Utility of integral representations for basic hypergeometric functions and orthogonal polynomials
We describe the utility of integral representations for sums of basic hypergeometric functions. In particular we use these to derive an infinite sequence of transformations for symmetrizations over certain variables which the functions possess. These integral representations were studied by Bailey,...
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| Veröffentlicht in: | The Ramanujan journal Jg. 61; H. 2; S. 649 - 674 |
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01.06.2023
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| ISSN: | 1382-4090, 1572-9303 |
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| Abstract | We describe the utility of integral representations for sums of basic hypergeometric functions. In particular we use these to derive an infinite sequence of transformations for symmetrizations over certain variables which the functions possess. These integral representations were studied by Bailey, Slater, Askey, Roy, Gasper and Rahman and were also used to facilitate the computation of certain outstanding problems in the theory of basic hypergeometric orthogonal polynomials in the
q
-Askey scheme. We also generalize and give consequences and transformation formulas for some fundamental integrals connected to nonterminating basic hypergeometric series and the Askey–Wilson polynomials. We express a certain integral of a ratio of infinite
q
-shifted factorials as a symmetric sum of two basic hypergeometric series with argument
q
. The result is then expressed as a
q
-integral. Examples of integral representations applied to the derivation of generating functions for the Askey–Wilson polynomials are given and as well the computation of a missing generating function for the continuous dual
q
-Hahn polynomials. |
|---|---|
| AbstractList | We describe the utility of integral representations for sums of basic hypergeometric functions. In particular we use these to derive an infinite sequence of transformations for symmetrizations over certain variables which the functions possess. These integral representations were studied by Bailey, Slater, Askey, Roy, Gasper and Rahman and were also used to facilitate the computation of certain outstanding problems in the theory of basic hypergeometric orthogonal polynomials in the
q
-Askey scheme. We also generalize and give consequences and transformation formulas for some fundamental integrals connected to nonterminating basic hypergeometric series and the Askey–Wilson polynomials. We express a certain integral of a ratio of infinite
q
-shifted factorials as a symmetric sum of two basic hypergeometric series with argument
q
. The result is then expressed as a
q
-integral. Examples of integral representations applied to the derivation of generating functions for the Askey–Wilson polynomials are given and as well the computation of a missing generating function for the continuous dual
q
-Hahn polynomials. |
| Author | Cohl, Howard S. Costas-Santos, Roberto S. |
| Author_xml | – sequence: 1 givenname: Howard S. orcidid: 0000-0002-9398-455X surname: Cohl fullname: Cohl, Howard S. email: howard.cohl@nist.gov organization: Applied and Computational Mathematics Division, National Institute of Standards and Technology – sequence: 2 givenname: Roberto S. orcidid: 0000-0002-9545-7411 surname: Costas-Santos fullname: Costas-Santos, Roberto S. organization: Departamento de Física y Matemáticas, Universidad de Alcalá |
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| Keywords | Basic hypergeometric functions Integral representations Basic hypergeometric orthogonal polynomials Generating functions 33D60 Transformations 33D15 |
| Language | English |
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| PublicationSubtitle | An International Journal Devoted to the Areas of Mathematics Influenced by Ramanujan |
