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Polynomials related to \(q\)-analog of the generalized derivative polynomials. (English) Zbl 1490.05018

Summary: The derivative polynomials introduced by D. E. Knuth and T. J. Buckholtz [Math. Comput. 21, 663–688 (1967; Zbl 0178.04401)] play an important role in calculating the tangent and secant numbers. In this paper, we study a class of polynomials defined by the Möbius transformation on \(q\)-analog of the generalized derivative polynomials for the secant function. In a certain sense, it can be looked on as a common \(q\)-analog of two kinds of alternating Eulerian polynomials. We derive many properties of such polynomials, such as symmetry, unimodality, strong \(x\)-log-convexity, Hurwitz stability, semi-\( \gamma \)-positivity and convolutional relation. Moreover, we also obtain its exponential generating function, the Jacobi continued fraction expansion of its ordinary generating function and its explicit formula.

MSC:

05A30 \(q\)-calculus and related topics
05A15 Exact enumeration problems, generating functions
26C10 Real polynomials: location of zeros
05A20 Combinatorial inequalities
30B70 Continued fractions; complex-analytic aspects
11B68 Bernoulli and Euler numbers and polynomials

Citations:

Zbl 0178.04401

Software:

OEIS
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Full Text: DOI

References:

[1] Athanasiadis, C. A., Gamma-positivity in combinatorics and geometry, Sémin. Lothar. Comb., 77 (2018), Article B77i, 64pp · Zbl 1440.05195
[2] Borcea, J.; Brändén, P., The Lee-Yang and Pólya-Schur programs. I. Linear operators preserving stability, Invent. math., 177, 541-569 (2009) · Zbl 1175.47032
[3] Brändrén, P., Unimodality, log-concavity, real-rootedness and beyond, Sidebook of enumerative combinatorics, 437-483 (2015), CRC Press · Zbl 1327.05051
[4] Brenti, F., Log-concave and unimodal sequences in algebra, combinatorics, and geometry: an update, Contemp. Math., 178, 71-89 (1994) · Zbl 0813.05007
[5] Chebikin, D., Variations on descents and inversions in permutations, Electron. J. Combin., 15, R132 (2008), 34pp · Zbl 1179.05004
[6] Chen, W. Y.C.; Tang, R. L.; Wang, L. X.W.; Yang, A. L.B., The \(q\)-log-convexity of the Narayana polynomials of type \(B\), Adv. Appl. Math., 44, 2, 85-110 (2010) · Zbl 1230.05276
[7] Chen, W. Y.C.; Wang, L. X.W.; Yang, A. L.B., Recurrence relations for strongly \(q\)-log-convex polynomials, Canad. Math. Bull., 54, 217-229 (2011) · Zbl 1239.05190
[8] M.-J. Ding, B.-X. Zhu, Stability of combinatorial polynomials and its applications, arXiv:2106.121.76v2, 2021.
[9] Flajolet, P., Combinatorial aspects of continued fractions, Discrete Math., 32, 125-161 (1980) · Zbl 0445.05014
[10] Foata, D.; Schützenberger, M. P., Théorie Géometrique des Polynômes Eulériens, Lecture Notes in Mathematics, 138 (1970), Springer-Verlag · Zbl 0214.26202
[11] Gal, S. R., Real root conjecture fails for five- and higher-dimensional spheres, Discrete Comput. Geom., 34, 269-284 (2005) · Zbl 1085.52005
[12] Han, B., Gamma-positivity of derangement polynomials and binomial Eulerian polynomials for colored permutations, J. Combin. Theory Ser. A, 182 (2021), Artical 105459, 22pp · Zbl 1467.05006
[13] Hoffman, M. E., Derivative polynomials for tangent and secant, Amer. Math. Monthly, 102, 23-30 (1995) · Zbl 0834.26002
[14] Hoffman, M. E., Derivative polynomials, Euler polynomials, and associated integer sequences, Electron. J. Combin., 6, R21 (1999), 13pp · Zbl 0933.11005
[15] Josuat-Vergès, M., Enumeration of snakes and cycle-alternating permutations, Australas. J. Combin., 60, 3, 279-305 (2014) · Zbl 1305.05011
[16] Knuth, D. E.; Buckholtz, T. J., Computation of tangent, Euler and Bernoulli numbers, Math. Comp., 21, 663-688 (1967) · Zbl 0178.04401
[17] Lin, Z.; Ma, S.-M.; Wang, D. G.L.; Wang, L., Positivity and divisibility of alternating descent polynomials, Ramanujan J. (2021), https://link.springer.com/article/10.1007/s11139-021-00460-5
[18] Liu, L. L.; Wang, Y., On the log-convexity of combinatorial sequences, Adv. Appl. Math., 39, 453-476 (2007) · Zbl 1131.05010
[19] S.-M. Ma, Q. Fang, T. Mansour, Y.-N. Yeh, Alternating Eulerian polynomials and left peak polynomials, arXiv:2104.09374v1, 2021.
[20] Ma, S.-M.; Ma, J.; Yeh, Y.-N., David-Barton type identities and alternating run polynomials, Adv. Appl. Math., 114 (2020), Artical 101978, 19pp · Zbl 1433.05013
[21] Ma, S.-M.; Yeh, Y.-N., Enumeration of permutations by number of alternating descents, Discrete Math., 339, 1362-1367 (2016) · Zbl 1329.05015
[22] Rahman, Q. I.; Schmeisser, G., Analytic Theory of Polynomials, London Math. Soc. Monographs (N.S.), 26 (2002), Oxford University Press: Oxford University Press New York · Zbl 1072.30006
[23] N.J.A. Sloane, The on-line encyclopedia of integer sequences, http://oeis.org. · Zbl 1274.11001
[24] Stanley, R. P., Log-concave and unimodal sequences in algebra, combinatorics, and geometry, Ann. New York Acad. Sci., 576, 500-534 (1989) · Zbl 0792.05008
[25] B.-X. Zhu, A unified approach to combinatorial triangles: a generalized Eulerian polynomial, arXiv:2007.12602, 2020.
[26] Zhu, B.-X., Log-convexity and strong \(q\)-log-convexity for some triangular arrays, Adv. Appl. Math., 50, 4, 595-606 (2013) · Zbl 1277.05014
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