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Differential transcendence of Bell numbers and relatives: a Galois theoretic approach (2012.15292v5)

Published 30 Dec 2020 in math.NT, math.CA, and math.CO

Abstract: In 2003 Klazar proved that the ordinary generating function of the sequence of Bell numbers is differentially transcendental over the field $\mathbb{C}({t})$ of meromorphic functions at $0$. We show that Klazar's result is an instance of a general phenomenon that can be proven in a compact way using difference Galois theory. We present the main principles of this theory in order to prove a general result about differential transcendence over $\mathbb{C}({t})$, that we apply to many other (infinite classes of) examples of generating functions, including as very special cases the ones considered by Klazar. Most of our examples belong to Sheffer's class, well studied notably in umbral calculus. They all bring concrete evidence in support to the Pak-Yeliussizov conjecture, according to which a sequence whose both ordinary and exponential generating functions satisfy nonlinear differential equations with polynomial coefficients necessarily satisfies a linear recurrence with polynomial coefficients.

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References (67)
  1. Hypertranscendence and linear difference equations. J. Amer. Math. Soc., 34(2):475–503, 2021. 10.1090/jams/960.
  2. Désiré André. Sur les permutations alternées. J. Math. Pures Appl., 7:167–184, 1881. URL http://eudml.org/doc/233984.
  3. M. Ali Özarslan. Unified Apostol-Bernoulli, Euler and Genocchi polynomials. Comput. Math. Appl., 62(6):2452–2462, 2011. 10.1016/j.camwa.2011.07.031.
  4. T. M. Apostol. On the Lerch zeta function. Pacific J. Math., 1:161–167, 1951. URL http://projecteuclid.org/euclid.pjm/1103052188.
  5. Tom M. Apostol. A primer on Bernoulli numbers and polynomials. Math. Mag., 81(3):178–190, 2008. 10.1080/0025570x.2008.11953547.
  6. P. Appell. Sur une classe de polynômes. Ann. Sci. École Norm. Sup. (2), 9:119–144, 1880. URL http://www.numdam.org/item?id=ASENS_1880_2_9__119_0.
  7. V. I. Arnol′d. Snake calculus and the combinatorics of the Bernoulli, Euler and Springer numbers of Coxeter groups. Uspekhi Mat. Nauk, 47(1(283)):3–45, 240, 1992. 10.1070/RM1992v047n01ABEH000861.
  8. V. I. Arnol′d. Springer numbers and Morsification spaces. J. Algebraic Geom., 1(2):197–214, 1992.
  9. Barred Preferential Arrangements. Electron. J. Combin., 20(2):Paper 55, 18, 2013. 10.37236/2482.
  10. Ralph P. Boas, Jr. and R. Creighton Buck. Polynomial expansions of analytic functions. Second printing, corrected. Ergebnisse der Mathematik und ihrer Grenzgebiete, N.F., Bd. 19. Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin, 1964.
  11. Generating functions for generating trees. Discrete Math., 246(1-3):29–55, 2002. 10.1016/S0012-365X(01)00250-3. Formal power series and algebraic combinatorics (Barcelona, 1999).
  12. Counting quadrant walks via Tutte’s invariant method. In DMTCS Proceedings of FPSAC’16, Vancouver, Canada, pages 203–214, 2016. URL https://fpsac2016.sciencesconf.org/113714.html.
  13. Counting quadrant walks via Tutte’s invariant method. Comb. Theory, 1:Paper No. 3, 77, 2021. 10.5070/C61055360.
  14. E. T. Bell. Exponential polynomials. Ann. of Math. (2), 35(2):258–277, 1934. 10.2307/1968431.
  15. A note on Hölder’s theorem concerning the gamma function. Math. Ann., 232(2):115–120, 1978. 10.1007/BF01421399.
  16. (𝟐+𝟐)22({\bf 2}+{\bf 2})( bold_2 + bold_2 )-free posets, ascent sequences and pattern avoiding permutations. J. Combin. Theory Ser. A, 117(7):884–909, 2010. 10.1016/j.jcta.2009.12.007.
  17. Coherent families of polynomials. Analysis, 6(4):339–389, 1986. 10.1524/anly.1986.6.4.339.
  18. L. Carlitz. A note on the multiplication formulas for the Bernoulli and Euler polynomials. Proc. Amer. Math. Soc., 4:184–188, 1953. 10.2307/2031788.
  19. L. Carlitz. Bernoulli and Euler numbers and orthogonal polynomials. Duke Math. J., 26:1–15, 1959. URL http://projecteuclid.org/euclid.dmj/1077468334.