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| References | GasperGq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q$$\end{document}-Extensions of Barnes’, Cauchy’s, and Euler’s beta integralsTopics in Mathematical Analysis1989TeaneckWorld Sci. Publ.29431410.1142/9789814434201_0013 SlaterLJGeneralized Hypergeometric Functions1966CambridgeCambridge University Press0135.28101 van de Bult, F.J., Rains, E.M.: Basic hypergeometric functions as limits of elliptic hypergeometric functions. SIGMA 5(59), 31 (2009) IsmailMEHStantonDq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q$$\end{document}-Integral and moment representations for q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q$$\end{document}-orthogonal polynomialsJournal Canadien de Mathématiques [Can. J. Math.]2002544709735191391610.4153/CJM-2002-027-21009.33015 Gasper, G., Rahman, M.: Basic Hypergeometric Series. Encyclopedia of Mathematics and its Applications, vol. 96. 2nd ed. Cambridge University Press, Cambridge, With a foreword by Richard Askey (2004) NassrallahBRahmanMProjection formulas, a reproducing kernel and a generating function for q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q$$\end{document}-Wilson polynomialsSIAM J. Math. Anal.198516118619777287810.1137/05160140564.33009 AskeyRRoyRMore q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q$$\end{document}-beta integralsRocky Mount. J. Math.198616236537284305710.1216/RMJ-1986-16-2-3650599.33002 CohlHSCostas-SantosRSGeLTerminating basic hypergeometric representations and transformations for the Askey–Wilson polynomialsSymmetry20201281410.3390/sym120812901456.33013 Koekoek, R., Lesky, P. A., Swarttouw, R. F.: Hypergeometric Orthogonal Polynomials and Their q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q$$\end{document}-Analogues. Springer Monographs in Mathematics. Springer, Berlin, With a foreword by Tom H. Koornwinder (2010) IsmailMEHWilsonJAAsymptotic and generating relations for the q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q$$\end{document}-Jacobi and 4φ3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{4}\varphi _{3}$$\end{document} polynomialsJ. Approx. Theory1982361435467385510.1016/0021-9045(82)90069-7 IsmailMEHLetessierJValentGWimpJTwo families of associated Wilson polynomialsJournal Canadien de Mathématiques [Can. J. Math.]1990424659695107422910.4153/CJM-1990-035-40712.33005 RahmanMSome generating functions for the associated Askey–Wilson polynomialsJ. Comput. Appl. Math.1996681–2287296141876210.1016/0377-0427(95)00249-90861.33015 BaileyWNGeneralized Hypergeometric Series. Cambridge Tracts in Mathematics and Mathematical Physics1964New YorkStechert-Hafner Inc. AndrewsGEAskeyRRoyRSpecial Functions. Encyclopedia of Mathematics and Its Applications1999CambridgeCambridge University Press RahmanMAn integral representation of a 10φ9\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{10}\varphi _9$$\end{document} and continuous bi-orthogonal 10φ9\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{10}\varphi _9$$\end{document} rational functionsCan. J. Math.198638360561878711510.4153/CJM-1986-030-6 IsmailMEHStantonDExpansions in the Askey–Wilson polynomialsJ. Math. Anal. Appl.20154241664674328658610.1016/j.jmaa.2014.11.0481309.33020 RahmanMSome extensions of Askey–Wilson’s q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q$$\end{document}-beta integral and the corresponding orthogonal systemsBulletin Canadien de. Mathématiques [Can. Math. Bull.]198831446747697157510.4153/CMB-1988-068-60673.33001 RahmanMA product formula for the continuous q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q$$\end{document}-Jacobi polynomialsJ. Math. Anal. Appl.1986118230932285216310.1016/0022-247X(86)90265-90589.33009 Ismail, M.E.H.: Classical and Quantum Orthogonal Polynomials in One Variable, Encyclopedia of Mathematics and it Applications, vol. 98. Cambridge University Press, Cambridge, With two chapters by Walter Van Assche (2005) AtakishiyevaMAtakishiyevNA non-standard generating function for continuous dual q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q$$\end{document}-Hahn polynomialsRevista de Matemática: Teoría y Aplicaciones20111811111201307.33009 NIST Digital Library of Mathematical Functions. https://dlmf.nist.gov, Release 1.1.5 of 2022-03-15. F.W.J. Olver, A.B. Olde Daalhuis, D.W. Lozier, B.I. Schneider, R.F. Boisvert, C.W. Clark, B.R.Miller, B.V. Saunders, H.S. Cohl, and M.A. McClain, Eds R Askey (509_CR2) 1986; 16 HS Cohl (509_CR5) 2020; 12 M Rahman (509_CR19) 1996; 68 GE Andrews (509_CR1) 1999 509_CR7 509_CR21 G Gasper (509_CR6) 1989 WN Bailey (509_CR4) 1964 509_CR8 M Rahman (509_CR16) 1986; 38 MEH Ismail (509_CR10) 2015; 424 M Rahman (509_CR18) 1988; 31 M Atakishiyeva (509_CR3) 2011; 18 LJ Slater (509_CR20) 1966 MEH Ismail (509_CR9) 2002; 54 MEH Ismail (509_CR12) 1990; 42 M Rahman (509_CR17) 1986; 118 B Nassrallah (509_CR14) 1985; 16 509_CR13 509_CR15 MEH Ismail (509_CR11) 1982; 36 |
