259-6},
NOTE = {ArXiv:math/9411229. MR:1418749. Zbl:0871.33008.},
ISSN = {0377-0427},
}
G. E. Andrews, G. Gasper, and S. Suslov :
“Preface ,”
pp. 5–6
in
Askey special issues ,
published as Ramanujan J.
13 : 1–3 .
Springer (New York ),
June 2007 .
MR
2281153
Zbl
1186.00042
incollection
People
BibTeX
@article {key2281153m,
AUTHOR = {Andrews, George E. and Gasper, George
and Suslov, Sergei},
TITLE = {Preface},
JOURNAL = {Ramanujan J.},
FJOURNAL = {The Ramanujan Journal},
VOLUME = {13},
NUMBER = {1--3},
MONTH = {June},
YEAR = {2007},
PAGES = {5--6},
DOI = {10.1007/s11139-006-0239-z},
NOTE = {\textit{Askey special issues}. MR:2281153.
Zbl:1186.00042.},
ISSN = {1382-4090},
}