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Several Remarks on q-Binomial Inverse Formula and Examples

Received: 22 January 2017     Accepted: 20 March 2017     Published: 13 April 2017
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Abstract

In this paper, we first give some comments on the paper [J. Goldman and G. C. Rota, on the foundations of combinatorial theory IV finite vector spaces and Eulerian generating functions, Stud. Appl. Math., 49: 239--258 (1970)]. In that paper, Goldman and Rota showed two incorrect inversion formulas in Section 3. We point out the formulas and give the correct versions with the proof in this this paper first. Then we give some remarks on -binomial inverse formula concerning its applications.

Published in International Journal of Discrete Mathematics (Volume 2, Issue 3)
DOI 10.11648/j.dmath.20170203.18
Page(s) 107-111
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2017. Published by Science Publishing Group

Keywords

Inverse Formula, Binomial Inverse Formula, q-Binomial Inverse Formula

References
[1] G. E. Andrews, On the difference of successive Gaussian polynomials, J. Stat. Planning and Inf., 1993 (34), pp. 19-22.
[2] Ch. A. Charalambides, Non-central generalized -factorial coefficients and -Stirling numbers, Discrete Math., 275 (2004), pp. 67-85.
[3] J. Goldman and G. C. Rota, On the foundations of combinatorial theory IV finite vector spaces and Eulerian generating functions, Stud. Appl. Math., 1970 (49), pp. 239-258.
[4] R. L. Graham, D. E. Knuth and O. Patashnik, Concrete mathematics, second ed., Addison-Wesley Publishing Company, Boston, 1994.
[5] M. Aigner, A course in enumeration, Springer, New York, 2007.
[6] L. Carlitz, Some inversion formulas, Rend. Circ. Mat. Palermo, 12 (1963), pp. 1-17.
[7] H. W. Goulo and L. C. Hsu, Some new inverse relations, Duke Math. J., 40 (1973), pp. 885-891.
[8] J. Roroa, Combinatorial Identities, New York, Wiley, 1968.
[9] L. Carlitz, Some inverse relations, Duke Math. J., 1973 (40), pp. 893-901.
[10] K. N. Boyadzhiev, Lah numbers, Laguerre polynomials of order negative one, and the nth derivative of , Acta Univ. Sapientiae, Mathematica, 2016 (8), pp. 22-31.
[11] B.-N. Guo and F. Qi, Six proofs for an identity of the Lah numbers, Online J. Anal. Comb., 2015 (10), pp. 1-5.
[12] J. F. Steffensen, Interpolation (2nd ed.), Dover Publications. (A reprint of the 1950 edition by Chelsea Publishing Co.)
[13] B. Osgood and W. Wu, Falling Factorials, Generating Functions, and Conjoint Ranking Tables, J. Integer Seq., 2009 (12), Article 09.7.8.
[14] R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed., Reading, MA: Addison-Wesley, 1994.
[15] Y.-X. Wang, A. Smola and R. J. Tibshirani, The falling factorial basis and its statistical applications, Proceedings of the 31st International Conference on Machine Learning.
[16] J. P. S. Kung, Curious Characterizations of Projective and Affine Geometries, Adv. Appl. Math., 2002 (28), pp. 523–543.
[17] Q. Zou, The -binomial inverse formula and a representation for the -Catalan-Qi numbers, submitted.
[18] Z. H. Sun, Invariant Sequences Under Binomial Transformations, The Fibonacci Quarterly 2001 (39), pp. 324-333.
[19] Y. Wang, Self-inverse sequences related to a binomial inverse pair, The Fibonacci Quarterly 2005 (43), pp. 46-52.
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  • APA Style

    Qing Zou. (2017). Several Remarks on q-Binomial Inverse Formula and Examples. International Journal of Discrete Mathematics, 2(3), 107-111. https://doi.org/10.11648/j.dmath.20170203.18

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    ACS Style

    Qing Zou. Several Remarks on q-Binomial Inverse Formula and Examples. Int. J. Discrete Math. 2017, 2(3), 107-111. doi: 10.11648/j.dmath.20170203.18

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    AMA Style

    Qing Zou. Several Remarks on q-Binomial Inverse Formula and Examples. Int J Discrete Math. 2017;2(3):107-111. doi: 10.11648/j.dmath.20170203.18

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  • @article{10.11648/j.dmath.20170203.18,
      author = {Qing Zou},
      title = {Several Remarks on q-Binomial Inverse Formula and Examples},
      journal = {International Journal of Discrete Mathematics},
      volume = {2},
      number = {3},
      pages = {107-111},
      doi = {10.11648/j.dmath.20170203.18},
      url = {https://doi.org/10.11648/j.dmath.20170203.18},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.dmath.20170203.18},
      abstract = {In this paper, we first give some comments on the paper [J. Goldman and G. C. Rota, on the foundations of combinatorial theory IV finite vector spaces and Eulerian generating functions, Stud. Appl. Math., 49: 239--258 (1970)]. In that paper, Goldman and Rota showed two incorrect inversion formulas in Section 3. We point out the formulas and give the correct versions with the proof in this this paper first. Then we give some remarks on  -binomial inverse formula concerning its applications.},
     year = {2017}
    }
    

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    T2  - International Journal of Discrete Mathematics
    JF  - International Journal of Discrete Mathematics
    JO  - International Journal of Discrete Mathematics
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    AB  - In this paper, we first give some comments on the paper [J. Goldman and G. C. Rota, on the foundations of combinatorial theory IV finite vector spaces and Eulerian generating functions, Stud. Appl. Math., 49: 239--258 (1970)]. In that paper, Goldman and Rota showed two incorrect inversion formulas in Section 3. We point out the formulas and give the correct versions with the proof in this this paper first. Then we give some remarks on  -binomial inverse formula concerning its applications.
    VL  - 2
    IS  - 3
    ER  - 

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Author Information
  • Department of Mathematics, the University of Iowa, Iowa City, USA

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