| References_xml | – reference: NIST Digital Library of Mathematical Functions. https://dlmf.nist.gov, Release 1.1.5 of 2022-03-15. F.W.J. Olver, A.B. Olde Daalhuis, D.W. Lozier, B.I. Schneider, R.F. Boisvert, C.W. Clark, B.R.Miller, B.V. Saunders, H.S. Cohl, and M.A. McClain, Eds – reference: BaileyWNGeneralized Hypergeometric Series. Cambridge Tracts in Mathematics and Mathematical Physics1964New YorkStechert-Hafner Inc. – reference: AskeyRRoyRMore q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q$$\end{document}-beta integralsRocky Mount. J. Math.198616236537284305710.1216/RMJ-1986-16-2-3650599.33002 – reference: RahmanMA product formula for the continuous q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q$$\end{document}-Jacobi polynomialsJ. Math. Anal. Appl.1986118230932285216310.1016/0022-247X(86)90265-90589.33009 – reference: Gasper, G., Rahman, M.: Basic Hypergeometric Series. Encyclopedia of Mathematics and its Applications, vol. 96. 2nd ed. Cambridge University Press, Cambridge, With a foreword by Richard Askey (2004) – reference: GasperGq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q$$\end{document}-Extensions of Barnes’, Cauchy’s, and Euler’s beta integralsTopics in Mathematical Analysis1989TeaneckWorld Sci. Publ.29431410.1142/9789814434201_0013 – reference: AtakishiyevaMAtakishiyevNA non-standard generating function for continuous dual q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q$$\end{document}-Hahn polynomialsRevista de Matemática: Teoría y Aplicaciones20111811111201307.33009 – reference: RahmanMSome generating functions for the associated Askey–Wilson polynomialsJ. Comput. Appl. Math.1996681–2287296141876210.1016/0377-0427(95)00249-90861.33015 – reference: AndrewsGEAskeyRRoyRSpecial Functions. Encyclopedia of Mathematics and Its Applications1999CambridgeCambridge University Press – reference: IsmailMEHLetessierJValentGWimpJTwo families of associated Wilson polynomialsJournal Canadien de Mathématiques [Can. J. Math.]1990424659695107422910.4153/CJM-1990-035-40712.33005 – reference: RahmanMAn integral representation of a 10φ9\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{10}\varphi _9$$\end{document} and continuous bi-orthogonal 10φ9\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{10}\varphi _9$$\end{document} rational functionsCan. J. Math.198638360561878711510.4153/CJM-1986-030-6 – reference: RahmanMSome extensions of Askey–Wilson’s q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q$$\end{document}-beta integral and the corresponding orthogonal systemsBulletin Canadien de. Mathématiques [Can. Math. Bull.]198831446747697157510.4153/CMB-1988-068-60673.33001 – reference: CohlHSCostas-SantosRSGeLTerminating basic hypergeometric representations and transformations for the Askey–Wilson polynomialsSymmetry20201281410.3390/sym120812901456.33013 – reference: IsmailMEHStantonDExpansions in the Askey–Wilson polynomialsJ. Math. Anal. Appl.20154241664674328658610.1016/j.jmaa.2014.11.0481309.33020 – reference: IsmailMEHStantonDq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q$$\end{document}-Integral and moment representations for q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q$$\end{document}-orthogonal polynomialsJournal Canadien de Mathématiques [Can. J. Math.]2002544709735191391610.4153/CJM-2002-027-21009.33015 – reference: Ismail, M.E.H.: Classical and Quantum Orthogonal Polynomials in One Variable, Encyclopedia of Mathematics and it Applications, vol. 98. Cambridge University Press, Cambridge, With two