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Asymptotics of bivariate analytic functions with algebraic singularities. (English) Zbl 1369.05012
Summary: In this paper, we use the multivariate analytic techniques of R. Pemantle and M. C. Wilson [Analytic combinatorics in several variables. Cambridge Studies in Advanced Mathematics 140. Cambridge: Cambridge University Press (2013; Zbl 1297.05004)] to derive asymptotic formulae for the coefficients of a broad class of multivariate generating functions with algebraic singularities. Then, we apply these results to a generating function encoding information about the stationary distributions of a graph coloring algorithm studied by S. Butler et al. [Adv. Appl. Math. 69, 46–64 (2015; Zbl 1327.05306)]. Historically, P. Flajolet and A. M. Odlyzko [SIAM J. Discrete Math. 3, No. 2, 216–240 (1990; Zbl 0712.05004)] analyzed the coefficients of a class of univariate generating functions with algebraic singularities. These results have been extended to classes of multivariate generating functions by Z. Gao and L. B. Richmond [J. Comput. Appl. Math. 41, No. 1–2, 177–186 (1992; Zbl 0755.05004)] and H. K. Hwang [Ann. Appl. Probab. 6, No. 1, 297–319 (1996; Zbl 0863.60013)], in both cases by immediately reducing the multivariate case to the univariate case. Pemantle and Wilson [loc. cit.] outlined new multivariate analytic techniques and used them to analyze the coefficients of rational generating functions. These multivariate techniques are used here to analyze functions with algebraic singularities.

05A15 Exact enumeration problems, generating functions
FGb; RAGlib
Full Text: DOI arXiv
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