Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



An infinite family of summation identities

Author: S. F. Keating
Journal: Proc. Amer. Math. Soc. 130 (2002), 3433-3437
MSC (2000): Primary 11-XX
Published electronically: May 29, 2002
MathSciNet review: 1913024
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Theta functions have historically played a prominent role in number theory. One such role is the construction of modular forms. In this work, a generalized theta function is used to construct an infinite family of summation identities. Our results grew out of some observations noted during a presentation given by the author at the 1992 AMS-MAA-SIAM Joint Meetings in Baltimore.

References [Enhancements On Off] (What's this?)

  • 1. Richard Bellman, A brief introduction to theta functions, Athena Series: Selected Topics in Mathematics, Holt, Rinehart and Winston, New York, 1961. MR 0125252
  • 2. Beyer, William H. (editor), CRC Standard Mathematical Tables, 26th ed., CRC Press, Inc., Boca Raton, Florida, 1981, p. 398.
  • 3. A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Tables of integral transforms. Vol. I, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1954. Based, in part, on notes left by Harry Bateman. MR 0061695
  • 4. I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, Fourth edition prepared by Ju. V. Geronimus and M. Ju. Ceĭtlin. Translated from the Russian by Scripta Technica, Inc. Translation edited by Alan Jeffrey, Academic Press, New York-London, 1965. MR 0197789
  • 5. S. F. Keating, An infinite family of zeta functions indexed by Hermite polynomials, J. Math. Anal. Appl. 140 (1989), no. 1, 218–223. MR 997851, 10.1016/0022-247X(89)90102-9
  • 6. N. N. Lebedev, Special functions and their applications, Dover Publications, Inc., New York, 1972. Revised edition, translated from the Russian and edited by Richard A. Silverman; Unabridged and corrected republication. MR 0350075
  • 7. Jens Marklof, Theta sums, Eisenstein series, and the semiclassical dynamics of a precessing spin, Emerging applications of number theory (Minneapolis, MN, 1996) IMA Vol. Math. Appl., vol. 109, Springer, New York, 1999, pp. 405–450. MR 1691543, 10.1007/978-1-4612-1544-8_17
  • 8. Andrew Ogg, Modular forms and Dirichlet series, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 0256993
  • 9. Peter Sarnak, Some applications of modular forms, Cambridge Tracts in Mathematics, vol. 99, Cambridge University Press, Cambridge, 1990. MR 1102679
  • 10. Goro Shimura, On the transformation formulas of theta series, Amer. J. Math. 115 (1993), no. 5, 1011–1052. MR 1246183, 10.2307/2375064
  • 11. Marie-France Vignéras, Séries thêta des formes quadratiques indéfinies, Séminaire Delange-Pisot-Poitou, 17e année (1975/76), Théorie des nombres: Fasc. 1, Exp. No. 20, Secrétariat Math., Paris, 1977, pp. 3 (French). MR 0480352

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 11-XX

Retrieve articles in all journals with MSC (2000): 11-XX

Additional Information

S. F. Keating
Affiliation: Department of Mathematics and Computer Science, Eastern Connecticut State University, Willimantic, Connecticut 06226

Received by editor(s): May 29, 2001
Published electronically: May 29, 2002
Communicated by: Dennis A. Hejhal
Article copyright: © Copyright 2002 American Mathematical Society