Multivariable Askey–Wilson function and bispectrality
For every positive integer d , we define a meromorphic function F d ( n ; z ), where n , z ∈ℂ d , which is a natural extension of the multivariable Askey–Wilson polynomials of Gasper and Rahman (Theory and Applications of Special Functions, Dev. Math., vol. 13, pp. 209–219, Springer, New York, 2005...
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| Vydáno v: | The Ramanujan journal Ročník 24; číslo 3; s. 273 - 287 |
|---|---|
| Hlavní autoři: | , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
Boston
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01.04.2011
|
| Témata: | |
| ISSN: | 1382-4090, 1572-9303 |
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| Abstract | For every positive integer
d
, we define a meromorphic function
F
d
(
n
;
z
), where
n
,
z
∈ℂ
d
, which is a natural extension of the multivariable Askey–Wilson polynomials of Gasper and Rahman (Theory and Applications of Special Functions, Dev. Math., vol. 13, pp. 209–219, Springer, New York,
2005
). It is defined as a product of very-well-poised
8
φ
7
series and we show that it is a common eigenfunction of two commutative algebras
and
of difference operators acting on
z
and
n
, with eigenvalues depending on
n
and
z
, respectively. In particular, this leads to certain identities connecting products of very-well-poised
8
φ
7
series. |
|---|---|
| AbstractList | For every positive integer
d
, we define a meromorphic function
F
d
(
n
;
z
), where
n
,
z
∈ℂ
d
, which is a natural extension of the multivariable Askey–Wilson polynomials of Gasper and Rahman (Theory and Applications of Special Functions, Dev. Math., vol. 13, pp. 209–219, Springer, New York,
2005
). It is defined as a product of very-well-poised
8
φ
7
series and we show that it is a common eigenfunction of two commutative algebras
and
of difference operators acting on
z
and
n
, with eigenvalues depending on
n
and
z
, respectively. In particular, this leads to certain identities connecting products of very-well-poised
8
φ
7
series. |
| Author | Geronimo, Jeffrey S. Iliev, Plamen |
| Author_xml | – sequence: 1 givenname: Jeffrey S. surname: Geronimo fullname: Geronimo, Jeffrey S. organization: School of Mathematics, Georgia Institute of Technology – sequence: 2 givenname: Plamen surname: Iliev fullname: Iliev, Plamen email: iliev@math.gatech.edu organization: School of Mathematics, Georgia Institute of Technology |
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| Cites_doi | 10.1063/1.529228 10.1016/j.aim.2006.09.012 10.1063/1.529158 10.1023/A:1011439924912 10.1090/S0002-9947-2010-05183-9 10.1007/BF01231441 10.1155/S1073792801000575 10.2307/2001881 10.1007/s002220050102 10.1007/s11139-006-8478-6 10.1007/BF01206937 10.1214/aoms/1177698308 10.1017/CBO9780511565717 10.1007/s11139-006-0259-8 10.2307/121102 10.1088/0305-4470/30/16/027 10.1007/s00365-009-9045-3 10.1007/0-387-24233-3_10 10.1016/B978-0-12-064850-4.50010-0 10.1090/conm/138/1199128 |
| ContentType | Journal Article |
| Copyright | Springer Science+Business Media, LLC 2011 |
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| Issue | 3 |
| Keywords | Bispectrality 33D50 Basic hypergeometric series Multivariable orthogonal polynomials 39A70 |
| Language | English |
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| PublicationSubtitle | An International Journal Devoted to the Areas of Mathematics Influenced by Ramanujan |
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| References | Milch (CR17) 1968; 39 van Diejen (CR23) 1996; 126 Askey, Wilson (CR1) 1985; 54 Duistermaat, Grünbaum (CR3) 1986; 103 Dunkl, Xu (CR4) 2001 Koelink, Stokman (CR14) 2001; 22 Gasper, Rahman (CR6) 2005 CR16 Iliev (CR10) 2011; 363 Suslov (CR19) 1997; 30 Koornwinder (CR15) 1992 Iliev, Xu (CR11) 2007; 212 Karlin, McGregor, Askey (CR13) 1975 Geronimo, Iliev (CR8) 2010; 31 Haine, Iliev (CR9) 2006; 11 Suslov (CR20) 2001; 5 Gasper, Rahman (CR7) 2007; 13 Tratnik (CR21) 1991; 32 Cherednik (CR2) 1995; 122 Sahi (CR18) 1999; 150 Gasper, Rahman (CR5) 1990 Ismail, Rahman (CR12) 1991; 328 Tratnik (CR22) 1991; 32 E. Koelink (9244_CR14) 2001; 22 R. Askey (9244_CR1) 1985; 54 I. Cherednik (9244_CR2) 1995; 122 L. Haine (9244_CR9) 2006; 11 P. Iliev (9244_CR11) 2007; 212 T. Koornwinder (9244_CR15) 1992 P.R. Milch (9244_CR17) 1968; 39 G. Gasper (9244_CR7) 2007; 13 J. Geronimo (9244_CR8) 2010; 31 J.F. Diejen van (9244_CR23) 1996; 126 P. Iliev (9244_CR10) 2011; 363 M.E.H. Ismail (9244_CR12) 1991; 328 9244_CR16 S. Suslov (9244_CR20) 2001; 5 M.V. Tratnik (9244_CR22) 1991; 32 G. Gasper (9244_CR6) 2005 S. Suslov (9244_CR19) 1997; 30 G. Gasper (9244_CR5) 1990 M.V. Tratnik (9244_CR21) 1991; 32 S. Sahi (9244_CR18) 1999; 150 C.F. Dunkl (9244_CR4) 2001 J.J. Duistermaat (9244_CR3) 1986; 103 S. Karlin (9244_CR13) 1975 |
