×

Log-concavity of \(P\)-recursive sequences. (English) Zbl 1466.05008

Summary: We consider the higher order Turán inequality and higher order log-concavity for sequences \(\{a_n\}_{n\geq 0}\) such that \[ \frac{a_{n-1}a_{n+1}}{a_n^2}=1+\sum\limits_{i=1}^m\frac{r_i(\log n)}{n^{\alpha_i}}+o\biggl(\frac{1}{n^\beta}\biggr), \] where \(m\) is a nonnegative integer, \(\alpha_i\) are real numbers, \(r_i(x)\) are rational functions of \(x\) and \[ 0<\alpha_1<\alpha_2<\cdots<\alpha_m<\beta. \] We will give a sufficient condition on the higher order Turán inequality and the \(\ell\)-log-concavity for \(n\) sufficiently large. Many \(P\)-recursive sequences fall in this frame. At last, we will give a method to find the \(N\) such that for any \(n>N\), the higher order Turán inequality holds.

MSC:

05A20 Combinatorial inequalities
05A10 Factorials, binomial coefficients, combinatorial functions
41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
41A58 Series expansions (e.g., Taylor, Lidstone series, but not Fourier series)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Birkhoff, G. D.; Trjitzinsky, W. J., Analytic theory of singular difference equations, Acta Math., 60, 1, 1-89 (1933) · JFM 59.0450.03
[2] Chen, W. Y.C.; Jia, D. X.Q.; Wang, L. X.W., Higher order Turán inequalities for the partition function, Trans. Am. Math. Soc., 372, 3, 2143-2165 (2019) · Zbl 1415.05020
[3] Csordas, G., Complex zero decreasing sequences and the Riemann hypothesis II, (Analysis and Applications — ISAAC 2001 (2003), Springer: Springer Boston, MA), 121-134 · Zbl 1070.11039
[4] DeSalvo, S.; Pak, I., Log-concavity of the partition function, Ramanujan J., 38, 1, 1-13 (2014)
[5] Dimitrov, D. K., Higher order Turán inequalities, Proc. Am. Math. Soc., 126, 7, 2033-2037 (1998) · Zbl 0891.30016
[6] Došlić, T., Log-balanced for combinatorial sequences, Int. J. Math. Sci., 2005, 507-522 (2005) · Zbl 1073.05005
[7] Griffin, M.; Ono, K.; Rolen, L.; Zagier, D., Jensen polynomials for the Riemann zeta function and other sequences, Proc. Natl. Acad. Sci. USA, 116, 23, 11103-11110 (2019) · Zbl 1431.11105
[8] Hou, Q. H. (2019)
[9] Hou, Q. H.; Zhang, Z. R., Asymptotic r-log-convexity and P-recursive sequences, J. Symb. Comput., 93, 21-33 (2018)
[10] Kauers, M., A Mathematica package for computing asymptotic expansions of solutions of P-finite recurrence equations (2011), Technical Report RISC 11-04
[11] McNamara, P. R.W.; Sagan, B. E., Infinite log-concavity: developments and conjectures, Adv. Appl. Math., 44, 1, 1-15 (2010) · Zbl 1184.05005
[12] Moll, V. H., Combinatorial sequences arising from a rational integral, Online J. Anal. Comb., 2, #4 (2007) · Zbl 1123.05003
[13] Niculescu, C. P., A new look at Newton’s inequalities, J. Inequal. Appl., 1, 2 (2013)
[14] Stanley, R. P., Log-concave and unimodal sequences in algebra, combinatorics, and geometry, Ann. N.Y. Acad. Sci., 576, 1, 500-535 (1989) · Zbl 0792.05008
[15] Stanley, R. P., Enumerative Combinatorics, Vol. 2 (1999), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0928.05001
[16] Szegő, G., On an inequality of P. Turán concerning Legendre polynomials, Bull. Am. Math. Soc., 54 (1948) · Zbl 0032.27502
[17] Wang, L. X.W., Higher order Turán inequalities for combinatorial sequences, Adv. Appl. Math., 110, 180-196 (2019) · Zbl 1421.05017
[18] Wagner, C. G., Newton’s inequality and a test for imaginary roots, Two-Year Coll. Math. J., 8, 3, 145-147 (1977)
[19] Wimp, J.; Zeilberger, D., Resurrecting the asymptotics of linear recurrences, J. Math. Anal. Appl., 111, 162-176 (1985) · Zbl 0579.05007
[20] Zeilberger, D., Asyrec: a Maple package for computing the asymptotics of solutions of linear recurrence equations with polynomial coefficients (2016)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.