  20. L. Carlitz. Some generalized multiplication formulas for the Bernoulli polynomials and related functions. Monatsh. Math., 66:1–8, 1962. 10.1007/BF01418872.
  21. L. Carlitz. Eulerian numbers and polynomials. Math. Mag., 32:247–260, 1958/59. 10.2307/3029225.
  22. On the nature of the generating series of walks in the quarter plane. Invent. Math., 213(1):139–203, 2018. 10.1007/s00222-018-0787-z.
  23. Walks in the quarter plane: genus zero case. J. Combin. Theory Ser. A, 174:105251, 25, 2020. 10.1016/j.jcta.2020.105251.
  24. Dominique Dumont. Interprétations combinatoires des nombres de Genocchi. Duke Math. J., 41:305–318, 1974. URL http://projecteuclid.org/euclid.dmj/1077310398.
  25. Lucia Di Vizio. Difference Galois Theory for the “Applied” Mathematician. In Arithmetic and geometry over local fields—VIASM 2018, volume 2275 of Lecture Notes in Math., pages 29–59. Springer, Cham, [2021] ©2021. 10.1007/978-3-030-66249-3_2.
  26. Descent for differential Galois theory of difference equations: confluence and q𝑞qitalic_q-dependence. Pacific J. Math., 256(1):79–104, 2012. 10.2140/pjm.2012.256.79.
  27. Higher transcendental functions. Vols. I, II. McGraw-Hill Book Company, Inc., New York-Toronto-London, 1953. Based, in part, on notes left by Harry Bateman.
  28. Higher transcendental functions. Vol. III. Robert E. Krieger Publishing Co., Inc., Melbourne, Fla., 1981. Based on notes left by Harry Bateman, Reprint of the 1955 original.
  29. Generalized digital trees and their difference-differential equations. Random Struct. Algorithms, 3(3):305–320, 1992. 10.1002/rsa.3240030309.
  30. G. Frobenius. Über die Bernoullischen Zahlen und die Eulerschen Polynome. Preussische Akademie der Wissenschaften Berlin: Sitzungsberichte der Preußischen Akademie der Wissenschaften zu Berlin, pages 809–847, 1910. URL https://books.google.fr/books?id=RPNanQEACAAJ.
  31. Mellin transforms and asymptotics: Finite differences and Rice’s integrals. Theoret. Comput. Sci., 144(1-2):101–124, 1995. 10.1016/0304-3975(94)00281-M. Special volume on mathematical analysis of algorithms.
  32. Analytic combinatorics. Cambridge University Press, Cambridge, 2009. 10.1017/CBO9780511801655.
  33. J. W. L. Glaisher. On the Bernoullian function. Quart. J. Pure and Applied Math., 29:1–168, 1898.
  34. J. W. L. Glaisher. On a Set of Coefficients analogous to the Eulerian Numbers. Proc. Lond. Math. Soc., 31(1):216–235, 1899. 10.1112/plms/s1-31.1.216.
  35. H. W. Gould. Series transformations for finding recurrences for sequences. Fibonacci Quart., 28(2):166–171, 1990.
  36. Structure and Enumeration of (𝟑+𝟏)31({\bf 3}+{\bf 1})( bold_3 + bold_1 )-Free Posets. Ann. Comb., 18(4):645–674, 2014. 10.1007/s00026-014-0249-2.
  37. O. Hölder. Ueber die Eigenschaft der Gammafunction keiner algebraischen Differentialgleichung zu genügen. Math. Ann., 28(1):1–13, 1886. 10.1007/BF02430507.
  38. Charlotte Hardouin. Hypertranscendance des systèmes aux différences diagonaux. Compos. Math., 144(3):565–581, 2008. 10.1112/S0010437X07003430.
  39. A. F. Horadam. Genocchi polynomials. In Applications of Fibonacci numbers, Vol. 4 (Winston-Salem, NC, 1990), pages 145–166. Kluwer Acad. Publ., Dordrecht, 1991.
  40. Differential Galois theory of linear difference equations. Math. Ann., 342(2):333–377, 2008. 10.1007/s00208-008-0238-z.
  41. On differentially algebraic generating series for walks in the quarter plane. Selecta Math. (N.S.), 27(5):Paper No. 89, 49, 2021. 10.1007/s00029-021-00703-9.
  42. B. Imschenetzky. Sur la généralisation des fonctions de Jacques Bernoulli. Mémoires de l’Académie Impériale des Sciences de St.-Pétersbourg, VII-ème série, 31(11):1–58, 1883.
