Abstract
In this paper, a modified approach to the multiparameter non-central Stirling numbers via differential operators, introduced by El-Desouky, and new explicit formulae of both kinds of these numbers are given. Also, some relations between these numbers and the generalized Hermite and Truesdel polynomials are obtained. Moreover, we investigate some new results for the generalized Stirling-type pair of Hsu and Shiue. Furthermore some interesting special cases, new combinatorial identities and a matrix representation are deduced.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Branson D.: Stirling numbers and Bell numbers: their role in combinatorics and probability. Math. Sci 25, 1–31 (2000)
Whei-Ching C. Chan, Kung-Yu Chen and H. M. Srivastava, Certain classes of generating functions for the Jacobi and related hypergeometric polynomials, Comput. Math. Appl. 44 (2002), 1539–1556.
Wenchang Chu, Chuanam Wei: Set partitions with restrictions. Discrete Math. 308, 3163–3168 (2008)
Comtet L.: Nombres de Stirling generaux et fonctions symetriques. C. R. Acad. Sci. Paris. (Ser. A) 275, 747–750 (1972)
L. Comtet, Advanced Combinatorics: The Art of Finite and Infinite Expansions, D. Reidel Publishing Company, Dordrecht, Holand, 1974.
Corcino R.B.: Some theorems on generalized Stirling numbers. Ars Combin. 60, 273–286 (2001)
Corcino M., Herrera R.B.,: Limit of the differences of the generalized factorial, its q- and p, q-analogue. Util. Math 72, 33–49 (2007)
Corcino R.B., Corcino C.B., Aldema R.: Asymptotic normality of the (r, β)-Stirling numbers. Ars Combin 81, 81–96 (2006)
Corcino R.B., Hsu L.C., Tan E.L.: A q-analogue of generalized Stirling numbers. Fibonacci Quart 44(2), 154–156 (2006)
Dattoli G., He M.X., Ricci P.E.: Eigenfunctions of Laguerre-type operators and generalized inevolution problems. Math. Comput. Modelling 42, 1263–1268 (2005)
El-Desouky B.S.: The multiparameter non-central Stirling numbers. Fibonacci Quart 32(3), 218–225 (1994)
Gould H.W., Hopper A.T.: Operational formulas connected with two generalizations of Hermite polynomials. Duke Math. J. 29, 51–63 (1962)
Howard F.T.: Degenerate weighted Stirling numbers. Discrete Math 57, 45–58 (1985)
Hsu L.C., Shiue P.J-Sh.: A unified approach to generalized Stirling numbers. Adv. in Appl. Math 20, 366–384 (1998)
A. O. Munagi, Labeled factorization of integers, Electron. J. Combin. 16 (2009), R50.
Shrivastava P.N.: On the polynomials of Truesdel type. Publ. Inst. Math. (Beograd) (N.S.) 9(23), 43–46 (1969)
Singh R.P.: On generalized Truesdel polynomials. Riv. Mat. Univ. Parma 2(8), 345–353 (1967)
Srivastava H.M., Manocha H.L.: A Treatise on Generating Functions. Ellis Horwood Limited, Chichester (1984)
Viskov O.V., Srivastava H.M.: New approaches to certain identities involving differential operators. J. Math. Anal. Appl 186, 1–10 (1994)
Wagner C.G.: Generalized Stirling and Lah numbers. Discrete Math. 160, 199–218 (1996)
C. G. Wagner, Partition statistics and q-Bell numbers (q = −1), J. Integer Seq. 7 (2004), Article 04.1.1.
H. S. Wilf, Generatingfunctionology, Academic Press Inc., New York, 1994.
Yu H.: generalization of Stirling numbers. Fibonacci Quart 36(3), 252–258 (1998)
Author information
Authors and Affiliations
Corresponding author
Additional information
This paper was supported in part by the Serbian Ministry of Education and Science (Cakić and Milovanović) and by the Egyptian Ministry of Higher Education (El-Desouky).
Rights and permissions
About this article
Cite this article
Cakić, N.P., El-Desouky, B.S. & Milovanović, G.V. Explicit Formulas and Combinatorial Identities for Generalized Stirling Numbers. Mediterr. J. Math. 10, 57–72 (2013). https://doi.org/10.1007/s00009-011-0169-x
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00009-011-0169-x