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Singularity Analysis Via the Iterated Kernel Method

Published online by Cambridge University Press:  10 June 2014

STEPHEN MELCZER  and
MARNI MISHNA
STEPHEN MELCZER
Affiliation:
Department of Mathematics, Simon Fraser University, Burnaby, BC V5A 1S6, Canada (e-mail: srm9@sfu.ca, mmishna@sfu.ca)
MARNI MISHNA
Affiliation:
Department of Mathematics, Simon Fraser University, Burnaby, BC V5A 1S6, Canada (e-mail: srm9@sfu.ca, mmishna@sfu.ca)
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Abstract

In the quarter plane, five lattice path models with unit steps have resisted the otherwise general approach featured in recent works by Fayolle, Kurkova and Raschel. Here we consider these five models, called the singular models, and prove that the univariate generating functions marking the number of walks of a given length are not D-finite. Furthermore, we provide exact and asymptotic enumerative formulas for the number of such walks, and describe an efficient algorithm for exact enumeration.

Keywords

Primary 05A15Secondary 05A16

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Type
Paper
Information
Combinatorics, Probability and Computing , Volume 23 , Issue 5: Honouring the Memory of Philippe Flajolet - Part 1 , September 2014 , pp. 861 - 888
Copyright
Copyright © Cambridge University Press 2014 

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References

[1]Banderier, C. and Flajolet, P. (2002) Basic analytic combinatorics of directed lattice paths. Theoret. Comput. Sci. 281 37–80.CrossRefGoogle Scholar
[2]Beraha, S., Kahane, J. and Weiss, N. (1978) Limits of zeros of recursively defined families of polynomials. Advances in Math., Supplementary Studies, Studies in Foundations and Combinatorics 1 213–232.Google Scholar
[3]Bostan, A. and Kauers, M. (2009) Automatic classification of restricted lattice walks. In 21st International Conference on Formal Power Series and Algebraic Combinatorics: FPSAC 2009, pp. 201–215.CrossRefGoogle Scholar
[4]Bostan, A., Raschel, K. and Salvy, B. (2014) Non-D-finite excursions in the quarter plane. (English summary) J. Combin. Theory Ser. A 121 45–63.CrossRefGoogle Scholar
[5]Bousquet-Mélou, M. (2005) Walks in the quarter plane: Kreweras' algebraic model. Ann. Appl. Probab. 15 1047–1589.CrossRefGoogle Scholar
[6]Bousquet-Mélou, M. and Mishna, M. (2010) Walks with small steps in the quarter plane. In Algorithmic Probability and Combinatorics, Vol. 520 of Contemporary Mathematics, AMS, pp. 1–39.Google Scholar
[7]Bousquet-Mélou, M. and Petkovšek, M. (2003) Walks confined in a quadrant are not always D-finite. Theoret. Comput. Sci. 307 257–276.CrossRefGoogle Scholar
[8]Christol, G. (1990) Globally bounded solutions of differential equations. In Analytic Number Theory: Tokyo, 1988, Vol. 1434 of Lecture Notes in Mathematics, Springer, pp. 45–64.Google Scholar
[9]Fayolle, G. and Raschel, K. (2010) On the holonomy or algebraicity of generating functions counting lattice walks in the quarter-plane. Markov Process. Rel. Fields 16 485–496.Google Scholar
[10]Fayolle, G. and Raschel, K. (2011) Random walks in the quarter-plane with zero drift: An explicit criterion for the finiteness of the associated group. Markov Process. Rel. Fields 17 619–636.Google Scholar
[11]Fayolle, G. and Raschel, K. (2012) Some exact asymptotics in the counting of walks in the quarter plane. In AofA, Discrete Mathematics and Theoretical Computer Science, pp. 109–124.CrossRefGoogle Scholar
[12]Fayolle, G., Iasnogorodski, R. and Malyshev, V. A. (1999) Random Walks in the Quarter-Plane: Algebraic Methods, Boundary Value Problems and Applications, Vol. 40 of Applications of Mathematics, Springer.CrossRefGoogle Scholar
[13]Flajolet, P. and Sedgewick, R. (2009) Analytic Combinatorics, Cambridge University Press.CrossRefGoogle Scholar
[14]Flajolet, P., Gerhold, S. and Salvy, B. (2004/06) On the non-holonomic character of logarithms, powers, and the nth prime function. Electron. J. Combin. 11 2.CrossRefGoogle Scholar
[15]Janse van Rensburg, E. J., Prellberg, T. and Rechnitzer, A. (2008) Partially directed paths in a wedge. J. Combin. Theory Ser. A 115 623–650.CrossRefGoogle Scholar
[16]Kauers, M., Koutschan, C. and Zeilberger, D. (2009) Proof of Ira Gessel's lattice path conjecture. Proc. Natl. Acad. Sci. USA 106 11502–11505.CrossRefGoogle Scholar
[17]Konvalina, J. and Matache, V. (2004) Palindrome-polynomials with roots on the unit circle. CR Math. Acad. Sci. Soc. R. Can. 26 39–44.Google Scholar
[18]Kurkova, I. and Raschel, K. (2011) Explicit expression for the generating function counting Gessel's walks. Adv. Appl. Math. 47 414–433.CrossRefGoogle Scholar
[19]Mishna, M. and Rechnitzer, A. (2009) Two non-holonomic lattice walks in the quarter plane. Theoret. Comput. Sci. 410 3616–3630.CrossRefGoogle Scholar
[20]OEIS Foundation Inc. (2011) The On-Line Encyclopedia of Integer Sequences. http://oeis.orgGoogle Scholar