| References_xml | – volume: 32 start-page: 2065 issue: 8 year: 1991 end-page: 2073 ident: CR21 article-title: Some multivariable orthogonal polynomials of the Askey tableau-continuous families publication-title: J. Math. Phys. doi: 10.1063/1.529228 – volume: 54 start-page: 55 issue: 319 year: 1985 ident: CR1 article-title: Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials publication-title: Mem. Am. Math. Soc. – year: 1990 ident: CR5 publication-title: Basic Hypergeometric Series – volume: 212 start-page: 1 issue: 1 year: 2007 end-page: 36 ident: CR11 article-title: Discrete orthogonal polynomials and difference equations of several variables publication-title: Adv. Math. doi: 10.1016/j.aim.2006.09.012 – volume: 32 start-page: 2337 issue: 9 year: 1991 end-page: 2342 ident: CR22 article-title: Some multivariable orthogonal polynomials of the Askey tableau-discrete families publication-title: J. Math. Phys. doi: 10.1063/1.529158 – volume: 5 start-page: 183 issue: 2 year: 2001 end-page: 218 ident: CR20 article-title: Some orthogonal very-well-poised -functions that generalize Askey–Wilson polynomials publication-title: Ramanujan J. doi: 10.1023/A:1011439924912 – start-page: 189 year: 1992 end-page: 204 ident: CR15 article-title: Askey–Wilson polynomials for root systems of type publication-title: Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications – ident: CR16 – volume: 363 start-page: 1577 issue: 3 year: 2011 end-page: 1598 ident: CR10 article-title: Bispectral commuting difference operators for multivariable Askey–Wilson polynomials publication-title: Trans. Am. Math. Soc. doi: 10.1090/S0002-9947-2010-05183-9 – volume: 122 start-page: 119 issue: 1 year: 1995 end-page: 145 ident: CR2 article-title: Macdonald’s evaluation conjectures and difference Fourier transform publication-title: Invent. 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Soc. – volume: 212 start-page: 1 issue: 1 year: 2007 ident: 9244_CR11 publication-title: Adv. Math. doi: 10.1016/j.aim.2006.09.012 – volume: 32 start-page: 2065 issue: 8 year: 1991 ident: 9244_CR21 publication-title: J. Math. Phys. doi: 10.1063/1.529228 – volume: 13 start-page: 389 issue: 1–3 year: 2007 ident: 9244_CR7 publication-title: Ramanujan J. doi: 10.1007/s11139-006-0259-8 – volume: 30 start-page: 5877 issue: 16 year: 1997 ident: 9244_CR19 publication-title: J. Phys. A doi: 10.1088/0305-4470/30/16/027 – volume: 31 start-page: 417 issue: 3 year: 2010 ident: 9244_CR8 publication-title: Constr. Approx. doi: 10.1007/s00365-009-9045-3 – volume: 150 start-page: 267 issue: 1 year: 1999 ident: 9244_CR18 publication-title: Ann. Math. (2) doi: 10.2307/121102 – volume-title: Orthogonal Polynomials of Several Variables year: 2001 ident: 9244_CR4 doi: 10.1017/CBO9780511565717 – volume: 103 start-page: 177 issue: 2 year: 1986 ident: 9244_CR3 publication-title: Commun. Math. Phys. doi: 10.1007/BF01206937 – volume: 363 start-page: 1577 issue: 3 year: 2011 ident: 9244_CR10 publication-title: Trans. Am. Math. Soc. doi: 10.1090/S0002-9947-2010-05183-9 – volume: 122 start-page: 119 issue: 1 year: 1995 ident: 9244_CR2 publication-title: Invent. Math. doi: 10.1007/BF01231441 – start-page: 209 volume-title: Theory and Applications of Special Functions year: 2005 ident: 9244_CR6 doi: 10.1007/0-387-24233-3_10 – volume: 5 start-page: 183 issue: 2 year: 2001 ident: 9244_CR20 publication-title: Ramanujan J. doi: 10.1023/A:1011439924912 – volume-title: Basic Hypergeometric Series year: 1990 ident: 9244_CR5 – volume: 32 start-page: 2337 issue: 9 year: 1991 ident: 9244_CR22 publication-title: J. Math. Phys. doi: 10.1063/1.529158 |
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| Snippet | For every positive integer
d
, we define a meromorphic function
F
d
(
n
;
z
), where
n
,
z
∈ℂ
d
, which is a natural extension of the multivariable... |
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| StartPage | 273 |
| SubjectTerms | Combinatorics Field Theory and Polynomials Fourier Analysis Functions of a Complex Variable Mathematics Mathematics and Statistics Number Theory |
| Title | Multivariable Askey–Wilson function and bispectrality |
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