R. Askey, J. Fitch, and G. Gasper :
“On a positive trigonometric sum ,”
Proc. Am. Math. Soc.
19 : 6
(December 1968 ),
pp. 1507 .
MR
231107
Zbl
0174.35704
article
Abstract
People
BibTeX
@article {key231107m,
AUTHOR = {Askey, Richard and Fitch, James and
Gasper, George},
TITLE = {On a positive trigonometric sum},
JOURNAL = {Proc. Am. Math. Soc.},
FJOURNAL = {Proceedings of the American Mathematical
Society},
VOLUME = {19},
NUMBER = {6},
MONTH = {December},
YEAR = {1968},
PAGES = {1507},
DOI = {10.2307/2036244},
NOTE = {MR:231107. Zbl:0174.35704.},
ISSN = {0002-9939},
}
R. Askey and G. Gasper :
“Linearization of the product of Jacobi polynomials, III ,”
Can. J. Math.
23 : 2
(1971 ),
pp. 332–338 .
MR
273273
Zbl
0212.40904
article
People
BibTeX
@article {key273273m,
AUTHOR = {Askey, Richard and Gasper, George},
TITLE = {Linearization of the product of {J}acobi
polynomials, {III}},
JOURNAL = {Can. J. Math.},
FJOURNAL = {Canadian Journal of Mathematics. Journal
Canadien de Math\'ematiques},
VOLUME = {23},
NUMBER = {2},
YEAR = {1971},
PAGES = {332--338},
DOI = {10.4153/CJM-1971-033-6},
NOTE = {MR:273273. Zbl:0212.40904.},
ISSN = {0008-414X},
}
R. Askey and G. Gasper :
“Jacobi polynomial expansions of Jacobi polynomials with non-negative coefficients ,”
Proc. Cambridge Philos. Soc.
70 : 2
(1971 ),
pp. 243–255 .
MR
296369
Zbl
0217.11402
article
People
BibTeX
@article {key296369m,
AUTHOR = {Askey, Richard and Gasper, George},
TITLE = {Jacobi polynomial expansions of {J}acobi
polynomials with non-negative coefficients},
JOURNAL = {Proc. Cambridge Philos. Soc.},
FJOURNAL = {Proceedings of the Cambridge Philosophical
Society},
VOLUME = {70},
NUMBER = {2},
YEAR = {1971},
PAGES = {243--255},
DOI = {10.1017/s0305004100049847},
NOTE = {MR:296369. Zbl:0217.11402.},
ISSN = {0008-1981},
}
G. Gasper :
“A positive integral (Richard Askey) ,”
SIAM Review
13 : 3
(1971 ),
pp. 396–397 .
article
People
BibTeX
@article {key91098074,
AUTHOR = {Gasper, George},
TITLE = {A positive integral (Richard Askey)},
JOURNAL = {SIAM Review},
VOLUME = {13},
NUMBER = {3},
YEAR = {1971},
PAGES = {396--397},
DOI = {10.1137/1013076},
}
R. Askey and G. Gasper :
“Certain rational functions whose power series have positive coefficients ,”
Am. Math. Monthly
79 : 4
(April 1972 ),
pp. 327–341 .
Part II was published in SIAM J. Math. Anal. 5 :1 (1974) . This also appeared as a 1970 technical report .
MR
301441
Zbl
0242.33023
article
People
BibTeX
@article {key301441m,
AUTHOR = {Askey, Richard and Gasper, George},
TITLE = {Certain rational functions whose power
series have positive coefficients},
JOURNAL = {Am. Math. Monthly},
FJOURNAL = {American Mathematical Monthly},
VOLUME = {79},
NUMBER = {4},
MONTH = {April},
YEAR = {1972},
PAGES = {327--341},
DOI = {10.2307/2978081},
NOTE = {Part II was published in \textit{SIAM
J. Math. Anal.} \textbf{5}:1 (1974).
This also appeared as a 1970 technical
report. MR:301441. Zbl:0242.33023.},
ISSN = {0002-9890},
}
C. C. Rousseau, R. Askey, and G. Gasper :
“Problem 71-13: ‘A positive integral’ by L. Carlitz ,”
SIAM Rev.
14
(1972 ),
pp. 372–378 .
MR
304731
article
People
BibTeX
@article {key304731m,
AUTHOR = {Rousseau, C. C. and Askey, R. and Gasper,
G.},
TITLE = {Problem 71-13: ``{A} positive integral''
by {L}. {C}arlitz},
JOURNAL = {SIAM Rev.},
FJOURNAL = {SIAM Review. A Publication of the Society
for Industrial and Applied Mathematics},
VOLUME = {14},
YEAR = {1972},
PAGES = {372--378},
DOI = {10.1137/1014061},
NOTE = {MR:304731.},
ISSN = {0036-1445},
}
R. Askey, G. Gasper, and M. E.-H. Ismail :
“A positive sum from summability theory ,”
J. Approximation Theory
13 : 4
(April 1975 ),
pp. 413–420 .