  43. Martin Klazar. Bell numbers, their relatives, and algebraic differential equations. J. Combin. Theory Ser. A, 102(1):63–87, 2003. 10.1016/S0097-3165(03)00014-1.
  44. Martin Klazar. Bell numbers—unordered and ordered—and algebraic differential equations, 2016. ICM 2006. Short Communications. Abstracts. Section 14. Combinatorics.
  45. On the unification of Bernoulli and Euler polynomials. Indian J. Pure Appl. Math., 6(1):98–107, 1975.
  46. Qiu-Ming Luo. Apostol-Euler polynomials of higher order and Gaussian hypergeometric functions. Taiwanese J. Math., 10(4):917–925, 2006. 10.11650/twjm/1500403883.
  47. Kurt Mahler. Zur Approximation der Exponentialfunktion und des Logarithmus. Teil I. J. Reine Angew. Math., 166:118–136, 1932. 10.1515/crll.1932.166.118.
  48. Keiji Nishioka. A note on differentially algebraic solutions of first order linear difference equations. Aequationes Math., 27(1-2):32–48, 1984. 10.1007/BF02192657.
  49. N. E. Nørlund. Mémoire sur les polynomes de Bernoulli. Acta Math., 43(1):121–196, 1922. 10.1007/BF02401755.
  50. Niels Erik Nørlund. Vorlesungen über Differenzenrechnung. Springer, 1924. URL http://eudml.org/doc/204170.
  51. A. M. Odlyzko. Asymptotic enumeration methods. In Handbook of combinatorics, Vol. 1, 2, pages 1063–1229. Elsevier Sci. B. V., Amsterdam, 1995.
  52. A unified presentation of the generating functions of the generalized Bernoulli, Euler and Genocchi polynomials. Comput. Math. Appl., 60(10):2779–2787, 2010. 10.1016/j.camwa.2010.09.031.
  53. Igor Pak. Complexity problems in enumerative combinatorics. In Proceedings of the International Congress of Mathematicians—Rio de Janeiro 2018. Vol. IV. Invited lectures, pages 3153–3180. World Sci. Publ., Hackensack, NJ, 2018. 10.1142/9789813272880_0176.
  54. Matthew A. Papanikolas. Tannakian duality for Anderson-Drinfeld motives and algebraic independence of Carlitz logarithms. Invent. Math., 171(1):123–174, 2008. 10.1007/s00222-007-0073-y.
  55. C. Praagman. Fundamental solutions for meromorphic linear difference equations in the complex plane, and related problems. J. Reine Angew. Math., 369:101–109, 1986. 10.1515/crll.1986.369.101.
  56. Helmut Prodinger. Some information about the binomial transform. Fibonacci Quart., 32(5):412–415, 1994.
  57. Steven Roman. The umbral calculus, volume 111 of Pure and Applied Mathematics. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1984.
  58. Ulrich Schmid. The average CRI-length of a tree collision resolution algorithm in presence of multiplicity-dependent capture effects. In Automata, languages and programming (Vienna, 1992), volume 623 of Lecture Notes in Comput. Sci., pages 223–234. Springer, Berlin, 1992. 10.1007/3-540-55719-9_76.
  59. Michael F. Singer. Algebraic relations among solutions of linear differential equations. Trans. Amer. Math. Soc., 295(2):753–763, 1986. 10.2307/2000062.
  60. T. A. Springer. Remarks on a combinatorial problem. Nieuw Arch. Wisk. (3), 19:30–36, 1971.
  61. Stephen M. Tanny. On some numbers related to the Bell numbers. Canad. Math. Bull., 17(5):733–738, 1974/75. 10.4153/CMB-1974-132-8.
  62. Letterio Toscano. Una classe di polinomi della matematica attuariale. Riv. Mat. Univ. Parma, 1:459–470, 1950.
  63. Jacques Touchard. Nombres exponentiels et nombres de Bernoulli. Canadian J. Math., 8:305–320, 1956. 10.4153/CJM-1956-034-1.
  64. Marius van der Put and Michael F. Singer. Galois Theory of Difference Equations, volume 1666 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1997. 10.1007/BFb0096118.
  65. Michael Wibmer. Existence of ∂\partial∂-parameterized Picard-Vessiot extensions over fields with algebraically closed constants. J. Algebra, 361:163–171, 2012. 10.1016/j.jalgebra.2012.03.035.
  66. D. Zagier. A modified Bernoulli number. Nieuw Arch. Wisk. (4), 16(1-2):63–72, 1998.
  67. Don Zagier. Vassiliev invariants and a strange identity related to the Dedekind eta-function. Topology, 40(5):945–960, 2001. 10.1016/S0040-9383(00)00005-7.
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