“Askey scheme”的意思、由来-开放百科全书

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Askey scheme

释义

Askey scheme for hypergeometric orthogonal polynomials
Askey scheme for basic hypergeometric orthogonal polynomials
References

In mathematics, the Askey scheme is a way of organizing orthogonal polynomials of hypergeometric or basic hypergeometric type into a hierarchy. For the classical orthogonal polynomials discussed in {{harvtxt|Andrews|Askey|1985}}, the Askey scheme was first drawn by {{harvtxt|Labelle|1985}} and by {{harvs|txt|authorlink=Richard Askey|last=Askey|last2=Wilson|year=1985}}, and has since been extended by {{harvtxt|Koekoek|Swarttouw|1998}} and {{harvtxt|Koekoek|Lesky|Swarttouw|2010}} to cover basic orthogonal polynomials.

Askey scheme for hypergeometric orthogonal polynomials

{{harvtxt|Koekoek|Lesky|Swarttouw|2010|loc=p.183}} give the following version of the Askey scheme:

₄F₃: Wilson | Racah

₃F₂: Continuous dual Hahn | Continuous Hahn | Hahn | dual Hahn

₂F₁: Meixner–Pollaczek | Jacobi | Pseudo Jacobi | Meixner | Krawtchouk

₂F₀/₁F₁: Laguerre | Bessel | Charlier

₁F₀: Hermite

Askey scheme for basic hypergeometric orthogonal polynomials

{{harvtxt|Koekoek|Lesky|Swarttouw|2010|loc=p.413}} give the following scheme for basic hypergeometric orthogonal polynomials:

₄₃: Askey–Wilson | q-Racah

₃₂: Continuous dual q-Hahn | Continuous q-Hahn | Big q-Jacobi | q-Hahn | dual q-Hahn

₂₁: Al-Salam–Chihara | q-Meixner–Pollaczek | Continuous q-Jacobi | Big q-Laguerre | Little q-Jacobi | q-Meixner | Quantum q-Krawtchouk | q-Krawtchouk | Affine q-Krawtchouk | Dual q-Krawtchouk

₂₀/₁₁: Continuous big q-Hermite | Continuous q-Laguerre | Little q-Laguerre | q-Laguerre | q-Bessel | q-Charlier | Al-Salam–Carlitz I | Al-Salam–Carlitz II

₁₀: Continuous q-Hermite | Stieltjes–Wigert | Discrete q-Hermite I | Discrete q-Hermite II

References

{{Citation | last1=Andrews | first1=George E. | last2=Askey | first2=Richard | editor1-last=Brezinski | editor1-first=C. | editor2-last=Draux | editor2-first=A. | editor3-last=Magnus | editor3-first=Alphonse P. | editor4-last=Maroni | editor4-first=Pascal | editor5-last=Ronveaux | editor5-first=A. | title=Polynômes orthogonaux et applications. Proceedings of the Laguerre symposium held at Bar-le-Duc, October 15–18, 1984. | publisher=Springer-Verlag | location=Berlin, New York | series=Lecture Notes in Math. | isbn=978-3-540-16059-5 | doi=10.1007/BFb0076530 | mr=838970 | year=1985 | volume=1171 | chapter=Classical orthogonal polynomials | pages=36–62}}
{{Citation | last1=Askey | first1=Richard | last2=Wilson | first2=James | title=Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials | url=https://books.google.com/books?id=9q9o03nD_xsC | isbn=978-0-8218-2321-7 | mr=783216 | year=1985 | journal=Memoirs of the American Mathematical Society | issn=0065-9266 | volume=54 | issue=319 | pages=iv+55 | doi=10.1090/memo/0319}}
{{Citation | last1=Koekoek | first1=Roelof | last2=Swarttouw | first2=René F. | title=The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue|url = http://homepage.tudelft.nl/11r49/askey/|year=1998 |publisher=Delft University of Technology, Faculty of Information Technology and Systems, Department of Technical Mathematics and Informatics

|volume= 98-17}}

{{Citation | last1=Koekoek | first1=Roelof | last2=Lesky | first2=Peter A. | last3=Swarttouw | first3=René F. | title=Hypergeometric orthogonal polynomials and their q-analogues | publisher=Springer-Verlag | location=Berlin, New York | series=Springer Monographs in Mathematics | isbn=978-3-642-05013-8 | doi=10.1007/978-3-642-05014-5 | mr=2656096 | year=2010}}
{{Citation | last1=Koornwinder | first1=Tom H. | title=Orthogonal polynomials and their applications (Segovia, 1986) | publisher=Springer-Verlag | location=Berlin, New York | series=Lecture Notes in Math. | isbn=978-3-540-19489-7 | doi=10.1007/BFb0083353 | mr=973421 | year=1988 | volume=1329 | chapter=Group theoretic interpretations of Askey's scheme of hypergeometric orthogonal polynomials | pages=46–72}}
{{Citation | last1=Labelle | first1=Jacques | editor1-last=Brezinski | editor1-first=C. | editor2-last=Draux | editor2-first=A. | editor3-last=Magnus | editor3-first=Alphonse P. | editor4-last=Maroni | editor4-first=Pascal | editor5-last=Ronveaux | editor5-first=A. | title=Polynômes Orthogonaux et Applications. Proceedings of the Laguerre Symposium held at Bar-le-Duc | publisher=Springer-Verlag | location=Berlin, New York | series=Lecture Notes in Math. | isbn=978-3-540-16059-5 | doi=10.1007/BFb0076527 | mr=838967 | year=1985 | volume=1171 | chapter=Tableau d'Askey | pages=xxxvi–xxxvii}}

3 : Orthogonal polynomials|Hypergeometric functions|Q-analogs

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