대한수학회보 (Bulletin of the Korean Mathematical Society)

  • 제59권5호
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  • Pages.1269-1277
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  • 2022
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  • 1015-8634(pISSN)
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  • 2234-3016(eISSN)
대한수학회 (Korean Mathematical Society)
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SOME POLYNOMIALS WITH UNIMODULAR ROOTS

  • Dubickas, Arturas (Institute of Mathematics Faculty of Mathematics and Informatics Vilnius University)
  • 투고 : 2021.10.01
  • 심사 : 2022.02.23
  • 발행 : 2022.09.30
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초록

In this paper we consider a sequence of polynomials defined by some recurrence relation. They include, for instance, Poupard polynomials and Kreweras polynomials whose coefficients have some combinatorial interpretation and have been investigated before. Extending a recent result of Chapoton and Han we show that each polynomial of this sequence is a self-reciprocal polynomial with positive coefficients whose all roots are unimodular. Moreover, we prove that their arguments are uniformly distributed in the interval [0, 2𝜋).

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참고문헌

  1. A. Bigeni, Combinatorial interpretations of the Kreweras triangle in terms of subset tuples, Electron. J. Combin. 25 (2018), no. 4, Paper No. 4.44, 11 pp.
  2. A. Bigeni, A generalization of the Kreweras triangle through the universal sl2 weight system, J. Combin. Theory Ser. A 161 (2019), 309-326. https://doi.org/10.1016/j.jcta.2018.08.005
  3. F. Chapoton and G.-N. Han, On the roots of the Poupard and Kreweras polynomials, Mosc. J. Comb. Number Theory 9 (2020), no. 2, 163-172. https://doi.org/10.2140/moscow.2020.9.163
  4. P. Drungilas, Unimodular roots of reciprocal Littlewood polynomials, J. Korean Math. Soc. 45 (2008), no. 3, 835-840. https://doi.org/10.4134/JKMS.2008.45.3.835
  5. T. Erdelyi, Improved lower bound for the number of unimodular zeros of self-reciprocal polynomials with coefficients in a finite set, Acta Arith. 192 (2020), no. 2, 189-210. https://doi.org/10.4064/aa190204-27-5
  6. D. Foata and G.-N. Han, The doubloon polynomial triangle, Ramanujan J. 23 (2010), no. 1-3, 107-126. https://doi.org/10.1007/s11139-009-9194-9
  7. D. Foata and G.-N. Han, Tree calculus for bivariate difference equations, J. Difference Equ. Appl. 20 (2014), no. 11, 1453-1488. https://doi.org/10.1080/10236198.2014.933820
  8. R. Graham and N. Zang, Enumerating split-pair arrangements, J. Combin. Theory Ser. A 115 (2008), no. 2, 293-303. https://doi.org/10.1016/j.jcta.2007.06.003
  9. E. Kim, A family of self-inversive polynomials with concyclic zeros, J. Math. Anal. Appl. 401 (2013), no. 2, 695-701. https://doi.org/10.1016/j.jmaa.2012.12.048
  10. S.-H. Kim and C. W. Park, On the zeros of certain self-reciprocal polynomials, J. Math. Anal. Appl. 339 (2008), no. 1, 240-247. https://doi.org/10.1016/j.jmaa.2007.06.055
  11. G. Kreweras, Sur les permutations comptees par les nombres de Genocchi de 1-iere et 2-ieme espece, European J. Combin. 18 (1997), no. 1, 49-58. https://doi.org/10.1006/eujc.1995.0081
  12. P. Lakatos and L. Losonczi, Self-inversive polynomials whose zeros are on the unit circle, Publ. Math. Debrecen 65 (2004), no. 3-4, 409-420. https://doi.org/10.5486/PMD.2004.3250
  13. M. N. Lalin and C. J. Smyth, Unimodularity of zeros of self-inversive polynomials, Acta Math. Hungar. 138 (2013), no. 1-2, 85-101. https://doi.org/10.1007/s10474-012-0225-4
  14. M. N. Lalin and C. J. Smyth, Addendum to: Unimodularity of zeros of self-inversive polynomials, Acta Math. Hungar. 147 (2015), no. 1, 255-257. https://doi.org/10.1007/s10474-015-0530-9
  15. L. Losonczi, Remarks to a theorem of Sinclair and Vaaler, Math. Inequal. Appl. 23 (2020), no. 2, 647-652. https://doi.org/10.7153/mia-2020-23-52
  16. I. D. Mercer, Unimodular roots of special Littlewood polynomials, Canad. Math. Bull. 49 (2006), no. 3, 438-447. https://doi.org/10.4153/CMB-2006-043-x
  17. C. Poupard, Deux proprietes des arbres binaires ordonnes stricts, European J. Combin. 10 (1989), no. 4, 369-374. https://doi.org/10.1016/S0195-6698(89)80009-5
  18. C. D. Sinclair and J. D. Vaaler, Self-inversive polynomials with all zeros on the unit circle, in Number theory and polynomials, 312-321, London Math. Soc. Lecture Note Ser., 352, Cambridge Univ. Press, Cambridge, 2008. https://doi.org/10.1017/CBO9780511721274.020
  19. D. Stankov, The number of unimodular roots of some reciprocal polynomials, C. R. Math. Acad. Sci. Paris 358 (2020), no. 2, 159-168. https://doi.org/10.5802/crmath.28
  20. The On-Line Encyclopedia of Integer Sequences, http://oeis.org.