Infinite log-concavity for polynomial Pólya frequency sequences
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- by Petter Brändén and Matthew Chasse PDF
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Abstract:
McNamara and Sagan conjectured that if $a_0,a_1, a_2, \ldots$ is a Pólya frequency (PF) sequence, then so is $a_0^2, a_1^2 -a_0a_2, a_2^2-a_1a_3, \ldots$. We prove this conjecture for a natural class of PF-sequences which are interpolated by polynomials. In particular, this proves that the columns of Pascal’s triangle are infinitely log-concave, as conjectured by McNamara and Sagan. We also give counterexamples to the first mentioned conjecture.
Our methods provide families of nonlinear operators that preserve the property of having only real and nonpositive zeros.
References
- Michael Aissen, I. J. Schoenberg, and A. M. Whitney, On the generating functions of totally positive sequences. I, J. Analyse Math. 2 (1952), 93–103 (English, with Hebrew summary). MR 53174, DOI 10.1007/BF02786970
- George Boros and Victor Moll, Irresistible integrals, Cambridge University Press, Cambridge, 2004. Symbolics, analysis and experiments in the evaluation of integrals. MR 2070237, DOI 10.1017/CBO9780511617041
- Petter Brändén, Iterated sequences and the geometry of zeros, J. Reine Angew. Math. 658 (2011), 115–131. MR 2831515, DOI 10.1515/CRELLE.2011.063
- Petter Brändén, On linear transformations preserving the Pólya frequency property, Trans. Amer. Math. Soc. 358 (2006), no. 8, 3697–3716. MR 2218995, DOI 10.1090/S0002-9947-06-03856-6
- Petter Brändén, Counterexamples to the Neggers-Stanley conjecture, Electron. Res. Announc. Amer. Math. Soc. 10 (2004), 155–158. MR 2119757, DOI 10.1090/S1079-6762-04-00140-4
- Francesco Brenti, Unimodal, log-concave and Pólya frequency sequences in combinatorics, Mem. Amer. Math. Soc. 81 (1989), no. 413, viii+106. MR 963833, DOI 10.1090/memo/0413
- W. Y. C. Chen, D. Q. J. Dou, A. L. Yang, Brändén’s conjectures on the Boros-Moll polynomials, Int. Math. Res. Not. IMRN. 20, (2013), 4819–4828.
- Thomas Craven and George Csordas, Jensen polynomials and the Turán and Laguerre inequalities, Pacific J. Math. 136 (1989), no. 2, 241–260. MR 978613
- Thomas Craven and George Csordas, Iterated Laguerre and Turán inequalities, JIPAM. J. Inequal. Pure Appl. Math. 3 (2002), no. 3, Article 39, 14. MR 1917798
- Albert Edrei, On the generating functions of totally positive sequences. II, J. Analyse Math. 2 (1952), 104–109 (English, with Hebrew summary). MR 53175, DOI 10.1007/BF02786971
- Lukasz Grabarek, A new class of non-linear stability preserving operators, Complex Var. Elliptic Equ. 58 (2013), no. 7, 887–898. MR 3170669, DOI 10.1080/17476933.2011.586696
- D. B. Karp, Positivity of Toeplitz determinants formed by rising factorial series and properties of related polynomials, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 404 (2012), no. Analiticheskaya Teoriya Chisel i Teoriya Funktsiĭ. 27, 184–198, 262–263 (English, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 193 (2013), no. 1, 106–114. MR 3029600, DOI 10.1007/s10958-013-1438-y
- B. Ja. Levin, Distribution of zeros of entire functions, American Mathematical Society, Providence, R.I., 1964. MR 0156975
- Peter R. W. McNamara and Bruce E. Sagan, Infinite log-concavity: developments and conjectures, Adv. in Appl. Math. 44 (2010), no. 1, 1–15. MR 2552652, DOI 10.1016/j.aam.2009.03.001
- Q. I. Rahman and G. Schmeisser, Analytic theory of polynomials, London Mathematical Society Monographs. New Series, vol. 26, The Clarendon Press, Oxford University Press, Oxford, 2002. MR 1954841
- Earl D. Rainville, Special functions, The Macmillan Company, New York, 1960. MR 0107725
- John Riordan, Combinatorial identities, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR 0231725
- Richard P. Stanley, Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, vol. 62, Cambridge University Press, Cambridge, 1999. With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin. MR 1676282, DOI 10.1017/CBO9780511609589
- John R. Stembridge, Counterexamples to the poset conjectures of Neggers, Stanley, and Stembridge, Trans. Amer. Math. Soc. 359 (2007), no. 3, 1115–1128. MR 2262844, DOI 10.1090/S0002-9947-06-04271-1
- Otto Szász, On sequences of polynomials and the distribution of their zeros, Bull. Amer. Math. Soc. 49 (1943), 377–383. MR 8274, DOI 10.1090/S0002-9904-1943-07919-0
- David G. Wagner, Total positivity of Hadamard products, J. Math. Anal. Appl. 163 (1992), no. 2, 459–483. MR 1145841, DOI 10.1016/0022-247X(92)90261-B
- Rintaro Yoshida, On some questions of Fisk and Brändén, Complex Var. Elliptic Equ. 58 (2013), no. 7, 933–945. MR 3170673, DOI 10.1080/17476933.2011.603418
- Yaming Yu, Confirming two conjectures of Su and Wang on binomial coefficients, Adv. in Appl. Math. 43 (2009), no. 4, 317–322. MR 2553542, DOI 10.1016/j.aam.2008.12.004
Additional Information
- Petter Brändén
- Affiliation: Department of Mathematics, Royal Institute of Technology, SE-100 44 Stockholm, Sweden
- MR Author ID: 721471
- Email: pbranden@kth.se
- Matthew Chasse
- Affiliation: Department of Mathematics, Royal Institute of Technology, SE-100 44 Stockholm, Sweden
- Email: chasse@kth.se
- Received by editor(s): June 23, 2014
- Received by editor(s) in revised form: October 14, 2014
- Published electronically: May 22, 2015
- Additional Notes: The first author is a Royal Swedish Academy of Sciences Research Fellow supported by a grant from the Knut and Alice Wallenberg Foundation. The research is also supported by the Göran Gustafsson Foundation.
- Communicated by: Patricia Hersh
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 143 (2015), 5147-5158
- MSC (2010): Primary 05A20, 26C10, 05E99, 30C15
- DOI: https://doi.org/10.1090/proc/12654
- MathSciNet review: 3411133