Rocky Mountain Journal of Mathematics

A class of Frobenius-type Eulerian polynomials

Abstract

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.

Article information

Source
Rocky Mountain J. Math., Volume 48, Number 3 (2018), 1003-1013.

Dates
First available in Project Euclid: 2 August 2018

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1533230836

Digital Object Identifier
doi:10.1216/RMJ-2018-48-3-1003

Mathematical Reviews number (MathSciNet)
MR3835583

Zentralblatt MATH identifier
06917359

Citation

Srivastava, H.M.; Boutiche, M.A.; Rahmani, M. A class of Frobenius-type Eulerian polynomials. Rocky Mountain J. Math. 48 (2018), no. 3, 1003--1013. doi:10.1216/RMJ-2018-48-3-1003. https://projecteuclid.org/euclid.rmjm/1533230836

References

• 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.
• K.N. Boyadzhiev, A series transformation formula and related polynomials, Inter. J. Math. Math. Sci. 23 (2005), 3849–3866.
• A.Z. Broder, The $r$-Stirling numbers, Discr. Math. 49 (1984), 241–259.
• L. Carlitz, Eulerian numbers and polynomials, Math. Mag. 32 (1958/1959), 247–260.
• ––––, Eulerian numbers and polynomials of higher order, Duke Math. J. 27 (1960), 401–423.
• L. Carlitz, Permutations, sequences and special functions, SIAM Rev. 17 (1975), 298–322.
• ––––, Weighted Stirling numbers of the first and second kind, I, Fibonacci Quart. 18 (1980), 147–162.
• ––––, Weighted Stirling numbers of the first and second kind, II, Fibonacci Quart. 18 (1980), 242–257.
• L. Carlitz and R. Scoville, Tangent numbers and operators, Duke Math. J. 39 (1972), 413–429.
• 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.
• D. Cvijović, Derivative polynomials and closed-form higher derivative formulae, Appl. Math. Comp. 215 (2009), 3002–3006.
• ––––, The Lerch zeta and related functions of non-positive integer order, Proc. Amer. Math. Soc. 138 (2010), 827–836.
• 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.
• D.S. Kim and T. Kim, Some new identities of Frobenius-Euler numbers and polynomials, J. Ineq. Appl. 2012 (2012), 1–10.
• 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.
• I. Mezö, A new formula for the Bernoulli polynomials, Result. Math. 58 (2010), 329–335.
• M. Mihoubi and M. Rahmani, The partial $r$-Bell polynomials, Afrika Mat. (2017), 1–17.
• M. Rahmani, Generalized Stirling transform, Miskolc Math. Notes 15 (2014), 677–690.
• ––––, Some results on Whitney numbers of Dowling lattices, Arab J. Math. Sci. 20 (2014), 11–27.
• H.M. Srivastava, Eulerian and other integral representations for some families of hypergeometric polynomials, Inter. J. Appl. Math. Stat. 11 (2007), 149–171.
• ––––, Some formulas for the Bernoulli and Euler polynomials at rational arguments, Math. Proc. Cambr. Philos. Soc. 129 (2000), 77–84.
• ––––, Some generalizations and basic $($or $q$-$)$ extensions of the Bernoulli, Euler and Genocchi polynomials, Appl. Math. Inf. Sci. 5 (2011), 390–444.
• H.M. Srivastava and J. Choi, Series associated with the zeta and related functions, Kluwer Academic Publishers, Dordrecht, 2001.
• ––––, Zeta and $q$-zeta functions and associated series and integrals, Elsevier Science Publishers, Amsterdam, 2012.