A Family of Generalized Stirling Numbers of the First Kind


A modified approach via differential operator is given to derive a new family of generalized Stirling numbers of the first kind. This approach gives us an extension of the techniques given by El-Desouky [1] and Gould [2]. Some new combinatorial identities and many relations between different types of Stirling numbers are found. Furthermore, some interesting special cases of the generalized Stirling numbers of the first kind are deduced. Also, a connection between these numbers and the generalized harmonic numbers is derived. Finally, some applications in coherent states and matrix representation of some results obtained are given.

Share and Cite:

El-Desouky, B. , El-Bedwehy, N. , Mustafa, A. and Menem, F. (2014) A Family of Generalized Stirling Numbers of the First Kind. Applied Mathematics, 5, 1573-1585. doi: 10.4236/am.2014.510150.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] El-Desouky, B.S. (2011) Generalized String Numbers of the First Kind: Modified Approach. Journal of Pure and Mathematics: Advances and Applications, 5, 43-59.
[2] Gould, H.W. (1964) The Operator and Stirling Numbers of the First Kind. The American Mathematical Monthly, 71, 850-858. http://dx.doi.org/10.2307/2312391
[3] Comtet, L. (1972) Nombres de stirling généraux et fonctions symétriques. Comptes Rendus de l’Académie des Sciences (Series A), 275, 747-750.
[4] Comtet, L. (1974) Ad-vanced Combinatorics: The Art of Finite and Infinite Expiations. D. Reidel Publishing Company, Dordrecht, Holand. http://dx.doi.org/10.1007/978-94-010-2196-8
[5] Comtet, L. (1973) Une formule explicite pour les puissances successive de l’operateur derivation de Lie. Comptes Rendus de l’Académie des Sciences (Series A), 276, 165-168.
[6] El-Desouky, B.S. and Cakic, N.P. (2011) Generalized Higher Order Stirling Numbers. Mathematical and Computer Modelling, 54, 2848-2857. http://dx.doi.org/10.1016/j.mcm.2011.07.005
[7] Blasiak, P. (2005) Combinatorics of Boson Normal Ordering and Some Applica-tions. PhD Thesis, University of Paris, Paris. http://arxiv.org/pdf/quant-ph/0507206.pdf
[8] Blasiak, P., Penson, K.A. and Solomon, A.I. (2003) The General Boson Normal Ordering Problem. Physics Letters A, 309, 198-205. http://dx.doi.org/10.1016/S0375-9601(03)00194-4
[9] Cakic, N.P. (1980) On Some Combinatorial Identities. Applicable Analysis and Discrete Mathematics, 678-715, 91-94.
[10] Carlitz, L. (1932) On Arrays of Numbers. American Journal of Mathematics, 54, 739-752.
[11] El-Desouky, B.S., Cakic, N.P. and Mansour, T. (2010) Modified Approach to Generalized Stirling Numbers via Differential Operators. Applied Mathematics Letters, 23, 115-120.
[12] Viskov, O.V. and Srivastava, H.M. (1994) New Approaches to Certain Identities Involving Differential Operators. Journal of Mathematical Analysis and Applications, 186, 1-10.
[13] Macdonald, I.G. (1979) Symmetric Functions and Hall Polynomials. Clarendon (Oxford University) Press, Oxford, London and New York.
[14] Choi, J. and Srivastava, H.M. (2011) Some Summation Formulas Involving Harmonic Numbers and Generalized Harmonic Numbers. Mathematical and Computer Modelling, 54, 2220-2234.

Copyright © 2023 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.