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On the set of zero coefficients of a function satisfying a linear differential equation

Published online by Cambridge University Press:  22 February 2012

JASON P. BELL
Affiliation:
Department of Mathematics, Simon Fraser University, 8888 University Dr., Burnaby, BC, V5A 1S6Canada. e-mail: jpb@sfu.ca
STANLEY N. BURRIS
Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, ON, N2L 3G1, Canada. e-mail: snburris@math.uwaterloo.ca
KAREN YEATS
Affiliation:
Department of Mathematics, Simon Fraser University, 8888 University Dr., Burnaby, BC, V5A 1S6, Canada. e-mail: karen_yeats@sfu.ca

Abstract

Let K be a field of characteristic zero and suppose that f: K satisfies a recurrence of the form

\[ f(n) = \sum_{i=1}^d P_i(n) f(n-i), \]
for n sufficiently large, where P1(z),. . .,Pd(z) are polynomials in K[z]. Given that Pd(z) is a nonzero constant polynomial, we show that the set of n for which f(n) = 0 is a union of finitely many arithmetic progressions and a finite set. This generalizes the Skolem–Mahler–Lech theorem, which assumes that f(n) satisfies a linear recurrence. We discuss examples and connections to the set of zero coefficients of a power series satisfying a homogeneous linear differential equation with rational function coefficients.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 2012

