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Multidimensional analog of the Bernoulli polynomials and its properties. (English) Zbl 07325294

Summary: We consider a generalization of the Bernoulli numbers and polynomials to several variables, namely, we define the Bernoulli numbers associated with a rational cone and the corresponding Bernoulli polynomials. Also, we prove some properties of the Bernoulli polynomials.

MSC:

11B68 Bernoulli and Euler numbers and polynomials
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References:

[1] J. Bernoulli, Ars Conjectandi, Basel, 1713
[2] L. Euler, Institutiones calculi differentialis, 1755; new printing: Birkhäuser, 1913
[3] J.-L. Raabe, Die Jacob Bernoullische Funktion, Zürich, 1848
[4] N. E. Nörlund, Differenzenrechnung, Berlin, 1924 · JFM 50.0318.04
[5] R. L. Graham, D. E. Knuth, O. Patashnik, Concrete mathematics, A foundation for computer science, Second edition, Addison-Wesley Publishing Company, Reading, MA, 1994 · Zbl 0836.00001
[6] J. Riordan, Combinatorial identities, Reprint of the 1968 original, Robert E. Krieger Publishing Co., Huntington, N.Y., 1979
[7] G.-C. Rota, B. D. Taylor, “The classical umbral calculus”, SIAM J. Math. Anal., 25:2 (1994), 694-711 · Zbl 0797.05006 · doi:10.1137/S0036141093245616
[8] C.-E. Fröberg, Lärobok i numerisk analys, Stockholm, 1962
[9] H. W. Gould, Combinatorial identities, Morgantown printing and binding co., Morgantown WV-USA, 1972 · Zbl 0241.05011
[10] T. Ernst, “\(q\)-Bernoulli and \(q\)-Euler polynomials, an umbral approach”, International Journal of Difference Equations, 1:1 (2006), 31-80 · Zbl 1116.39013
[11] L. Carlitz, “Bernoulli and Euler numbers and orthogonal polynomials”, Duke Math. J., 26 (1959), 694-711
[12] T. Ernst, The history of \(q\)-calculus and a new method, Uppsala, 2000
[13] D. H. Lehmer, “Lacunary recurrence formulas for the numbers of Bernoulli and Euler”, Ann. of Math., 36:3 (1935), 637-649 · Zbl 0012.15103 · doi:10.2307/1968647
[14] H. M. Srivastava, A. Pintér, “Remarks on some relationships between the Bernoulli and Euler polynomials”, Appl. Math. Lett., 17:4 (2004), 375-380 · Zbl 1070.33012 · doi:10.1016/S0893-9659(04)90077-8
[15] H. S. Vandiver, “Simple explicit expressions for generalized Bernoulli numbers of the first order”, Duke Math. J., 8 (1941), 575-584 · Zbl 0063.07952 · doi:10.1215/S0012-7094-41-00849-9
[16] N. M. Temme, “Bernoulli polynomials old and new: Generalization and asymptotics”, CWI Quarterly, 1995, no. 1, 47-66 · Zbl 1044.11547
[17] P. Appell, “Sur une classe de polynômes”, Annales Scientifiques de l’École Normale Supérieure Sér., 2:9 (1880), 119-144 · JFM 12.0342.02
[18] M. Lenz, “Lattice points in polytopes, box splines, and Todd operators”, International Mathematics Research Notices, 2015, no. 14, 5289-5310 · Zbl 1360.52009 · doi:10.1093/imrn/rnu095
[19] A. V. Pukhlikov, A. G. Khovanskii, “The Riemann-Roch theorem for integrals and sums of quasipolynomials on virtual polytopes”, St. Petersburg Mathematical Journal, 4:4 (1993), 789-812 · Zbl 0798.52010
[20] M. Brion, M. Vergne, “Lattice points in simple polytopes”, Journal of the American Mathematical Society, 10:2 (1997), 371-392 · Zbl 0871.52009 · doi:10.1090/S0894-0347-97-00229-4
[21] M. Brion, M. Vergne, “Residue formulae, vector partition functions and lattice points in rational polytopes”, Journal of the American Mathematical Society, 10:4 (1997), 797-833 · Zbl 0926.52016 · doi:10.1090/S0894-0347-97-00242-7
[22] M. Vergne, “Residue formulae for Verlinde sums, and for number of integral points in convex rational polytopes”, European women in mathematics (Malta, 2001), World Sci. Publ., River Edge, NJ, 2003, 225-285 · Zbl 1122.11065 · doi:10.1142/9789812704276_0012
[23] M. Brion, N. Berline, “Local Euler-Maclaurin formula for polytopes”, Moscow Mathematical Society Journal, 7 (2007), 355-383 · Zbl 1146.52006
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