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Original Articles

How To Generalize (and Not To Generalize) the Chu–Vandermonde Identity

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Pages 54-62 | Received 31 Jan 2018, Accepted 23 Apr 2018, Published online: 19 Dec 2019

References

  • Aceto, L., Trigiante, D. (2001). The matrices of Pascal and other greats. Amer. Math. Monthly. 108(3): 232–245. DOI: 10.2307/2695384.
  • Bala, P. (2015). Notes on generalized Riordan arrays. oeis.org/A260492/a260492.pdf
  • Barry, P. (2016). Riordan Arrays: A Primer. County Kildare, Ireland: Logic Press.
  • Bergeron, F., Labelle, G., Leroux, P. (1998). Combinatorial Species and Tree-Like Structures. Cambridge–New York: Cambridge Univ. Press.
  • Call, G. S., Velleman, D. J. (1993). Pascal’s matrices. Amer. Math. Monthly. 100(4): 372–376. DOI:10.2307/2324960.
  • Edelman, A., Strang, G. (2004). Pascal matrices. Amer. Math. Monthly. 111(3): 189–197. DOI:10.2307/4145127.
  • Fillmore, J. P., Williamson, S. G. (1973). A linear algebra setting for the Rota–Mullin theory of polynomials of binomial type. Linear Multilinear Algebra. 1(1): 67–80. DOI:10.1080/03081087308817006.
  • Garsia, A. M. (1973). An exposé of the Mullin–Rota theory of polynomials of binomial type. Linear Multilinear Algebra. 1(1): 47–65. DOI:10.1080/03081087308817005.
  • Gessel, I. M. (2003). Applications of the classical umbral calculus. Algebra Universalis. 49(4): 397–434. DOI:10.1007/s00012-003-1813-5.
  • Graham, R. L, Knuth, D. E., Patashnik, O. (1994). Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison-Wesley.
  • Hoggatt, V. E., Jr., Bergum, G. E. (1975). Generalized convolution arrays. Fibonacci Quart. 13(3): 193–198.
  • Knuth, D. E. (1992). Convolution polynomials. Math. J. 2(4): 67–78.
  • Labelle, G. (1980). Sur l’inversion et l’itération continue des séries formelles. Eur. J. Combin. 1(2): 113–138. DOI:10.1016/S0195-6698(80)80047-3.
  • Labelle, G. (1981). Une nouvelle démonstration combinatoire des formules d’inversion de Lagrange. Adv. Math. 42(3): 217–247. DOI:10.1016/0001-8708(81)90041-4.
  • Mullin, R., Rota, G.-C. (1970). On the foundations of combinatorial theory. III. Theory of binomial enumeration. In: Harris, B., ed. Graph Theory and Its Applications. New York: Academic Press, pp. 167–213.
  • Olive, G. (1979). Binomial functions and combinatorial mathematics. J. Math. Anal. Appl. 70(2): 460–473. DOI:10.1016/0022-247X(79)90058-1.
  • Pemantle, R., Wilson, M. C. (2013). Analytic Combinatorics in Several Variables. Cambridge: Cambridge Univ. Press.
  • Pétréolle, M. Sokal, A. D., Zhu, B.-X. (2018). Non-triangular linear transforms preserving Hankel-total positivity. In preparation.
  • Roman, S. M. (1984). The Umbral Calculus. New York: Academic Press.
  • Roman, S. M., Rota, G.-C. (1978). The umbral calculus. Adv. Math. 27(2): 95–188. DOI:10.1016/0001-8708(78)90087-7.
  • Rota, G.-C., Kahaner, D., Odlyzko, A. (1973). On the foundations of combinatorial theory. VIII. Finite operator calculus. J. Math. Anal. Appl. 42(3): 684–760. DOI:10.1016/0022-247X(73)90172-8.
  • Rota, G.-C., Taylor, B. D. (1994). The classical umbral calculus. SIAM J. Math. Anal. 25(2): 694–711. DOI:10.1137/S0036141093245616.
  • Scott, A. D., Sokal, A. D. (2009). Some variants of the exponential formula, with application to the multivariate Tutte polynomial (alias Potts model). Séminaire Lotharingien de Combinatoire. 61A: article 61Ae.
  • Shapiro, L. W., Getu, S., Woan, W. J., Woodson, L. C. (1991). The Riordan group. Discrete Appl. Math. 34(1–3): 229–239. DOI:10.1016/0166-218X(91)90088-E.
  • Sokal, A. D. (2018). Coefficientwise total positivity (via continued fractions) for some Hankel matrices of combinatorial polynomials. In preparation.
  • Sprugnoli, R. (1994). Riordan arrays and combinatorial sums. Discrete Math. 132(1–3): 267–290. DOI:10.1016/0012-365X(92)00570-H.
  • Stanley, R. P. (1999). Enumerative Combinatorics, Vol. 2. Cambridge–New York: Cambridge Univ. Press.
  • Vandermonde, A.-T. (1772). Mémoire sur des irrationnelles de différens ordres avec une application au cercle. Mémoires de Mathématique et de Physique, Tirés des Registres de l’Académie Royale des Sciences. 1772: 489–498.
  • Wang, W., Wang, T. (2008). Generalized Riordan arrays. Discrete Math. 308(24): 6466–6500. DOI:10.1016/j.disc.2007.12.037.
  • Wilf, H. S. (1994). generatingfunctionology, 2nd ed. San Diego–London: Academic Press.
  • Zeng, J. (1996). Multinomial convolution polynomials. Discrete Math. 160(1–3): 219–228. DOI:10.1016/0012-365X(95)00160-X.
  • Zhū Shìjié [= Chu Shih-chieh]. (2006). Jade Mirror of the Four Unknowns, vols. IandII. Translated into Modern Chinese by Guo Shuchun, translated into English by Ch’en Tsai Hsin, revised and supplemented by Guo Jinhai. Shenyang: Liaoning Education Press. Originally published in Chinese at Yangzhou, 1303.

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