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References

REFERENCES

[1]Adamczewski, B. and Bell, J. P.On the set of zero coefficients of algebraic power series. Invent. Math. 187, no. 2 (2012), 343393.CrossRefGoogle Scholar
[2]Allouche, J.-P. and Shallit, J.Automatic Sequences. Theory, Applications, Generalizations (Cambridge University Press, 2003).CrossRefGoogle Scholar
[3]Bell, J. P.A generalised Skolem–Mahler–Lech Theorem for affine varieties. J. London Math. Soc. 73 (2006), 367379.CrossRefGoogle Scholar
[4]Bell, J. P.Corrigendum to “A generalised Skolem–Mahler–Lech Theorem for affine varieties”. J. London Math. Soc. 78 (2008), 267272.CrossRefGoogle Scholar
[5]Bell, J. P., Burris, S. N. and Yeats, K.Spectra and Systems of Equations. arXiv:0911.2494.Google Scholar
[6]Bell, J. P., Ghioca, D. and Tucker, T. J.The dynamical Mordell–Lang problem for étale maps. Amer. J. Math. 132, no. 6 (2010), 16551675.CrossRefGoogle Scholar
[7]Bézivin, J.-P.Une généralisation du théorème de Skolem–Mahler–Lech. Quart. J. Math. Oxford Ser. (2). 40 (1989), no. 158, 133138.CrossRefGoogle Scholar
[8]Bézivin, J.-P. and Laohakosol, V.On the theorem of Skolem–Mahler–Lech. Exposition. Math. 9 (1991), no. 1, 8996.Google Scholar
[9]Bousquet–Mélou, M.Walks in the quarter plane: Kreweras' algebraic model. Ann. Appl. Probab. 15 (2005), no. 2, 14511491.CrossRefGoogle Scholar
[10]Chyzak, F. and Salvy, B.Non-commutative elimination in Ore algebras proves multivariate identities. J. Symbolic Comput. 26 (1998), no. 2, 187227.CrossRefGoogle Scholar
[11]Chyzak, F., Mishna, M. and Bruno, B. SalvyEffective scalar products of D-finite symmetric functions. J. Combin. Theory Ser. A 112 (2005), no. 1, 143.CrossRefGoogle Scholar
[12]Derksen, H.A Skolem–Mahler–Lech theorem in positive characteristic and finite automata. Invent. Math. 168 (2007), 175224.CrossRefGoogle Scholar
[13]Everest, G., van der Poorten, A., Shparlinski, Alf I. and Ward, T.Recurrence sequences. Mathematical Surveys and Monographs, 104 (Amer. Math. Soc. 2003).CrossRefGoogle Scholar
[14]Evertse, J.–H., Schlickewei, H. P. and Schmidt, W. M.Linear equations in variables which lie in a multiplicative group. Ann. of Math. (2) 155 (2002), no. 3, 807836.CrossRefGoogle Scholar
[15]Garoufalidis, S.G-functions and multisum versus holonomic sequences. Adv. Math. 220 (2009), no. 6, 19451955.CrossRefGoogle Scholar
[16]Gessel, I.Symmetric functions and P-recursiveness. J. Combin. Theory Ser. A 53 (1990), no. 2, 257285.CrossRefGoogle Scholar
[17]Hansel, G.Une démonstration simple du théorème de Skolem–Mahler–Lech. Theoret. Comput. Sci. 43 (1986), no. 1, 9198.CrossRefGoogle Scholar
[18]Lang, S.Algebraic number theory. Second edition. Graduate Texts in Mathematics, 110 (Springer-Verlag, 1994).CrossRefGoogle Scholar
[19]Laohakosol, V.Some extensions of the Skolem–Mahler–Lech theorem. Exposition. Math. 7 (1989), no. 2, 137187.Google Scholar
[20]Lech, C.A note on recurring series. Ark. Mat. 2 (1953), 417421.CrossRefGoogle Scholar
[21]Li, H. and Van Oystaeyen, F.Elimination of variables in linear solvable polynomial algebras and ∂-holonomicity. J. Algebra 234 (2000), no. 1, 101127.Google Scholar
[22]Mahler, K.Eine arithmetische Eigenshaft der Taylor–Koeffizienten rationaler Funktionen. Proc. Kon. Nederlandsche Akad. v. Wetenschappen 38 (1935), 5060.Google Scholar
[23]Mahler, K.On the Taylor coefficients of rational functions. Proc. Camb. Phil. Soc. 52 (1956), 3948.CrossRefGoogle Scholar
[24]Mahler, K.Addendum to the paper “On the Taylor coefficients of rational functions". Proc. Camb. Phil. Soc. 53 (1957), 544.CrossRefGoogle Scholar
[25]Mahler, K.p-adic Numbers and Their Functions, Second ed. (Cambridge University Press, Cambridge, New York, 1981).Google Scholar
[26]Methfessel, C.On the zeros of recurrence sequences with non-constant coefficients. Arch. Math. (Basel) 74 (2000), no. 3, 201206.CrossRefGoogle Scholar
[27]van der Poorten, A. J.A proof that Euler missed . . . Apéry's proof of the irrationality of ζ(3). Math. Intelligencer 1 (1979), 195203.CrossRefGoogle Scholar
[28]van der Poorten, A. J.Some facts that should be better known; especially about rational functions. Number Theory and Applications ed. Mollin, Richard A., (NATO–Advanced Study Institute, Banff, 1988), (Kluwer Academic Publishers, 1989), 497528.Google Scholar
[29]van der Poorten, A. J. and Tijdeman, R.On common zeros of exponential polynomials. Enseign. Math. (2) 21 (1975), no. 1, 5767.Google Scholar
[30]Robert, A.A course in p-adic analysis. Graduate Texts in Mathematics, 198 (Springer-Verlag, 2000).CrossRefGoogle Scholar
[31]Rubel, L. A.Some research problems about algebraic differential equations. Trans. Amer. Math. Soc. 280 (1983), no. 1, 4352 (Problem 16).CrossRefGoogle Scholar
[32]Skolem, T. Ein Verfahren zur Behandlung gewisser exponentialer Gleichungen und diophantischer Gleichungen. C. r. 8 Congr. Scand. à Stockholm (1934), 163–188.Google Scholar
[33]Stanley, R.Differentiably finite power series. European J. Combin. 1 (1980), 175188.CrossRefGoogle Scholar
[34]Strassman, R.Über den Wertevorrat von Potenzreihen im Gebiet der p-adischen Zahlen. J. Reine Angew. Math. 159 (1928), 13–28; 6566.CrossRefGoogle Scholar
[35]Wilf, H. S. and Zeilberger, D.An algorithmic proof theory for hypergeometric (ordinary and “q”) multisum/integral identities. Invent. Math. 108 (1992), no. 3, 575633.CrossRefGoogle Scholar