Hostname: page-component-857557d7f7-nbs69 Total loading time: 0 Render date: 2025-12-12T01:24:35.640Z Has data issue: false hasContentIssue false
  • English
  • Français

Inequalities for the sequences with unified form

Part of: Combinatorics Forms and linear algebraic groups Zeta and $L$-functions: analytic theory Additive number theory; partitions

Published online by Cambridge University Press:  20 November 2025

Li-Mei Dou ,
Zhen-Yu Gao  and
Larry X.W. Wang
Li-Mei Dou
Affiliation:
Center for Combinatorics, LPMC, Nankai University, China e-mail: 1120220002@mail.nankai.edu.cn 2120230007@mail.nankai.edu.cn
Zhen-Yu Gao
Affiliation:
Center for Combinatorics, LPMC, Nankai University, China e-mail: 1120220002@mail.nankai.edu.cn 2120230007@mail.nankai.edu.cn
Larry X.W. Wang*
Affiliation:
Center for Combinatorics, LPMC, Nankai University, China e-mail: 1120220002@mail.nankai.edu.cn 2120230007@mail.nankai.edu.cn
*
e-mail: wsw82@nankai.edu.cn
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider a class of sequence $c(n)$ with the following asymptotic form:

$$ \begin{align*}c(n) \sim \frac C{n^\kappa} \exp\left(\sum_{\lambda\in\mathcal S}A_\lambda n^\lambda \right) \sum_{\mu\in\mathcal T} \frac{\beta_\mu}{n^\mu} \qquad (n \to \infty).\end{align*} $$
We give criteria for the Turán inequality of any order, the double Turán inequality, and the Laguerre inequality of any order of $c(n)$ for sufficiently large n. We also give the companion inequalities for the Turán inequality and the Laguerre inequality of any order for $c(n)$. As applications, we will show that the numbers of commuting $\ell $-tuples in $S_n$, the partition without sequence, the plane partition, the partition into k-gonal numbers, the finite-dimensional representations of groups $\mathfrak {su}(3)$ and $\mathfrak {so}(5),$ and the coefficients of infinite product generating functions asymptotically satisfy these inequalities. Some of them settle open problems proposed by Bringmann, Franke, and Heim.

Keywords

Turán inequalityLaguerre inequalitysymmetric groupplane partitionfinite representation of group

MSC classification

Primary: 11P82: Analytic theory of partitions 11E45: Analytic theory (Epstein zeta functions; relations with automorphic forms and functions) 11M41: Other Dirichlet series and zeta functions 05A20: Combinatorial inequalities

Information

Type
Article
Information
Canadian Journal of Mathematics , First View , pp. 1 - 20
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

Footnotes

This work was supported by the National Science Foundation of China (Grant No. 12171254).

