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On two kinds of q-analogues of generalized Stirling numbers

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Abstract

We investigate two kinds of q-analogues of generalized Stirling numbers. One is a q-analogue of Hsu-Shiue’s generalized Stirling numbers and the other is a q-analogue of Comtet’s numbers. In particular, we derive a q-analogue of an equality of Dobinski-type. Moreover, a determinant of the matrix consisting of the q-analogue of Comtet’s numbers is evaluated.

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References

  1. Carlitz, L.: \(q\)-Bernoulli numbers and polynomials. Duke Math. J. 15, 987–1000 (1948)

    Article  MathSciNet  MATH  Google Scholar 

  2. Carlitz, L.: Degenerate Stirling, Bernoulli and Eulerian numbers. Utilitas Math. 15, 51–88 (1979)

    MathSciNet  MATH  Google Scholar 

  3. Carlitz, L.: Weighted Stirling numbers of the first and second kind-I, II. Fibonacci Q. 18, 242–257 (1980)

    MathSciNet  MATH  Google Scholar 

  4. Carlitz, L., Klamkin, M.S.: Stirling operators. Collect. Math. XXV(2), 186–211 (1974)

    MathSciNet  MATH  Google Scholar 

  5. Charalambides, ChA, Koutras, M.: On the differences of the generalized factorials at an arbitrary point and their combinatorial applications. Discret. Math. 47, 183–201 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  6. Charalambides, ChA: On the \(q\)-differences of the generalized \(q\)-factorials. J. Stat. Plann. Inference 54, 31–43 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  7. Charalambides, ChA: Non-central generalized \(q\)-factorial coefficients and \(q\)-Stirling numbers. Discret. Math. 275, 67–85 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chu, W.C., Wei, C.: Set partitions with restrictions. Discret. Math. 20, 1097–1108 (2008)

    MathSciNet  MATH  Google Scholar 

  9. Comtet, L.: Nombres de Stirling generaux et fonctions symetriques. C. R. Acad. Sci. Paris Ser. A 275, 747–750 (1972)

    MATH  Google Scholar 

  10. Comtet, L.: Advanced Combinatorics. D. Reidel Publishing Co., Dordrecht (1974)

    Book  MATH  Google Scholar 

  11. Ehrenborg, R.: Determinants involving \(q\)-Stirling numbers. Adv. Appl. Math. 31, 630–642 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  12. El-Desouky, B.S.: Multiparameter non-central Stirling numbers. Fibonacci Q. 32(3), 218–225 (1994)

    MathSciNet  MATH  Google Scholar 

  13. El-Desouky, B.S., Cakić, N.P.: Generalized higher order Stirling numbers. Math. Comput. Model. 54, 2848–2857 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. El-Desouky, B.S., Cakić, N.P., Mansour, T.: Modified approach to generalized Stirling numbers via differential operators. Appl. Math. Lett. 23, 115–120 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  15. El-Desouky, B.S., Gomaa, R.S., Cakić, N.P.: \(q\)-Analogues of multiparameter non-central Stirling and generalized harmonic numbers. Appl. Math. Comput. 232, 132–143 (2014)

    MathSciNet  Google Scholar 

  16. Euler, L.: Introductio in Analysisn Infinitorum, vol. 2. M.-M. Bousquet & Soc., Lausanne (1748)

    Google Scholar 

  17. Gould, H.W.: The \(q\)-Stirling numbers of the first and second kinds. Duke Math. J. 28, 281–289 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  18. Gould, H.W., Hopper, A.T.: Operational formulas connected with two generalizations of Hermite polynomials. Duke Math. J. 29, 51–63 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  19. Howard, F.T.: Degenerate weighted Stirling numbers. Discret. Math. 57, 45–58 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  20. Hsu, L.C., Shiue, P.J.-S.: A unified approach to generalized Stirling numbers. Adv. Appl. Math. 20, 366–384 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  21. Johnson, W.P.: Some applications of the \(q\)-exponential formula. Discret. Math. 157, 207–225 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  22. Knuth, D.E.: Two notes on notation. Am. Math. Mon. 99, 403–422 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  23. Knuth, D.E.: Selected Papers on Discrete Mathematics. Center for the Study of Language and Information, Stanford (2003)

    MATH  Google Scholar 

  24. Koutras, M.: Non-central Stirling numbers and some applications. Discret. Math. 42, 73–89 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  25. Lang, W.: On generalizations of the Stirling number triangles. J. Integer Seq. 3, Art. 00.2.4 (2000)

  26. Lang, W.: Combinatorial interpretation of generalized Stirling numbers. J. Integer Seq. 12, Art. 09.3.3 (2009)

  27. Lumelskii, Y.P., Feigin, P.D., Tsilova, E.G.: Pólya distributions, combinatorial identities, and generalized Stirling numbers. J. Math. Sci. 127(4), 2073–2081 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  28. Mansour, T., Schork, M., Shattuck, M.: On a new family of generalized Stirling and Bell numbers. Electron. J. Comb. 18, 77 (2011)

    MathSciNet  MATH  Google Scholar 

  29. Milne, S.C.: A \(q\)-analog of restricted growth functions, Dobinski’s equality, and Charlier polynomials. Trans. Am. Math. Soc. 245, 89–118 (1978)

    MathSciNet  MATH  Google Scholar 

  30. Oruc, H., Akmaz, H.K.: Symmetric functions and the Vandermonde matrix. J. Comput. Appl. Math. 172(1), 49–64 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  31. Song, S.Z., Cheon, G.S., Jun, Y.B., Beasley, L.B.: A \(q\)-analogue of the generalized factorial numbers. J. Korean Math. Soc. 47(3), 645–657 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  32. Tsylova, E.G.: The asymptotic behavior of generalized Stirling numbers. Combinatorial-Algebraic Methods in Applied Mathematics, pp. 143–154, 158. Gor’kov Gos. University, Gorkin (1985). in Russian

    Google Scholar 

  33. Wagner, C.G.: Generalized Stirling and Lah numbers. Discret. Math. 160, 199–218 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  34. Xu, A.: A Newton interpolation approach to generalized Stirling numbers. J. Appl. Math. 2012, 796814:1–796814:12 (2012)

    MathSciNet  Google Scholar 

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Acknowledgments

We sincerely appreciate the anonymous referees and the editors for their very helpful comments and suggestions, which greatly improved the quality of this paper.

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Authors and Affiliations

  1. Institute of Mathematics, Zhejiang Wanli University, Ningbo, 315100, China

    Aimin Xu

  2. School of Computing Science, Zhejiang University City College, Hangzhou, 310015, China

    Peipei Tang

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  1. Aimin Xu
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  2. Peipei Tang
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Correspondence to Peipei Tang.

Additional information

This work was supported by the National Natural Science Foundation (Grant Nos. 11201430 and 11226297) of China, the Ningbo Natural Science Foundation (Grant No. 2014A610021) and the Natural Science Foundation of Zhejiang Province of China (Grant No. LQ13A010006).

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Xu, A., Tang, P. On two kinds of q-analogues of generalized Stirling numbers. Ramanujan J 43, 371–381 (2017). https://doi.org/10.1007/s11139-016-9792-2

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