A class of Frobenius-type Eulerian polynomials. (English) Zbl 1431.11036
Summary: The aim of this paper is to prove several explicit formulas associated with the Frobenius-type Eulerian polynomials in terms of the weighted Stirling numbers of the second kind. As a consequence, we derive an explicit formula for the tangent numbers of higher order. We also give a recursive method for the calculation of the Frobenius-type Eulerian numbers and polynomials.
MSC:
| 11B68 | Bernoulli and Euler numbers and polynomials |
| 11B73 | Bell and Stirling numbers |
| 11B65 | Binomial coefficients; factorials; \(q\)-identities |
Keywords:
algorithm; Frobenius-type Eulerian polynomials; tangent numbers; explicit formulas; recurrence relations; Stirling numbers of the first and second kind; Whitney numbers of the second kindReferences:
| [1] | M.A. Boutiche, M. Rahmani and H.M. Srivastava, Explicit formulas associated with some families of generalized Bernoulli and Euler polynomials, Mediterr. J. Math. 14 (2017), 1-10. · Zbl 1402.11034 · doi:10.1007/s00009-017-0891-0 |
| [2] | K.N. Boyadzhiev, A series transformation formula and related polynomials, Inter. J. Math. Math. Sci. 23 (2005), 3849-3866. · Zbl 1086.05006 · doi:10.1155/IJMMS.2005.3849 |
| [3] | A.Z. Broder, The \(r\)-Stirling numbers, Discr. Math. 49 (1984), 241-259. · Zbl 0535.05006 |
| [4] | L. Carlitz, Eulerian numbers and polynomials, Math. Mag. 32 (1958/1959), 247-260. · Zbl 0092.06601 · doi:10.2307/3029225 |
| [5] | —-, Eulerian numbers and polynomials of higher order, Duke Math. J. 27 (1960), 401-423. · Zbl 0104.29003 |
| [6] | L. Carlitz, Permutations, sequences and special functions, SIAM Rev. 17 (1975), 298-322. · Zbl 0301.05001 · doi:10.1137/1017034 |
| [7] | —-, Weighted Stirling numbers of the first and second kind, I, Fibonacci Quart. 18 (1980), 147-162. · Zbl 0428.05003 |
| [8] | —-, Weighted Stirling numbers of the first and second kind, II, Fibonacci Quart. 18 (1980), 242-257. · Zbl 0441.05003 |
| [9] | L. Carlitz and R. Scoville, Tangent numbers and operators, Duke Math. J. 39 (1972), 413-429. · Zbl 0243.05009 |
| [10] | J. Choi, D.S. Kim, T. Kim and Y.H. Kim, A note on some identities of Frobenius-Euler numbers and polynomials, Inter. J. Math. Math. Sci. 2012 (2012), 1-9. · Zbl 1333.11016 |
| [11] | D. Cvijović, Derivative polynomials and closed-form higher derivative formulae, Appl. Math. Comp. 215 (2009), 3002-3006. · Zbl 1181.26003 |
| [12] | —-, The Lerch zeta and related functions of non-positive integer order, Proc. Amer. Math. Soc. 138 (2010), 827-836. · Zbl 1196.11119 · doi:10.1090/S0002-9939-09-10116-8 |
| [13] | B.N. Guo, I. Mezö and F. Qi, An explicit formula for Bernoulli polynomials in terms of \(r\)-Stirling numbers of the second kind, Rocky Mountain J. Math. 46 (2016), 1919-1923. · Zbl 1371.11045 |
| [14] | D.S. Kim and T. Kim, Some new identities of Frobenius-Euler numbers and polynomials, J. Ineq. Appl. 2012 (2012), 1-10. · Zbl 1332.11025 |
| [15] | B. Kurt, A note on the Apostol type \(q\)-Frobenius-Euler polynomials and generalizations of the Srivastava-Pintér addition theorems, Filomat. 30 (2016), 65-72. · Zbl 1432.11022 |
| [16] | I. Mezö, A new formula for the Bernoulli polynomials, Result. Math. 58 (2010), 329-335. · Zbl 1237.11010 |
| [17] | M. Mihoubi and M. Rahmani, The partial \(r\)-Bell polynomials, Afrika Mat. (2017), 1-17. · Zbl 1387.05018 · doi:10.1007/s13370-017-0510-z |
| [18] | M. Rahmani, Generalized Stirling transform, Miskolc Math. Notes 15 (2014), 677-690. · Zbl 1324.11024 |
| [19] | —-, Some results on Whitney numbers of Dowling lattices, Arab J. Math. Sci. 20 (2014), 11-27. · Zbl 1377.11032 |
| [20] | H.M. Srivastava, Eulerian and other integral representations for some families of hypergeometric polynomials, Inter. J. Appl. Math. Stat. 11 (2007), 149-171. · Zbl 1157.33309 |
| [21] | —-, Some formulas for the Bernoulli and Euler polynomials at rational arguments, Math. Proc. Cambr. Philos. Soc. 129 (2000), 77-84. · Zbl 0978.11004 · doi:10.1017/S0305004100004412 |
| [22] | —-, Some generalizations and basic \((\)or \(q\)-\()\) extensions of the Bernoulli, Euler and Genocchi polynomials, Appl. Math. Inf. Sci. 5 (2011), 390-444. · Zbl 1231.33013 · doi:10.1016/j.camwa.2011.07.031 |
| [23] | H.M. Srivastava and J. Choi, Series associated with the zeta and related functions, Kluwer Academic Publishers, Dordrecht, 2001. · Zbl 1014.33001 |
| [24] | —-, Zeta and \(q\)-zeta functions and associated series and integrals, Elsevier Science Publishers, Amsterdam, 2012. · Zbl 1239.33002 |
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.