References

Banerjee, K., Invariants of the quartic binary form and proofs of Chen’s conjectures on inequalities for the partition function and the Andrews’ spt function . Res. Math. Sci. 12(2025), no. 1.Google Scholar
Bridges, W., Brindle, B., Bringmann, K., and Franke, J., Asymptotic expansions for partitions generated by infinite products . Math. Ann. 390(2024), 2593–2632.Google Scholar
Bridges, W. and Bringmann, K., Log-concavity for unimodal sequences . Res. Number Theory 10(2023), no. 6.Google Scholar
Bringmann, K. and Franke, J., An asymptotic formula for the number of $n$ -dimensional representations of $\mathfrak{su}(3)$ . Rev. Mat. Ibero. 39(2023), no. 5, 1599–1640.Google Scholar
Bringmann, K., Franke, J., and Heim, B., Asymptotics of commuting $l$ -tuples in symmetric groups and log-concavity . Res. Number Theory 10(2024), no. 83.Google Scholar
Bringmann, K. and Mahlburg, K., An extension of the Hardy-Ramanujan circle method and applications to partitions without sequences . Am. J. Math. 133(2011), no. 4, 1157–1178.Google Scholar
Chen, W. Y. C., The spt-function of Andrews . In: A. Claesson, M. Dukes, S. Kitaev, D. Manlove, and K. Meeks (eds.), Surveys in combinatorics 2017, London Mathematical Society Lecture Note Series, 440, Cambridge University Press, Cambridge, 2017, pp. 141–203.Google Scholar
Csordas, G., Norfolk, T. S., and Varga, R. S., The Riemann hypothesis and the Turán inequalities . Trans. Amer. Math. Soc. 296(1986), no. 2, 521–541.Google Scholar
Csordas, G. and Vishnyakova, A., The generalized Laguerre inequalities and functions in the Laguerre-Pólya class . Cent. Eur. J. Math. 11(2013), no. 9, 1643–1650.Google Scholar
Dou, L.-M., Wang, L. X. W., and Yang, N. N. Y., Unified coefficients for the reversed inequalities associated with class. submitted.Google Scholar
Gasper, G., Positivity and special functions . In: A. Askey (ed.), Proceedings of an advanced seminar sponsored by the mathematics research center, The University of Wisconsin-Madison, New York-London, 1975, pp. 375–433.Google Scholar
Griffin, M., Ono, K., Rolen, L., and Zagier, D., Jensen polynomials for the Riemann zeta function and other sequences . Proc. Nat. Acad. Sci. USA 116(2019), no. 23, 11103–11110.Google Scholar
Heim, B. and Neuhauser, M., Log-concavity of infinite product generating functions . Res. Number Theory 8(2022), no. 53, 1–14.Google Scholar
Heim, B., Neuhauser, M., and Tröger, R., Inequalities for plane partitions . Ann. Comb. 27(2023), no. 1, 87–108.Google Scholar
Laguerre, E., Oeuvres. Vol. 1. Gaauthier-Villars, Paris, 1989.Google Scholar
Matsumoto, K., On Mordell–Tornheim and other multiple zeta-functions . In: D. R. Heath-Brown and B. Z. Moroz (eds.), Proceedings of the session in analytic number theory and Diophantine equations, Bonner Mathematische Schriften, The University of Bonn, Bonn, 360, 2003, p. 17.Google Scholar
Matsumoto, K. and Tsumura, H., On Witten multiple zeta-functions associated with semisimple lie algebras I . Ann. Inst. Fourier 56(2006), no. 5, 1457–1504.Google Scholar
Mauth, L., Log-concavity for partitions without sequences. Preprint. arXiv:2306.07459.Google Scholar
Ono, K., Pujahari, S., and Rolen, L., Turán inequalities for the plane partition function . Adv. Math. 409(2022), 108692.Google Scholar
Pandey, B. V., Higher Turán inequalities for the plane partition function . Res. Number Theory 10(2024), no. 69.Google Scholar
Patrick, M. L., Some inequalities concerning Jacobi polynomials . SIAM J. Math. Anal. 2(1971), no. 2, 213–220.Google Scholar
Patrick, M. L., Extensions of inequalities of the Laguerre and Turán type . Pacific J. Math. 44(1973), no. 2, 675–682.Google Scholar
Rhoades, R., Asymptotics for the number of strongly unimodal sequences . Int. Math. Res. Notices 3(2014), 700–719.Google Scholar
Romik, D., On the number of $n$ -dimensional representations of $\mathfrak{su}(3)$ , the Bernoulli numbers, and the Witten zeta function . Acta Arith. 180(2017), no. 2, 111–159.Google Scholar
Wagner, I., On a new class of Lagueree-Pólya type functions with applications in number theory . Pacific J. Math. 320(2022), 177–192.Google Scholar
Wang, L. X. W. and Yang, E. Y. Y., Laguerre inequalities for discrete sequences . Adv. Appl. Math. 139(2022), 120357.Google Scholar
Wang, L. X. W. and Yang, N. N. Y., Laguerre inequalities and complete monotonicity for the Riemann xi-function and the partition function . Trans. Amer. Math. Soc. 377(2024), 4703–4725.Google Scholar
Wright, E. M., Asymptotic partition formulae: I. Plane partitions . Q. J. Math., Oxf. Ser. 2(1931), 177–189.Google Scholar