Binomial sequences

Nowicki, Andrzej

doi:10.2478/aupcsm-2019-0008

Abstract

We present a description of all binomial sequences of polynomials in one variable over a field of characteristic zero.

References

[1] Aceto, Lidia and Isabel Cação. “A matrix approach to Sheffer polynomials.” J. Math. Anal. Appl. 446, no. 1 (2017): 87-100. Cited on 105.10.1016/j.jmaa.2016.08.038
Search in Google Scholar Back to article
[2] Berthelot, Pierre Cohomologie cristalline des schémas de caractéristique p > 0. Vol. 407 of Lecture Notes in Mathematics. Berlin-New York: Springer-Verlag, 1974. Cited on 95.
Search in Google Scholar Back to article
[3] Brand, Louis. “Binomial expansions in factorial powers.” Amer. Math. Monthly 67 (1960): 953-957. Cited on 94.10.1080/00029890.1960.11992031
Search in Google Scholar Back to article
[4] Brown, J.W. “On zero type sets of Laguerre polynomials.” Duke Math. J. 35 (1968): 821-823. Cited on 117.10.1215/S0012-7094-68-03586-2
Search in Google Scholar Back to article
[5] Di Bucchianico, A. Probabilistic and analytical aspects of the umbral calculus. Vol. 119 CWI Tract. Amsterdam: Stichting Mathematisch Centrum, Centrum voor Wiskunde en Informatica, 1997. Cited on 94.
Search in Google Scholar Back to article
[6] Carlitz, L. “Some generating functions for Laguerre polynomials.” Duke Math. J. 35 (1968), 825-827. Cited on 117.10.1215/S0012-7094-68-03587-4
Search in Google Scholar Back to article
[7] Comtet, Louis. Advanced combinatorics. The art of finite and infinite expansions. Dordrecht: D. Reidel Publishing Co., 1974. Cited on 115.
Search in Google Scholar Back to article
[8] Ekhad, Shalosh B. and John E. Majewicz. “A short WZ-style proof of Abel’s identity.” Electron. J. Combin. 3, no. 2 (1996): Research Paper 16, approx. 1 p. Cited on 115.10.37236/1274
Search in Google Scholar Back to article
[9] Foata, Dominique. “Enumerating k-trees.” Discrete Math. 1, no. 2 (1971/72): 181-186. Cited on 115.10.1016/0012-365X(71)90023-9
Search in Google Scholar Back to article
[10] Graham, Ronald L., Donald E. Knuth and Oren Patashnik. Concrete mathematics. A foundation for computer science. Reading, MA: Addison-Wesley Publishing Company, 1989. Cited on 95.
Search in Google Scholar Back to article
[11] Huang, Fengying and Bolian Liu. “The Abel-type polynomial identities.” Electron. J. Combin. 17, no. 1, (2010): Research Paper 10, 7 pp. Cited on 115.10.37236/282
Search in Google Scholar Back to article
[12] Kisil, Vladimir V. “Polynomial sequences of binomial type and path integrals.” Ann. Comb. 6, no. 1, (2002): 45-56. Cited on 94 and 117.10.1007/s00026-002-8029-9
Search in Google Scholar Back to article
[13] Krall, H.L. “Polynomials with the binomial property.” Amer. Math. Monthly 64 (1957): 342-343. Cited on 94, 103, 104, 105, 107, 109, 110 and 118.
Search in Google Scholar Back to article
[14] Lipnowski, M. “A solution of Problem 310.” Vol. 5 of Olymon – Mathematical Olympiads’ Correspondence Program. Canada: Canadian Mathematical Society, 2004. Cited on 115.
Search in Google Scholar Back to article
[15] Mihoubi, Miloud. “Bell polynomials and binomial type sequences.” Discrete Math. 308, no. 12 (2008): 2450-2459. Cited on 94.10.1016/j.disc.2007.05.010
Search in Google Scholar Back to article
[16] Młotkowski, Wojciech and Romanowicz, Anna. “A family of sequences of binomial type.” Probab. Math. Statist. 33, no. 2 (2013): 401-408. Cited on 119.
Search in Google Scholar Back to article
[17] Nowicki, Andrzej. Arithmetic functions Vol.5 of Podróże po Imperium Liczb. Toruń, Olsztyn: Wydawnictwo OWSIiZ, 2012. Cited on 94.
Search in Google Scholar Back to article
[18] Nowicki, Andrzej. Factorials and binomial coefficiens, Vol. 11 of Podróże po Imperium Liczb. Toruń, Olsztyn: Wydawnictwo OWSIiZ, 2013. Cited on 95.
Search in Google Scholar Back to article
[19] Petrullo, Pasquale. “Outcomes of the Abel identity.” Mediterr. J. Math. 10, no. 3 (2013): 1141-1150. Cited on 115.10.1007/s00009-013-0271-3
Search in Google Scholar Back to article
[20] Roman, Steven M. and Gian-Carlo Rota. “The umbral calculus.” Advances in Math. 27, no. 2 (1978): 95-188. Cited on 94 and 117.10.1016/0001-8708(78)90087-7
Search in Google Scholar Back to article
[21] Rota, Gian-Carlo, D. Kahaner and A. Odlyzko, “On the foundations of combinatorial theory. VIII. Finite operator calculus.” J. Math. Anal. Appl. 42 (1973): 684-760. Cited on 94.10.1016/0022-247X(73)90172-8
Search in Google Scholar Back to article
[22] Rota, Gian-Carlo, Jianhong Shen and Brian D. Taylor. “All polynomials of binomial type are represented by Abel polynomials.” Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25, no. 3-4 (1997): 731-738. Cited on 115.
Search in Google Scholar Back to article
[23] Schneider, Jonathan. “Polynomial sequences of binomial-type arising in graph theory.” Electron. J. Combin. 21, no. 1 (2014): Paper 1.43, 17 pp. Cited on 94.10.37236/3702
Search in Google Scholar Back to article
[24] Sheffer, I.M. “Some properties of polynomial sets of type zero.” Duke Math. J. 5 (1939): 590-622. Cited on 94, 104, 105, 107 and 109.
Search in Google Scholar Back to article
[25] Shukla, A.K. and S.J. Rapeli. “An extension of Sheffer polynomials.” Proyecciones 30, no. 2 (2011): 265-275. Cited on 105.10.4067/S0716-09172011000200009
Search in Google Scholar Back to article
[26] Sykora, S. “An Abel’s identity and its corollaries.” Preprint, 2014. Cited on 115.
Search in Google Scholar Back to article
[27] Zheng, G. “A solution of Problem 310.” Vol. 5 of Olymon – Mathematical Olympiads’ Correspondence Program. Canada: Canadian Mathematical Society, 2004. Cited on 115.
Search in Google Scholar Back to article

Binomial sequences

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