R. Askey, M. E. H. Ismail, and T. Koornwinder :
“Weighted permutation problems and Laguerre polynomials J. Combin. Theory Ser. A
25 : 3
(November 1978 ),
pp. 277–287 .
MR
514623 Zbl
0405.05008 article
Abstract
People
BibTeX
The connection between integrals of products of Laguerre polynomials, power series coefficients of certain rational functions of several variables, and certain numbers of weighted permutation problems is investigated. A combinatorial proof of our main result would be very desirable since this could lead the way to more general result, \( q \) -analogs, and perhaps even a \( q \) -analog of MacMahon’s Master Theorem.
@article {key514623m,
AUTHOR = {Askey, Richard and Ismail, Mourad E.
H. and Koornwinder, Tom},
TITLE = {Weighted permutation problems and {L}aguerre
polynomials},
JOURNAL = {J. Combin. Theory Ser. A},
FJOURNAL = {Journal of Combinatorial Theory. Series
A},
VOLUME = {25},
NUMBER = {3},
MONTH = {November},
YEAR = {1978},
PAGES = {277--287},
DOI = {10.1016/0097-3165(78)90020-1},
NOTE = {MR:514623. Zbl:0405.05008.},
ISSN = {0097-3165},
}
Special functions: Group theoretical aspects and applications R. A. Askey, T. H. Koornwinder, and W. Schempp .
Mathematics and Its Applications 18 .
Reidel (Dordrecht ),
1984 .
MR
774053 Zbl
0543.00007 book
People
BibTeX
@book {key774053m,
TITLE = {Special functions: {G}roup theoretical
aspects and applications},
EDITOR = {Askey, R. A. and Koornwinder, T. H.
and Schempp, W.},
SERIES = {Mathematics and Its Applications},
NUMBER = {18},
PUBLISHER = {Reidel},
ADDRESS = {Dordrecht},
YEAR = {1984},
PAGES = {xxiv+311},
DOI = {10.1007/978-94-010-9787-1},
NOTE = {MR:774053. Zbl:0543.00007.},
ISSN = {0921-3791},
ISBN = {9781402003196},
}
R. Askey :
“Preface ,”
pp. xi–xxxiv
in
Special functions: Group theoretical aspects and applications .
Edited by R. A. Askey, T. H. Koornwinder, and W. Schempp .
Mathematics and Its Applications 18 .
Reidel (Dordrecht ),
1984 .
MR
774054 incollection
People
BibTeX
@incollection {key774054m,
AUTHOR = {Askey, Richard},
TITLE = {Preface},
BOOKTITLE = {Special functions: {G}roup theoretical
aspects and applications},
EDITOR = {Askey, R. A. and Koornwinder, T. H.
and Schempp, W.},
SERIES = {Mathematics and Its Applications},
NUMBER = {18},
PUBLISHER = {Reidel},
ADDRESS = {Dordrecht},
YEAR = {1984},
PAGES = {xi--xxxiv},
NOTE = {MR:774054.},
ISSN = {0921-3791},
ISBN = {9781402003196},
}
R. Askey, T. H. Koornwinder, and M. Rahman :
“An integral of products of ultraspherical functions and a \( q \) -extension J. London Math. Soc. (2)
33 : 1
(1986 ),
pp. 133–148 .
MR
829394 Zbl
0564.33008 article
Abstract
People
BibTeX
Let \( P_n(x) \) and \( Q_n(x) \) denote the Legendre polynomial of degree \( n \) and the usual second solution to the differential equation, respectively. Din showed that
\[ \int_{-1}^1 Q_n(x)P_m(x)P_l(x)\,dx \]
vanishes when \( |l-m| < n < l + m \) , and Askey evaluated the integral for arbitrary integral values of \( l \) , \( m \) and \( n \) . We extend this to the evaluation of
\[ \int_{-1}^1 D_n^{\lambda}(x)C_m^{\lambda}(x)C_l^{\lambda}(x)(1 - x^2)^{2\lambda - 1}\,dx ,\]
where \( C_n^{\lambda}(x) \) is the ultraspherical polynomial and \( C_n^{\lambda}(x) \) is the appropriate second solution to the ultraspherical differential equation. A \( q \) -extension is found using the continuous \( q \) -ultraspherical polynomials of Rogers. Again the integral vanishes when \( |l - m| < n < l + m \) . It is shown that this vanishing phenomenon holds for quite general orthogonal polynomials. A related integral of the product of three Bessel functions is also evaluated.
@article {key829394m,
AUTHOR = {Askey, Richard and Koornwinder, Tom
H. and Rahman, Mizan},
TITLE = {An integral of products of ultraspherical
functions and a \$q\$-extension},
JOURNAL = {J. London Math. Soc. (2)},
FJOURNAL = {Journal of the London Mathematical Society.
Second Series},
VOLUME = {33},
NUMBER = {1},
YEAR = {1986},
PAGES = {133--148},
DOI = {10.1112/jlms/s2-33.1.133},
NOTE = {MR:829394. Zbl:0564.33008.},
ISSN = {0024-6107},
}
R. A. Askey, W. Koepf, and T. H. Koornwinder :
“Foreword ,”
pp. 737–738
in
Special issue on orthogonal polynomials and computer algebra ,
published as J. Symb. Comput.