Dedicated to G. G. Lorentz on the occasion of his sixty-fifth birthday.
MR
369978
Zbl
0295.40008
article
Abstract
People
BibTeX
@article {key369978m,
AUTHOR = {Askey, Richard and Gasper, George and
Ismail, Mourad E.-H.},
TITLE = {A positive sum from summability theory},
JOURNAL = {J. Approximation Theory},
FJOURNAL = {Journal of Approximation Theory},
VOLUME = {13},
NUMBER = {4},
MONTH = {April},
YEAR = {1975},
PAGES = {413--420},
DOI = {10.1016/0021-9045(75)90024-6},
NOTE = {Dedicated to G. G. Lorentz on the occasion
of his sixty-fifth birthday. MR:369978.
Zbl:0295.40008.},
ISSN = {0021-9045},
}
R. Askey, G. Gasper, and L. A. Harris :
“An inequality for Tchebycheff polynomials and extensions ,”
J. Approximation Theory
14 : 1
(May 1975 ),
pp. 1–11 .
MR
372285
Zbl
0297.33018
article
Abstract
People
BibTeX
The inequality
\[ T_n(xy) \leq T_n(x)T_n(y) ,\]
\( x,y \geq 1 \) , where \( T_n(x) \) is the Tchebycheff polynomial of the first kind, can be proven very easily by use of one of the extremal properties of these polynomials. It also follows from
\[ (d^2 du^2)\log T_n(e^u) \leq 0 ,\]
\( u\geq 0 \) . Various proofs are given for these inequalities and for generalizations to other classes of polynomials.
@article {key372285m,
AUTHOR = {Askey, Richard and Gasper, George and
Harris, Lawrence A.},
TITLE = {An inequality for {T}chebycheff polynomials
and extensions},
JOURNAL = {J. Approximation Theory},
FJOURNAL = {Journal of Approximation Theory},
VOLUME = {14},
NUMBER = {1},
MONTH = {May},
YEAR = {1975},
PAGES = {1--11},
DOI = {10.1016/0021-9045(75)90061-1},
NOTE = {MR:372285. Zbl:0297.33018.},
ISSN = {0021-9045},
}
R. Askey and G. Gasper :
“Positive Jacobi polynomial sums, II ,”
Am. J. Math.
98 : 3
(1976 ),
pp. 709–737 .
Part I was published in Tohoku Math. J. 24 :2 (1972) . Part III was published in Linear operators and approximation, II (1974) .
MR
430358
Zbl
0355.33005
article
Abstract
People
BibTeX
Among the positive polynomial sums of Jacobi polynomials, there are two which have been very useful, the Cesàro means of the formal reproducing kernel and the sum
\[ \sum_{k=0}^n \frac{P_k^{(\alpha,\beta)}(x)}{P_k^{(\beta,\alpha)}(1)} , \tag{*} \]
which was first considered by Fejér when \( \alpha = 1/2 \) , \( \beta = \pm 1/2 \) and when \( \alpha = \beta = 0 \) . A conjecture is given which connects these two sets of inequalities and this conjecture is proven for many values of \( (\alpha,\beta) \) . In particular, it is shown that if \( \beta\geq 0 \) , then the sum (*) is nonnegative for \( -1 \leq x \leq 1 \) if and only if \( \alpha + \beta \geq -2 \) . It is also shown that
\[ \sum_{k=0}^n
\frac{(\lambda+1)_{n-k}}{(n-k)!} \frac{(\lambda+1)_k}{k!} \frac{\sin(k+1)\theta}{k+1} > 0 ,\]
\( 0 < \theta < \pi \) , \( -1 < \lambda \leq 1 \) , and that for real \( \alpha \) the function
\[ (1 - r)^{-|\alpha|}[1 \pm r + (1 - 2xr + r^2)^{1/2}]^{\alpha} \]
is absolutely monotonic for \( -1 \leq x \leq 1 \) , i.e., it has nonnegative power series coefficients when it is expanded in a power series in \( r \) . Limiting cases involving Laguerre and Hermite polynomials are also considered.
@article {key430358m,
AUTHOR = {Askey, Richard and Gasper, George},
TITLE = {Positive {J}acobi polynomial sums, {II}},
JOURNAL = {Am. J. Math.},
FJOURNAL = {American Journal of Mathematics},
VOLUME = {98},
NUMBER = {3},
YEAR = {1976},
PAGES = {709--737},
DOI = {10.2307/2373813},
NOTE = {Part I was published in \textit{Tohoku
Math. J.} \textbf{24}:2 (1972). Part
III was published in \textit{Linear
operators and approximation, II} (1974).
