Vol. 17 (2023): CNU-JHE

The SM r-Stirling Numbers: An Algebraic Approach

Published 23-06-2023


  • Stirling numbers, r-Stirling numbers, generating functions, orthogonality,
  • relations,recursive formula, explicit formula, Schlömilch formula


The SM r-Stirling numbers by Broder were initially defined through their combinatorial interpretation, and all essential properties and identities were obtained using a combinatorial approach. This paper introduces a slightly modified version of the -Stirling numbers through their exponential generating functions and derives all necessary properties and identities using an algebraic approach.


  1. Abramowitz, M. and Stegun, I., 1970. Handbook of Mathematical Functions. Dover, New York.
  2. Agoh, T., 2014. Convolution identities for Bernoulli and Genocchi polynomials. Electronic J. Combin., 21:Article ID P1.65.
  3. Araci, S., 2012. Novel identities for q-Genocchi numbers and polynomials. J. Funct.Spaces Appl., 2012:Article ID 214961.
  4. Araci, S., 2014. Novel identities involving Genocchi numbers and polynomials arising from application of umbral calculus. Appl. Math. Comput., 247:780–785.
  5. Broder, A.Z., 1984. The r-Stirling numbers,” Discrete Mathematics, 49(3), pp. 241–259.
  6. Chen, C-C. and Koh, K-M., 1992. Principles and Techniques in Combinatorics, World Scientific.
  7. Comtet, L., 1974. Advanced Combinatorics, D. Reidel Publishing Company, Dordrecht, Holland.
  8. Corcino, R.B., 2020. Multi Poly-Bernoulli and Multi Poly-Euler Polynomials. In: Dutta, H., Peters, J. (eds) Applied Mathematical Analysis: Theory, Methods, and Applications. Studies in Systems, Decision and Control, vol 177. Springer, Cham. https://doi.org/10.1007/978-3-319-99918-0_21
  9. Corcino, R. B. and Barrientos, C., 2011. Some Theorems on the q-Analogue of the Generalized Stirling Numbers, Bulletin of the Malaysian Mathematical Sciences Society, 34(3), pp. 487-501.
  10. Corcino, R.B. and Corcino, C.B., 2012. The Hankel Transform of Generalized Bell Numbers and Its q-Analogue, Utilitas Mathematica, 89, pp. 297-309.
  11. Corcino, R. B. and Montero, C.B., 2012. A q-Analogue of Rucinski-Voigt Numbers, ISRN Discrete Mathematics, 2012, Article ID 592818, 18 pages.
  12. Corcino, R.B., Montero, M.B. and Ballenas, S., 2014. Schlömilch-Type Formula for r-Whitney numbers of the First Kind, Matimyas Matematika, 37(1-2), pp. 1-10.
  13. De Medicis, A. and Leroux, P., 1995. Generalized Stirling Numbers, Convolution Formulae and p,q-Analogues, Can. J. Math 47(3), pp. 474-499.
  14. Kim, T., Kim, D.S., Dolgy, D.V. and Rim, S.H., 2012. Some formula for the product of two Bernoulli and Euler polynomials. Abstr. Appl. Anal., 2012:Article ID 784307, 15 pages.
  15. Stirling, J., 1730. Methodus Differentialis, Sire Tractatus De Summatione et Interpolazione Serierum Infinitorum, London.