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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
Posted:
September 10, 2007
MathSciNet review:
2341935
Full-text PDF Free Access
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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.
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- 1.
- George Boros and Victor H. Moll.
An integral hidden in Gradshteyn and Ryzhik.
Journal of Computational and Applied Mathematics, 106:361-368, 1999. MR 1696417 (2000c:33024)
- 2.
- George Boros and Victor H. Moll.
Irresistible Integrals.
Cambridge University Press, Cambridge, 2004.
- 3.
- Bob F. Caviness and Jeremy R. Johnson, editors.
Quantifier Elimination and Cylindrical Algebraic Decomposition, Texts and Monographs in Symbolic Computation. Springer, 1998. MR 1634186 (99b:03007)
- 4.
- George E. Collins.
Quantifier elimination for the elementary theory of real closed fields by cylindrical algebraic decomposition.
Lecture Notes in Computer Science, 33:134-183, 1975. MR 0403962 (53:7771)
- 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.
Journal of Symbolic Computation, 41:1039-1057, 2006. MR 2251819
- 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 of the AMS, 49(3):311-317, 2002. MR 1879857 (2002m:11105)
- 9.
- Carsten Schneider.
The summation package Sigma: Underlying principles and a rhombus tiling application.
Discrete Mathematics and Theoretical Computer Science, 6(2):365-386, 2004. MR 2081481 (2005e:68270)
- 10.
- Kurt Wegschaider.
Computer generated proofs of binomial multi-sum identities.
Master's thesis, RISC-Linz, May 1997.
- 11.
- Herb S. Wilf and Doron Zeilberger.
An algorithmic proof theory for hypergeometric (ordinary and  ) multisum/integral identities.
Invent. Math., 108:575-633, 1992. MR 1163239 (93k:33010)
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Additional Information
Manuel Kauers
Affiliation:
Research Institute for Symbolic Computation (RISC-Linz), Johannes Kepler University Linz, Austria
Email:
mkauers@risc.uni-linz.ac.at
Peter Paule
Affiliation:
Research Institute for Symbolic Computation (RISC-Linz), Johannes Kepler University Linz, Austria
Email:
ppaule@risc.uni-linz.ac.at
DOI:
http://dx.doi.org/10.1090/S0002-9939-07-08912-5
PII:
S 0002-9939(07)08912-5
Received by editor(s):
June 19, 2006
Posted:
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.
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