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Recurrence Relations for Strongly q-Log-Convex Polynomials

Published online by Cambridge University Press:  20 November 2018

William Y. C. Chen ,
Larry X. W. Wang  and
Arthur L. B. Yang
William Y. C. Chen
Affiliation:
Center for Combinatorics, LPMC-TJKLC, Nankai University, Tianjin 300071, P. R. Chinae-mail: chen@nankai.edu.cnwxw@cfc.nankai.edu.cnyang@nankai.edu.cn
Larry X. W. Wang
Affiliation:
Center for Combinatorics, LPMC-TJKLC, Nankai University, Tianjin 300071, P. R. Chinae-mail: chen@nankai.edu.cnwxw@cfc.nankai.edu.cnyang@nankai.edu.cn
Arthur L. B. Yang
Affiliation:
Center for Combinatorics, LPMC-TJKLC, Nankai University, Tianjin 300071, P. R. Chinae-mail: chen@nankai.edu.cnwxw@cfc.nankai.edu.cnyang@nankai.edu.cn
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Abstract

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We consider a class of strongly $q$-log-convex polynomials based on a triangular recurrence relation with linear coefficients, and we show that the Bell polynomials, the Bessel polynomials, the Ramanujan polynomials and the Dowling polynomials are strongly $q$-log-convex. We also prove that the Bessel transformation preserves log-convexity.

Keywords

05A2005E99log-concavityq-log-convexitystrongq-log-convexityBell polynomialsBessel polynomialsRamanujan polynomialsDowling polynomials
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 54 , Issue 2 , 01 June 2011 , pp. 217 - 229
Copyright
Copyright © Canadian Mathematical Society 2011

References

[1] Bell, E. T., Exponential polynomials. Ann. of Math. 35(1934), no. 2, 258–277. doi:10.2307/1968431Google Scholar
[2] Benoumhani, M., On some numbers related to Whitney numbers of Dowling lattices. Adv. in Appl. Math. 19(1997), no. 1, 106–116. doi:10.1006/aama.1997.0529Google Scholar
[3] Berndt, B. C., Ramanujan's Notebooks. Part I. Chpt. 3. Springer-Verlag, New York, 1985.Google Scholar
[4] Brenti, F., Unimodal, log-concave, and Pólya frequency sequences in combinatorics. Mem. Amer. Math. Soc. 81(1989), no. 413, 106 pp.Google Scholar
[5] Brenti, F., Log-concave and unimodal sequences in algebra, combinatorics, and geometry: an update. Contemp. Math., 178, American Mathematical Society, 1994, pp. 71–89.Google Scholar
[6] Burchnall, J. L., The Bessel polynomials. Canad. J. Math. 3(1951), 62–68.Google Scholar
[7] Butler, L. M., The q-log-concavity of q-binomial coefficients. J. Combin. Theory Ser. A 54(1990), no. 1, 54–63. doi:10.1016/0097-3165(90)90005-HGoogle Scholar
[8] Butler, L. M. and Flanigan, W. P., A note on log-convexity of q-Catalan numbers. Ann. Comb. 11(2007), no. 3–4, 369–373. doi:10.1007/s00026-007-0324-zGoogle Scholar
[9] Carlitz, L., A note on the Bessel polynomials. Duke. Math. J. 24(1957), 151–162. doi:10.1215/S0012-7094-57-02421-3Google Scholar
[10] Choi, J. Y. and Smith, J. D. H., On the unimodality and combinatorics of Bessel numbers. The 2000 om2MaC Conference on Association Schemes, Codes and Designs (Pohang). Discrete Math. 264(2003) no. 1–3, 45–53. doi:10.1016/S0012-365X(02)00549-6Google Scholar
[11] Dowling, T. A., A class of geometric lattices based on finite groups. J. Combinatorial Theory Ser. B 14(1973), 61–86; Erratum. J. Combin. Theory Ser. B 15(1973), 211. doi:10.1016/S0095-8956(73)80007-3Google Scholar
[12] Dumont, D. and Ramamonjisoa, A., Grammaire de Ramanujan et arbres de Cayley. Electron. J. Combin. 3(1996), no. 2, Research Paper 17, 18 pp.Google Scholar
[13] Grosswald, E., On some algebraic properties of the Bessel polynomials. Trans. Amer. Math. Soc. 71(1951), 197–210.Google Scholar
[14] Han, H. and Seo, S., Combinatorial proofs of inverse relations and log-concavity for Bessel numbers. European J. Combin. 29(2008), no. 7, 1544–1554. doi:10.1016/j.ejc.2007.12.002Google Scholar
[15] Krall, H. L. and Frink, O., A new class of orthogonal polynomials: The Bessel polynomials. Trans. Amer. Math. Soc. 65(1949), 100–115.Google Scholar
[16] Krattenthaler, C., On the q-log-concavity of Gaussian binomial coefficients. Monatsh. Math. 107(1989), no. 4, 333–339. doi:10.1007/BF01517360Google Scholar
[17] Kurtz, D. C., A note on concavity properties of triangular arrays of numbers. J. Combinatorial Theory Ser. A 13(1972), 135–139. doi:10.1016/0097-3165(72)90017-9Google Scholar
[18] Liu, L. L. and Wang, Y., On the log-convexity of combinatorial sequences. Adv. Appl. Math. 39(2007), no. 4, 453–476. doi:10.1016/j.aam.2006.11.002Google Scholar
[19] Leroux, P., Reduced matrices and q-log concavity properties of q-Stirling numbers. J. Combin. Theory Ser. A 54(1990), no. 1, 64–84. doi:10.1016/0097-3165(90)90006-IGoogle Scholar
[20] Sagan, B. E., Inductive proofs of q-log concavity. Discrete Math. 99(1992), no. 1–3, 298–306. doi:10.1016/0012-365X(92)90377-RGoogle Scholar
[21] Sagan, B. E., Log concave sequences of symmetric functions and analogs of the Jacobi-Trudi determinants. Trans. Amer. Math. Soc. 329(1992), no. 2, 795–811. doi:10.2307/2153964Google Scholar
[22] Shor, P.W., A new proof of Cayley's formula for counting labeled trees. J. Combin. Theory Ser. A 71(1995), no. 1, 154–158. doi:10.1016/0097-3165(95)90022-5Google Scholar
[23] Stanley, R. P., Log-concave and unimodal sequences in algebra, combinatorics and geometry. In: Graph theory and its applications: East andWest (Jinan, 1986), Ann. New York Acad. Sci, 576, New York Acad. Sci., New York, 1989, pp. 500–535.Google Scholar
[24] Tanny, S., On some numbers related to the Bell numbers. Canad. Math. Bull. 17(1974/75), no. 5, 733–738.Google Scholar
[25] Zeng, J., A Ramanujan sequence that refines the Cayley formula for trees. Ramanujan J. 3(1999), no. 1, 45–54. doi:10.1023/A:1009809224933Google Scholar