chapters by Walter Van Assche (2005) – reference: IsmailMEHWilsonJAAsymptotic and generating relations for the q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q$$\end{document}-Jacobi and 4φ3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{4}\varphi _{3}$$\end{document} polynomialsJ. Approx. Theory1982361435467385510.1016/0021-9045(82)90069-7 – reference: van de Bult, F.J., Rains, E.M.: Basic hypergeometric functions as limits of elliptic hypergeometric functions. SIGMA 5(59), 31 (2009) – reference: Koekoek, R., Lesky, P. A., Swarttouw, R. F.: Hypergeometric Orthogonal Polynomials and Their q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q$$\end{document}-Analogues. Springer Monographs in Mathematics. Springer, Berlin, With a foreword by Tom H. Koornwinder (2010) – reference: SlaterLJGeneralized Hypergeometric Functions1966CambridgeCambridge University Press0135.28101 – reference: NassrallahBRahmanMProjection formulas, a reproducing kernel and a generating function for q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q$$\end{document}-Wilson polynomialsSIAM J. Math. Anal.198516118619777287810.1137/05160140564.33009 – volume-title: Generalized Hypergeometric Functions year: 1966 ident: 509_CR20 – volume: 18 start-page: 111 issue: 1 year: 2011 ident: 509_CR3 publication-title: Revista de Matemática: Teoría y Aplicaciones – volume: 16 start-page: 186 issue: 1 year: 1985 ident: 509_CR14 publication-title: SIAM J. Math. Anal. doi: 10.1137/0516014 – volume: 12 start-page: 14 issue: 8 year: 2020 ident: 509_CR5 publication-title: Symmetry doi: 10.3390/sym12081290 – ident: 509_CR8 doi: 10.1017/CBO9781107325982 – volume: 38 start-page: 605 issue: 3 year: 1986 ident: 509_CR16 publication-title: Can. J. Math. doi: 10.4153/CJM-1986-030-6 – start-page: 294 volume-title: Topics in Mathematical Analysis year: 1989 ident: 509_CR6 doi: 10.1142/9789814434201_0013 – ident: 509_CR13 doi: 10.1007/978-3-642-05014-5 – volume: 68 start-page: 287 issue: 1–2 year: 1996 ident: 509_CR19 publication-title: J. Comput. Appl. Math. doi: 10.1016/0377-0427(95)00249-9 – volume: 118 start-page: 309 issue: 2 year: 1986 ident: 509_CR17 publication-title: J. Math. Anal. Appl. doi: 10.1016/0022-247X(86)90265-9 – volume: 36 start-page: 43 issue: 1 year: 1982 ident: 509_CR11 publication-title: J. Approx. Theory doi: 10.1016/0021-9045(82)90069-7 – volume: 424 start-page: 664 issue: 1 year: 2015 ident: 509_CR10 publication-title: J. Math. Anal. Appl. doi: 10.1016/j.jmaa.2014.11.048 – ident: 509_CR15 – volume: 54 start-page: 709 issue: 4 year: 2002 ident: 509_CR9 publication-title: Journal Canadien de Mathématiques [Can. J. Math.] doi: 10.4153/CJM-2002-027-2 – ident: 509_CR21 doi: 10.3842/SIGMA.2009.059 – volume-title: Special Functions. Encyclopedia of Mathematics and Its Applications year: 1999 ident: 509_CR1 – volume: 16 start-page: 365 issue: 2 year: 1986 ident: 509_CR2 publication-title: Rocky Mount. J. Math. doi: 10.1216/RMJ-1986-16-2-365 – volume: 31 start-page: 467 issue: 4 year: 1988 ident: 509_CR18 publication-title: Bulletin Canadien de. Mathématiques [Can. Math. Bull.] doi: 10.4153/CMB-1988-068-6 – volume-title: Generalized Hypergeometric Series. Cambridge Tracts in Mathematics and Mathematical Physics year: 1964 ident: 509_CR4 – ident: 509_CR7 – volume: 42 start-page: 659 issue: 4 year: 1990 ident: 509_CR12 publication-title: Journal Canadien de Mathématiques [Can. J. Math.] doi: 10.4153/CJM-1990-035-4 |
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| Snippet | We describe the utility of integral representations for sums of basic hypergeometric functions. In particular we use these to derive an infinite sequence of... |
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| SubjectTerms | Combinatorics Field Theory and Polynomials Fourier Analysis Functions of a Complex Variable Mathematics Mathematics and Statistics Number Theory |
| Title | Utility of integral representations for basic hypergeometric functions and orthogonal polynomials |
| URI | https://link.springer.com/article/10.1007/s11139-021-00509-5 |
| Volume | 61 |
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