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Euler transformation formula for multiple basic hypergeometric series of type A and some applications: Euler transformation formula for multiple basic hypergeometric series of type \(A\) and some applications

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Title: Euler transformation formula for multiple basic hypergeometric series of type A and some applications: Euler transformation formula for multiple basic hypergeometric series of type \(A\) and some applications
Authors: Yasushi Kajihara
Source: Advances in Mathematics. 187:53-97
Publisher Information: Elsevier BV, 2004.
Publication Year: 2004
Subject Terms: Mathematics(all), Heine transformation, Pfaff-Saalschütz summation formula, Hypergeometric functions associated with root systems, Multiple basic hypergeometric series of type A, Gauß summation formula, 01 natural sciences, Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.), Watson type transformation, Pfaff–Saalschutz summation, basic hypergeometric series associated to root systems, Euler transformation, Macdonald polynomials, 0101 mathematics, Bailey–Jackson type transformation–summation formula, Bailey's transformation formula, Basic hypergeometric functions associated with root systems, Watson's transformation formula
Description: In this remarkable paper, the author establishes a rather general transformation formula between two basic hypergeometric series associated to root systems of type \(A\) which are \textit{of different dimensions}. This is remarkable because, up to this date, there do not appear many identities of this type in the literature. The only other examples that the reviewer is aware of are [\textit{I.~M.~Gessel} and the reviewer, Trans. Am. Math. Soc. 349, 429--479 (1997; Zbl 0865.05003); Sec.~8], the reviewer, in: [\(q\)-Series with Applications to Combinatorics, Number Theory, and Physics, Urbana-Champaign, Oct.~26--28, 2000, \textit{B.~C.~Berndt, K.~Ono} (eds.), Contemporary Mathematics. 291. Providence, RI: American Mathematical Society (AMS), 153--161 (2001; Zbl 0980.00024); Conjecture], [\textit{H.~Rosengren}, J. Math. Anal. Appl. 281, 332--345 (2003; Zbl 1032.33013), and Constructive Approximation 20, 525--548 (2004; Zbl 1077.33031)]. The author's formula reduces to one of Heine's transformation formulae between two \(_2\phi_1\)-series in its simplest case. The proof of the formula is based on results from the theory of Macdonald polynomials. The author provides numerous applications of his formula, such as proofs of Pfaff--Saalschütz and Gauß summation formulae for multiple basic hypergeometric series of type \(A_n\) (some of them having been found earlier by Milne), multiple series generalizations of Watson's transformation formula between series of different dimensions, a transformation formula between series of different dimensions generalizing at the same time Bailey's transformation formula between two very-well-poised \(_{10}\phi_9\)-series and Jackson's summation formula for a very-well-poised \(_8\phi_7\)-series, and a transformation formula between series of different dimensions which reduces to Sears' transformation formula between two balanced \(_4\phi_3\)-series in its simplest case. Reviewer's remark: H.~Rosengren [``New transformations for elliptic hypergeometric series on the root system \(A_n\),'' preprint \url{arXiv:math.CA/0305379}] has generalized the main formula of the paper under review to elliptic hypergeometric series. In a special case, this has also been done by the author and \textit{M.~Noumi} [Indag. Math., New Ser. 14, 395--421 (2003; Zbl 1051.33009)].
Document Type: Article
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Language: English
ISSN: 0001-8708
DOI: 10.1016/j.aim.2003.08.012
Access URL: https://zbmath.org/2093004
https://doi.org/10.1016/j.aim.2003.08.012
https://dialnet.unirioja.es/servlet/articulo?codigo=928457
https://www.sciencedirect.com/science/article/pii/S0001870803002603
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https://www.sciencedirect.com/science/article/abs/pii/S0001870803002603
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  Data: Euler transformation formula for multiple basic hypergeometric series of type A and some applications: Euler transformation formula for multiple basic hypergeometric series of type \(A\) and some applications
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  Data: In this remarkable paper, the author establishes a rather general transformation formula between two basic hypergeometric series associated to root systems of type \(A\) which are \textit{of different dimensions}. This is remarkable because, up to this date, there do not appear many identities of this type in the literature. The only other examples that the reviewer is aware of are [\textit{I.~M.~Gessel} and the reviewer, Trans. Am. Math. Soc. 349, 429--479 (1997; Zbl 0865.05003); Sec.~8], the reviewer, in: [\(q\)-Series with Applications to Combinatorics, Number Theory, and Physics, Urbana-Champaign, Oct.~26--28, 2000, \textit{B.~C.~Berndt, K.~Ono} (eds.), Contemporary Mathematics. 291. Providence, RI: American Mathematical Society (AMS), 153--161 (2001; Zbl 0980.00024); Conjecture], [\textit{H.~Rosengren}, J. Math. Anal. Appl. 281, 332--345 (2003; Zbl 1032.33013), and Constructive Approximation 20, 525--548 (2004; Zbl 1077.33031)]. The author's formula reduces to one of Heine's transformation formulae between two \(_2\phi_1\)-series in its simplest case. The proof of the formula is based on results from the theory of Macdonald polynomials. The author provides numerous applications of his formula, such as proofs of Pfaff--Saalschütz and Gauß summation formulae for multiple basic hypergeometric series of type \(A_n\) (some of them having been found earlier by Milne), multiple series generalizations of Watson's transformation formula between series of different dimensions, a transformation formula between series of different dimensions generalizing at the same time Bailey's transformation formula between two very-well-poised \(_{10}\phi_9\)-series and Jackson's summation formula for a very-well-poised \(_8\phi_7\)-series, and a transformation formula between series of different dimensions which reduces to Sears' transformation formula between two balanced \(_4\phi_3\)-series in its simplest case. Reviewer's remark: H.~Rosengren [``New transformations for elliptic hypergeometric series on the root system \(A_n\),'' preprint \url{arXiv:math.CA/0305379}] has generalized the main formula of the paper under review to elliptic hypergeometric series. In a special case, this has also been done by the author and \textit{M.~Noumi} [Indag. Math., New Ser. 14, 395--421 (2003; Zbl 1051.33009)].
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        Value: 10.1016/j.aim.2003.08.012
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      – Text: English
    PhysicalDescription:
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        PageCount: 45
        StartPage: 53
    Subjects:
      – SubjectFull: Mathematics(all)
        Type: general
      – SubjectFull: Heine transformation
        Type: general
      – SubjectFull: Pfaff-Saalschütz summation formula
        Type: general
      – SubjectFull: Hypergeometric functions associated with root systems
        Type: general
      – SubjectFull: Multiple basic hypergeometric series of type A
        Type: general
      – SubjectFull: Gauß summation formula
        Type: general
      – SubjectFull: 01 natural sciences
        Type: general
      – SubjectFull: Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.)
        Type: general
      – SubjectFull: Watson type transformation
        Type: general
      – SubjectFull: Pfaff–Saalschutz summation
        Type: general
      – SubjectFull: basic hypergeometric series associated to root systems
        Type: general
      – SubjectFull: Euler transformation
        Type: general
      – SubjectFull: Macdonald polynomials
        Type: general
      – SubjectFull: 0101 mathematics
        Type: general
      – SubjectFull: Bailey–Jackson type transformation–summation formula
        Type: general
      – SubjectFull: Bailey's transformation formula
        Type: general
      – SubjectFull: Basic hypergeometric functions associated with root systems
        Type: general
      – SubjectFull: Watson's transformation formula
        Type: general
    Titles:
      – TitleFull: Euler transformation formula for multiple basic hypergeometric series of type A and some applications: Euler transformation formula for multiple basic hypergeometric series of type \(A\) and some applications
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              M: 09
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