MR:430358. Zbl:0355.33005.},
ISSN = {0002-9327},
}
R. Askey and G. Gasper :
“Convolution structures for Laguerre polynomials ,”
J. Analyse Math.
31
(1977 ),
pp. 48–68 .
MR
486692
Zbl
0347.33006
article
People
BibTeX
@article {key486692m,
AUTHOR = {Askey, Richard and Gasper, George},
TITLE = {Convolution structures for {L}aguerre
polynomials},
JOURNAL = {J. Analyse Math.},
FJOURNAL = {Journal d'Analyse Math\'{e}matique},
VOLUME = {31},
YEAR = {1977},
PAGES = {48--68},
DOI = {10.1007/BF02813297},
NOTE = {MR:486692. Zbl:0347.33006.},
ISSN = {0021-7670},
}
R. Askey and G. Gasper :
“Inequalities for polynomials ,”
pp. 7–32
in
The Bieberbach conjecture
(West Lafayette, IN, 11–14 March 1985 ).
Edited by D. Drasin, P. Duren, and A. Marden .
Mathematical Surveys and Monographs 21 .
American Mathematical Society (Providence, RI ),
1986 .
MR
875228
incollection
Abstract
People
BibTeX
The polynomial inequality that de Branges used in his solution of the Bieberbach conjecture is equivalent to the positivity of the sum of certain Jacobi polynomials. There are many other positive sums of Jacobi polynomials, including some that are fractional derivatives of order one-half of the ones de Branges used. A historical summary of some of these inequalities is given, Koornwinder’s extension of the de Branges–Milin type inequality is outlined as an example of a refinement suggested by inequalities for hypergeometric functions, and a few places to look for other refinements are mentioned.
@incollection {key875228m,
AUTHOR = {Askey, Richard and Gasper, George},
TITLE = {Inequalities for polynomials},
BOOKTITLE = {The {B}ieberbach conjecture},
EDITOR = {Drasin, David and Duren, Peter and Marden,
Albert},
SERIES = {Mathematical Surveys and Monographs},
NUMBER = {21},
PUBLISHER = {American Mathematical Society},
ADDRESS = {Providence, RI},
YEAR = {1986},
PAGES = {7--32},
DOI = {10.1090/surv/021/02},
NOTE = {(West Lafayette, IN, 11--14 March 1985).
MR:875228.},
ISSN = {0076-5376},
ISBN = {9780821815212},
}
G. Gasper and M. Rahman :
Basic hypergeometric series .
Encyclopedia of Mathematics and its Applications 35 .
Cambridge University Press ,
1990 .
With a foreword by Richard Askey.
A Russian translation was published in 1993 . 2nd edition published in 2004 .
MR
1052153
Zbl
0695.33001
book
People
BibTeX
@book {key1052153m,
AUTHOR = {Gasper, G. and Rahman, M.},
TITLE = {Basic hypergeometric series},
SERIES = {Encyclopedia of Mathematics and its
Applications},
NUMBER = {35},
PUBLISHER = {Cambridge University Press},
YEAR = {1990},
PAGES = {xx+287},
NOTE = {With a foreword by Richard Askey. A
Russian translation was published in
1993. 2nd edition published in 2004.
MR:1052153. Zbl:0695.33001.},
ISSN = {0953-4806},
ISBN = {9780521350495},
}
G. Gasper and M. Rahman :
Basic hypergeometric series ,
2nd edition.
Encyclopedia of Mathematics and its Applications 96 .
Cambridge University Press ,
1990 .
With a foreword by Richard Askey.
Republication of 1990 original .
MR
2128719
Zbl
1129.33005
book
People
BibTeX
@book {key2128719m,
AUTHOR = {Gasper, G. and Rahman, M.},
TITLE = {Basic hypergeometric series},
EDITION = {2nd},
SERIES = {Encyclopedia of Mathematics and its
Applications},
NUMBER = {96},
PUBLISHER = {Cambridge University Press},
YEAR = {1990},
PAGES = {xxvi+428},
NOTE = {With a foreword by Richard Askey. Republication
of 1990 original. MR:2128719. Zbl:1129.33005.},
ISSN = {0953-4806},
ISBN = {9780521833578},
}
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.},
}
G. Gasper, M. E. H. Ismail, T. Koornwinder, P. Nevai, and D. Stanton :
“The mathematical contributions of Richard Askey ,”
pp. 1–18
in
-Series from a contemporary perspective
(South 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},
}
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},
}