Abstract
In this paper, we study some properties of several polynomials arising from umbral calculus. In particular, we investigate the properties of orthogonality type of the Frobeniustype Eulerian polynomials which are derived from umbral calculus. By using our properties, we can derive many interesting identities of special polynomials associated with Frobeniustype Eulerian polynomials. An application to normal ordering is presented.
Similar content being viewed by others
References
L.C. Biedenharn, R.A. Gustafson, M.A. Lohe, J.D. Louck, and S.C. Milne, “Special Functions and Group Theory in Theoretical Physics. In Special Functions: Group Theoretical Aspects and Applications,” Math. Appl. (Reidel, Dordrecht, 1984), pp. 129–162.
L.C. Biedenharn, R.A. Gustafson, and S.C. Milne, “An Umbral Calculus for Polynomials Characterizing U(n) Tensor Products,” Adv. Math. 51, 36–90 (1984).
P. Blasiak, “Combinatorics of Boson Normal Ordering and Some Applications,” Concepts of Phyiscs 1, 177–278 (2004).
P. Blasiak, G. Dattoli, A. Horzela, and K.A. Penson, “Representations of Monomiality Principle with Sheffer-Type Polynomials and Boson Normal Ordering,” Phys. Lett. A 352, 7–12 (2006).
M. Can, M. Cenkci, V. Kurt, and Y. Simsek, “Twisted Dedekind Type Sums Associated with Barnes’ Type Multiple Frobenius-Euler l-Functions,” Adv. Stud. Contemp. Math. (Kyungshang) 18(2), 135–160 (2009).
I.N. Cangul, and Y. Simsek, “A Note on Interpolation Functions of the Frobenious-Euler Numbers, Application of Mathematics in Technical and Natural Sciences,” AIP Conf. Proc., 1301 (Amer. Inst. Phys., Melville, NY, 2010), pp. 59–67.
L. Carlitz, “Eulerian Numbers and Polynomials of Higher Order,” Duke Math. J. 27, 401–423 (1960).
L. Carlitz, “Eulerian Numbers and Polynomials,” Math. Mag. 32, 247–260 (1958/1959).
Ch. Charalambides, “On a Generalized Eulerian Distribution,” Ann. Inst. Statist. Math. 43, 197–206 (1991).
Ch. Charalambides, and M. Koutras, “On a Generalization of Morisita’s Model for Estimating the Habitat Preferencem,” Ann. Inst. Statist. Math. 46, 201–210 (1993).
P.S. Dwyer, “The Computation of Moments with the Use of Cumulative Totals,” Ann. Math Stat 9, 288–304 (1938).
P.S. Dwyer, “The Cumulative Numbers and Their Polynomials,” Ann. Math. Stat. 11, 66–71 (1940).
D. Foata, “Eulerian Polynomials: from Euler’s Time to the Present,” The legacy of Alladi Ramakrishnan in the mathematical sciences (Springer, New York, 2010), pp. 253–273.
J. Haglund, and M. Visontai, “Stable Multivariate Eulerian Polynomials and Generalized Stirling Permutations,” Eur. J. Combin. 33(4), 447–487 (2012).
T.-X. He, “Eulerian Polynomials and B-Splines,” J. Comput. Appl. Math. 236(15), 3763–3773 (2012).
K.G. Janardan, “Relationship between Morisita’s Model for Estimating the Environmental Density and the Generalized Eulerian Numbers,” Ann. Inst. Statist. Math. 40, 43–50 (1988).
J. Katriel, “Combinatorial Aspects of Boson Algebra,” Lett. Nuovo Cimento 10, 565–567 (1974).
J. Katriel, and M. Kibler, “Normal Ordering for Deformed Boson Operators and Operator-Valued Deformed Stirling Numbers,” J. Phys. A: Math. Gen. 25, 2683–2691 (1992).
D.S. Kim, T. Kim, and S.-H. Rim, “Frobenius-Type Eulerian Polynomials,” Proc. Jangjeon Math. Soc. 16(2), 156–163 (2013).
D.S. Kim, T. Kim, Y.H. Kim, and D.V. Dolgy, “A Note on Eulerian Polynomials, Associated with Bernoulli and Euler Numbers and Polynomials,” Adv. Stud. Contemp. Math. 22(3), 379–389 (2012).
D.S. Kim, T. Kim, S.-H. Lee, and S.-H. Rim, “Frobenius-Euler Polynomials and Umbral Calculus in the p-Adic Case,” Adv. Difference Equ. 2012, 222 (2012).
D.S. Kim, T. Kim, T. Mansour, S.-H. Rim, and M. Schork, “Umbral Calculus and Sheffer Sequences of Polynomials,” J. Math. Physics 54, 083504 (2013).
T. Kim, “An Identity of the Symmetry for the Frobenius-Euler Polynomials Associated with the Fermionic p-Adic Invariant q-Integrals on ℤp,” Rocky Mountain J. Math. 41(1), 239–247 (2011).
T. Kim, “Symmetry p-Adic Invariant Itegral on ℤp for Bernoulli and Euler Polynomials,” J. Difference Equ. Appl. 14(12), 1267–1277 (2008).
S. M. Ma, “A Summation Formula Related to q-Eulerian Polynomails,” Ars Combin. 94, 299–301 (2010).
T. Mansour, M. Schork, and S. Severini, “Wick’s Theorem for q-Deformed Boson Operators,” J. Phys. A 40, 8393–8401 (2007).
T. Mansour, M. Schork, and S. Severini, “A Generalization of the Boson Normal Ordering,” Phys. Lett. A 364, 214–220 (2007).
T. Mansour, and M. Schork, “On the Normal Ordering of Multi-Mode Boson Operators,” Russ. J. Math. Phys. 15, 51–60 (2008).
V. P. Maslov, Operational Methods [in Russian] (Nauka, Moscow, 1973); V. P. Maslov, Operational Methods (Mir, Moscow, 1976).
S. Roman, “More on the Umbral Calculus, with Emphasis on the q-Umbral Calculus,” J. Math. Anal. Appl. 107, 222–254 (1985).
S. Roman, The Umbral Calculus (Dover Publ. Inc. New York, 2005).
M. Schork, “On the Combinatorics of Normal Ordering Bosonic Operators and Deformations of It,” J. Phys. A: Math. Gen. 36, 4651–4665 (2003).
M. Schork, “Generalized Heisenberg Algebras and K-Generalized Fibonacci Numbers,” J. Phys. A 40(15), 4207–4214 (2007).
M. Schork, “Normal Ordering q-Bosons and Combinatorics,” Phys. Lett. A 355(4–5), 293–297 (2006).
B.G. Wilson, and F.G. Rogers, “Umbral Calculus and the Theory of Multispecies Nonideal Gases,” Phys. A 139, 359–386 (1986).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kim, T., Mansour, T. Umbral calculus associated with Frobenius-type Eulerian polynomials. Russ. J. Math. Phys. 21, 484–493 (2014). https://doi.org/10.1134/S1061920814040062
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1061920814040062