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Ramanujan’s Master Theorem for the Hypergeometric Fourier Transform Associated with Root Systems

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Abstract

Ramanujan’s Master theorem states that, under suitable conditions, the Mellin transform of an alternating power series provides an interpolation formula for the coefficients of this series. Ramanujan applied this theorem to compute several definite integrals and power series, which explains why it is referred to as the “Master Theorem”. In this paper we prove an analogue of Ramanujan’s Master theorem for the hypergeometric Fourier transform associated with root systems. This theorem generalizes to arbitrary positive multiplicity functions the results previously proven by the same authors for the spherical Fourier transform on semisimple Riemannian symmetric spaces.

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Notes

  1. The constant (i/2)l of [28, formulas (37) and (53)] should be corrected as 2−l.

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Authors and Affiliations

  1. Department of Mathematics, Louisiana State University, Baton Rouge, LA, USA

    G. Ólafsson

  2. Institut Elie Cartan de Lorraine, UMR CNRS 7502, Université de Lorraine, 57045, Metz, France

    A. Pasquale

Authors
  1. G. Ólafsson
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  2. A. Pasquale
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Correspondence to G. Ólafsson.

Additional information

Communicated by Arieh Iserles.

The research of G. Ólafsson was supported by NSF grant DMS-1101337. A. Pasquale would like to thank the Louisiana State University, Baton Rouge, for hospitality and financial support. She also gratefully acknowledges financial support from Tufts University and travel support from the Commission de Colloques et Congrès Internationaux (CCCI).

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Ólafsson, G., Pasquale, A. Ramanujan’s Master Theorem for the Hypergeometric Fourier Transform Associated with Root Systems. J Fourier Anal Appl 19, 1150–1183 (2013). https://doi.org/10.1007/s00041-013-9290-5

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