Abstract
In previous work we introduced and studied a function \(R(a_{+},a_{-},{\bf c};v,\hat{v})\) that generalizes the hypergeometric function. In this paper we focus on a similarity-transformed function \({\mathcal E} (a_{+},a_{-},\gamma ;v,\hat{v})\), with parameters γ∈ℂ4 related to the couplings c∈ℂ4 by a shift depending on a + , a − . We show that the ℰ-function is invariant under all maps γ↦w(γ), with w in the Weyl group of type D 4 . Choosing a + , a − positive and \({\bf \gamma},\hat{v}\) real, we obtain detailed information on the |Re v|→∞ asymptotics of the ℰ-function. In particular, we explicitly determine the leading asymptotics in terms of plane waves and the c-function that implements the similarity R→ℰ.
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Ruijsenaars, S. N. M.: A generalized hypergeometric function satisfying four analytic difference equations of Askey-Wilson type. Commun. Math. Phys. 206, 639–690 (1999)
Ruijsenaars, S. N. M.: Systems of Calogero-Moser type. In: Proceedings of the 1994 Banff summer school Particles and fields. CRM Ser. in Math. Phys., Semenoff, G., Vinet, L., eds., New York: Springer, 1999, pp. 251–352
Askey, R., Wilson, J.: Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials. Mem. Am. Math. Soc. 319, (1985)
Gasper, G., Rahman, M.: Basic hypergeometric series. In: Encyclopedia of Mathematics and its Applications. 35, Cambridge: Cambridge Univ. Press 1990
Ruijsenaars, S. N. M.: Special functions defined by analytic difference equations. In: Proceedings of the Tempe NATO Advanced Study Institute ‘‘Special Functions 2000’‘, NATO Science Series Vol. 30, Bustoz, J., Ismail, M., Suslov, S., eds., Dordrecht: Kluwer, 2001, pp. 281–333
Ruijsenaars, S. N. M.: Sine-Gordon solitons vs. relativistic Calogero-Moser particles. In: Proceedings of the Kiev NATO Advanced Study Institute ‘‘Integrable structures of exactly solvable two-dimensional models of quantum field theory’‘, NATO Science Series Vol. 35, Pakuliak, S., von~Gehlen, G., eds., Dordrecht: Kluwer, 2001, pp. 273–292
Ruijsenaars, S. N. M.: A generalized hypergeometric function III. Associated Hilbert space transform. To appear in Commun. Math. Phys. - DOI 10.1007/s00220-003-0970-x
Koornwinder, T. H.: Askey-Wilson polynomials for root systems of type BC. Contemp. Math. 138, 189–204 (1992)
van Diejen, J. F.: Self-dual Koornwinder-Macdonald polynomials. Invent. Math. 126, 319–339 (1996)
Grünbaum, F. A., Haine, L.: Some functions that generalize the Askey-Wilson polynomials. Commun. Math. Phys. 184, 173–202 (1997)
Ismail, M. E. H., Rahman, M.: The associated Askey-Wilson polynomials. Trans. Am. Math. Soc. 328, 201–237 (1991)
Suslov, S. K.: Some orthogonal very-well-poised 8φ7-functions. J. Phys. A: Math. Gen. 30, 5877–5885 (1997)
Suslov, S. K.: Some orthogonal very-well-poised 8φ7-functions that generalize Askey-Wilson polynomials. The Ramanujan Journal 5, 183–218 (2001)
Koelink, E., Stokman, J. V.: The Askey-Wilson function transform. Int. Math. Res. Notes No. 22, 1203–1227 (2001)
Stokman, J. V.: Askey-Wilson functions and quantum groups. Preprint, math.QA/0301330
Ruijsenaars, S. N. M.: Relativistic Lamé functions revisited. J. Phys. A: Math. Gen. 34, 10595–10612 (2001)
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Communicated by L. Takhtajan
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Ruijsenaars, S. A Generalized Hypergeometric Function II. Asymptotics and D 4 Symmetry. Commun. Math. Phys. 243, 389–412 (2003). https://doi.org/10.1007/s00220-003-0969-3
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DOI: https://doi.org/10.1007/s00220-003-0969-3