51
Views
5
CrossRef citations to date
0
Altmetric
Original Articles

Computations of Green's Function and Its Fourier Coefficients on Fuchsian Groups

Pages 317-334 | Received 01 Jun 2007, Accepted 18 Aug 2009, Published online: 11 Feb 2011

REFERENCES

  • Abramowitz , M. and Stegun , I. A. , eds. 1964 . Handbook of Mathematical Functions New York : Dover. . [Abramowitz and Stegun 64]
  • Avelin , H. 2007 . “Deformation of Γ0(5)-Cusp Forms.” . Math. Comp. , 76 : 361 – 384 . [Avelin 07]
  • Avelin , H. 2010 . “Numerical Computations of the Green's Function and Its Fourier Coefficients on PSL(2, Z).” . Exp. Math. , 19 ( 3 ) : 335 – 343 . [Avelin 10]
  • Barnard , A. 2003 . “The Singular Theta Correspondence, Lorentzian Lattices and Borcherds-Kac-Moody Algebras.” , PhD thesis U.C. Berkeley. . [Barnard 03]
  • Bender , C. M. and Orszag , S. A. 1999 . Advanced Mathematical Methods for Scientists and Engineers, Asymptotic Methods and Perturbation Theory New York : Springer. . [Bender and Orszag 99]
  • Borcherds , R. E. 1998 . “Automorphic Forms with Singularities on Grassmannians.” . Invent. math. , 132 : 491 – 562 . [Borcherds 98]
  • Boyd , J. P. 2005 . “Hyperasymptotics and the Linear Boundary Layer Problem: Why Asymptotic Series Diverge.” . SIAM Review , 47 ( 3 ) : 553 – 575 . [Boyd 05]
  • Bruinier , J. H. 2002 . Borhcerds Products on O(2, l) and Chern Classes of Heegner Divisors, Lecture Notes in Math. 1780 New York : Springer. . [Brunier 02]
  • Conrey , J. B. 2003 . “The Riemann Hypothesis.” . Notices of the AMS , 50 ( 3 ) : 341 – 353 . [Conrey 03]
  • Erdélyi , A. , Magnus , W. , Oberhettinger , F. and Tricomi , F. G. 1953 . Higher Transcendental Functions vol. 2 , New York : McGraw-Hill. . [Erdélyi et al. 53]
  • Fay , J. D. 1977 . “Fourier Coefficients of the Resolvent for a Fuchsian Group.” . J. Reine Angew. Math. , : 143 – 203 . [Fay 77]
  • Gil , A. , Segura , J. and Temme , N. M. 2002 . “Evaluation of the Modified Bessel Function of the Third Kind of Imaginary Order.” . Journal of Computational Physics , 715 : 398 – 411 . [Gil et al. 02]
  • Gil , A. , Segura , J. and Temme , N. M. 2003 . “Computation of the Modified Bessel Function of the Third Kind of Imaginary Orders: Uniform Airy-Type Asypmtotic Expansion.” . Journal of Computational and Applied Mathematics , 153 : 225 – 234 . [Gil et al. 03]
  • Gil , A. , Segura , J. and Temme , N. M. 2004 . “Algorithm 831: Modified Bessel Functions of Imaginary Order and Positive Argument.” . ACM Trans. Math. Softw. , 30 : 159 – 164 . [Gil et al. 04a]
  • Gil , A. , Segura , J. and Temme , N. M. 2004 . “Computating Solutions of the Modified Bessel Differential Equation for Imaginary Orders and Positive Arguments.” . ACM Trans. Math. Softw. , 30 : 145 – 158 . [Gil et al. 04b]
  • Hejhal , D. A. 1976 . The Selberg Trace Formula for PSL(2, R), vol. 1, Lecture Notes in Math. 548 New York : Springer. . [Hejhal 76]
  • Hejhal , D. A. 1981 . “Some Observations Concerning Eigenvalues of the Laplacian and Dirichlet L-Series.” . In Recent Progress in Analytic Number Theory vol. 2 , 95 – 110 . New York : Academic Press. . [Hejhal 81]
  • Hejhal , D. A. 1983 . The Selberg Trace Formula for PSL(2, R), vol. 2, Lecture Notes in Math. 1001 Berlin : Springer. . [Hejhal 83]
  • Hejhal , D. A. 1992 . Eigenvalues of the Laplacian for Hecke Triangle Groups . Mem. Amer. Math. Soc. , : 469 [Hejhal 92]
  • Hejhal , D. A. 1999 . “On Eigenfunctions of the Laplacian for Hecke Triangle Groups.” . In Emerging Applications of Number Theory Edited by: Hejhal , D. , Friedman , J. , Gutzwiller , M. and Odlyzko , A. 291 – 315 . New York : Springer. . [Hejhal 99]
  • Iwaniec , H. 1997 . Topics in Classical Automorphic Forms Providence : American Mathematical Society. . [Iwaniec 97]
  • Iwaniec , H. 2002 . Spectral Methods of Automorphic Forms, , 2nd ed Providence : American Mathematical Society. . [Iwaniec 02]
  • Marklof , J. and Strömbergsson , A. 2003 . “Equidistribution of Kronecker sequences along Closed Horocycles. . Geom. Funct. Anal. , 13 : 1239 – 1280 . [Marklof and Strömbergsson 03]
  • Niebur , D. 1973 . “A Class of Nonanalytic Automorphic Functions.” . Nagoya Math. J. , 52 : 133 – 145 . [Niebur 73]
  • Odlyzko , A. M. 2009 . “Correspondence about the Origins of the Hilbert-Pólya Conjecture.” Available online http://www.dtc.umn.edu/∼odlyzko/polya/index.htmlaccessed. [Odlyzko 09]
  • Olver , F. W. J. 1954 . “The Asymptotic Expansion of Bessel Functions of Large Order.” . Philos. Trans. Roy. Soc. London Ser. A , 247 : 328 – 368 . [Olver 54]
  • Olver , F. W.J. 1974 . Asymptotics and Special Functions. New York : Academic Press. . [Olver 74]
  • Selberg , A. 1956 . “Harmonic Analysis and Discontinuous Groups in Weakly Symmetric Riemannian Spaces with Applications to Dirichlet Series.” . J. Indian Math. Soc. B , 20 : 47 – 87 . [Selberg 56]
  • Shimizu , H. 1963 . “On Discontinuous Subgroups Operating on the Oroduct of the Upper Half Planes.” . Ann. of Math. , 77 : 33 – 71 . [Shimizu 63]
  • Strömbergsson , A. 2004 . “On the Uniform Equidistribution of Long Closed Horocycles.” . Duke Math. J. , 123 ( 3 ) : 507 – 543 . [Strömbergsson 04]
  • Watson , G. N. 1944 . A Treatise on the Theory of Bessel Functions, , 2nd ed. Cambridge, , UK : Cambridge University Press. . [Watson 44]

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.