28 : 6 .
Issue edited by R. A. Askey, W. Koepf, and T. H. Koornwinder .
Elsevier (London ),
December 1999 .
incollection
People
BibTeX
@article {key71200140,
AUTHOR = {Askey, R. A. and Koepf, W. and Koornwinder,
T. H.},
TITLE = {Foreword},
JOURNAL = {J. Symb. Comput.},
FJOURNAL = {Journal of Symbolic Computation},
VOLUME = {28},
NUMBER = {6},
MONTH = {December},
YEAR = {1999},
PAGES = {737--738},
NOTE = {\textit{Special issue on orthogonal
polynomials and computer algebra}. Issue
edited by R. A. Askey,
W. Koepf, and T. H. Koornwinder.},
ISSN = {0747-7171},
}
Special issue on orthogonal polynomials and computer algebra ,
published as J. Symb. Comput.
28 : 6 .
Issue edited by R. A. Askey, W. Koepf, and T. H. Koornwinder .
Elsevier (London ),
December 1999 .
MR
1750542 Zbl
0944.00022 book
People
BibTeX
@book {key1750542m,
TITLE = {Special issue on orthogonal polynomials
and computer algebra},
EDITOR = {Askey, R. A. and Koepf, W. and Koornwinder,
T. H.},
PUBLISHER = {Elsevier},
ADDRESS = {London},
MONTH = {December},
YEAR = {1999},
PAGES = {737--845},
NOTE = {Published as \textit{J. Symb. Comput.}
\textbf{28}:6. MR:1750542. Zbl:0944.00022.},
ISSN = {0747-7171},
}
G. Gasper, M. E. H. Ismail, T. Koornwinder, P. Nevai, and D. Stanton :
“The mathematical contributions of Richard Askey ,”
pp. 1–18
in
\( q \) -Series from a contemporary perspectiveSouth Hadley, MA, 21–25 June 1998 ).
Edited by M. E. H. Ismail and D. W. Stanton .
Contemporary Mathematics 254 .
American Mathematical Society (Providence, RI ),
2000 .
MR
1768919 Zbl
0953.01015 incollection
People
BibTeX
@incollection {key1768919m,
AUTHOR = {Gasper, George and Ismail, Mourad E.
H. and Koornwinder, Tom and Nevai, Paul
and Stanton, Dennis},
TITLE = {The mathematical contributions of {R}ichard
{A}skey},
BOOKTITLE = {\$q\$-{S}eries from a contemporary perspective},
EDITOR = {Ismail, Mourad E. H. and Stanton, Dennis
W.},
SERIES = {Contemporary Mathematics},
NUMBER = {254},
PUBLISHER = {American Mathematical Society},
ADDRESS = {Providence, RI},
YEAR = {2000},
PAGES = {1--18},
NOTE = {(South Hadley, MA, 21--25 June 1998).
MR:1768919. Zbl:0953.01015.},
ISSN = {0271-4132},
ISBN = {9780821811504},
}
Special issue dedicated to Dick Askey on the occasion of his 80th birthday J. Approx. Theory
193 .
Issue edited by M. Ismail, T. Koornwinder, P. Nevai, W. Van Assche, and O. Warnaar .
Elsevier (San Diego, CA ),
May 2015 .
book
People
BibTeX
@book {key35068102,
TITLE = {Special issue dedicated to {D}ick {A}skey
on the occasion of his 80th birthday},
EDITOR = {Ismail, Mourad and Koornwinder, Tom
and Nevai, Paul and Van Assche, Walter
and Warnaar, Ole},
PUBLISHER = {Elsevier},
ADDRESS = {San Diego, CA},
MONTH = {May},
YEAR = {2015},
PAGES = {210},
URL = {https://www.sciencedirect.com/journal/journal-of-approximation-theory/vol/193/suppl/C},
NOTE = {Published as \textit{J. Approx. Theory}
\textbf{193}.},
ISSN = {0021-9045},
}
T. Koornwinder, W. Van Assche, and O. Warnaar :
“A tribute to Dick Askey 1–3
in
Special issue dedicated to Dick Askey on the occasion of his 80th birthday ,
published as J. Approx. Theory
193 .
Issue edited by M. Ismail, T. Koornwinder, P. Nevai, W. Van Assche, and O. Warnaar .
Elsevier (San Diego, CA ),
May 2015 .
MR
3324559 Zbl
1315.01039 ArXiv
1909.10508 incollection
People
BibTeX
@article {key3324559m,
AUTHOR = {Koornwinder, Tom and Van Assche, Walter
and Warnaar, Ole},
TITLE = {A tribute to {D}ick {A}skey},
JOURNAL = {J. Approx. Theory},
FJOURNAL = {Journal of Approximation Theory},
VOLUME = {193},
MONTH = {May},
YEAR = {2015},
PAGES = {1--3},
DOI = {10.1016/j.jat.2015.03.001},
NOTE = {\textit{Special issue dedicated to {D}ick
{A}skey on the occasion of his 80th
birthday}. Issue edited by M. Ismail,
T. Koornwinder, P. Nevai,
W. Van Assche, and O. Warnaar.
ArXiv:1909.10508. MR:3324559. Zbl:1315.01039.},
ISSN = {0021-9045},
}
