Proceedings of the American Mathematical Society

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



A computer proof of Moll's log-concavity conjecture

Authors: Manuel Kauers and Peter Paule
Journal: Proc. Amer. Math. Soc. 135 (2007), 3847-3856
MSC (2000): Primary 33F10, 05A20
Published electronically: September 10, 2007
MathSciNet review: 2341935
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In a study on quartic integrals, Moll met a specialized family of Jacobi polynomials. He conjectured that the corresponding coefficient sequences are log-concave. In this paper we settle Moll's conjecture by a nontrivial usage of computer algebra.

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

  • 1. George Boros and Victor H. Moll, An integral hidden in Gradshteyn and Ryzhik, J. Comput. Appl. Math. 106 (1999), no. 2, 361–368. MR 1696417, 10.1016/S0377-0427(99)00081-3
  • 2. George Boros and Victor H. Moll.
    Irresistible Integrals.
    Cambridge University Press, Cambridge, 2004.
  • 3. B. F. Caviness and J. R. Johnson (eds.), Quantifier elimination and cylindrical algebraic decomposition, Texts and Monographs in Symbolic Computation, Springer-Verlag, Vienna, 1998. MR 1634186
  • 4. George E. Collins, Quantifier elimination for real closed fields by cylindrical algebraic decomposition, Automata theory and formal languages (Second GI Conf., Kaiserslautern, 1975), Springer, Berlin, 1975, pp. 134–183. Lecture Notes in Comput. Sci., Vol. 33. MR 0403962
  • 5. Stefan Gerhold and Manuel Kauers.
    A procedure for proving special function inequalities involving a discrete parameter.
    In Proceedings of ISSAC'05, pages 156-162, 2005.
  • 6. Manuel Kauers, SumCracker: a package for manipulating symbolic sums and related objects, J. Symbolic Comput. 41 (2006), no. 9, 1039–1057. MR 2251819, 10.1016/j.jsc.2006.06.005
  • 7. Christian Mallinger.
    Algorithmic manipulations and transformations of univariate holonomic functions and sequences.
    Master's thesis, J. Kepler University, Linz, August 1996.
  • 8. Victor H. Moll, The evaluation of integrals: a personal story, Notices Amer. Math. Soc. 49 (2002), no. 3, 311–317. MR 1879857
  • 9. Carsten Schneider, The summation package Sigma: underlying principles and a rhombus tiling application, Discrete Math. Theor. Comput. Sci. 6 (2004), no. 2, 365–386 (electronic). MR 2081481
  • 10. Kurt Wegschaider.
    Computer generated proofs of binomial multi-sum identities.
    Master's thesis, RISC-Linz, May 1997.
  • 11. Herbert S. Wilf and Doron Zeilberger, An algorithmic proof theory for hypergeometric (ordinary and “𝑞”) multisum/integral identities, Invent. Math. 108 (1992), no. 3, 575–633. MR 1163239, 10.1007/BF02100618

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 33F10, 05A20

Retrieve articles in all journals with MSC (2000): 33F10, 05A20

Additional Information

Manuel Kauers
Affiliation: Research Institute for Symbolic Computation (RISC-Linz), Johannes Kepler University Linz, Austria

Peter Paule
Affiliation: Research Institute for Symbolic Computation (RISC-Linz), Johannes Kepler University Linz, Austria

Received by editor(s): June 19, 2006
Published electronically: September 10, 2007
Additional Notes: The first author was partially supported by FWF grants SFB F1305 and P16613-N12
The second author was partially supported by FWF grant SFB F1301
Communicated by: Jim Haglund
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.