Extended admissible functions and Gaussian limiting distributions

Drmota, Michael; Gittenberger, Bernhard; Klausner, Thomas

doi:10.1090/S0025-5718-05-01744-8

Extended admissible functions and Gaussian limiting distributions
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by Michael Drmota, Bernhard Gittenberger and Thomas Klausner;

Math. Comp. 74 (2005), 1953-1966

DOI: https://doi.org/10.1090/S0025-5718-05-01744-8

Published electronically: March 14, 2005

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Abstract:

We consider an extension of Hayman’s notion of admissibility to bivariate generating functions $f(z,u)$ that have the property that the coefficients $a_{nk}$ satisfy a central limit theorem. It is shown that these admissible functions have certain closure properties. Thus, there is a large class of functions for which it is possible to check this kind of admissibility automatically. This is realized with help of a MAPLE program that is also presented. We apply this concept to various combinatorial examples.

References

Edward A. Bender and L. Bruce Richmond, Central and local limit theorems applied to asymptotic enumeration. II. Multivariate generating functions, J. Combin. Theory Ser. A 34 (1983), no. 3, 255–265. MR 700034, DOI 10.1016/0097-3165(83)90062-6
Edward A. Bender and L. Bruce Richmond, A generalisation of Canfield’s formula, J. Combin. Theory Ser. A 41 (1986), no. 1, 50–60. MR 826937, DOI 10.1016/0097-3165(86)90114-7
Edward A. Bender and L. Bruce Richmond, Admissible functions and asymptotics for labelled structures by number of components, Electron. J. Combin. 3 (1996), no. 1, Research Paper 34, approx. 23. MR 1418482, DOI 10.37236/1258
Michael Drmota, A bivariate asymptotic expansion of coefficients of powers of generating functions, European J. Combin. 15 (1994), no. 2, 139–152. MR 1261060, DOI 10.1006/eujc.1994.1016
Philippe Flajolet and Michèle Soria, Gaussian limiting distributions for the number of components in combinatorial structures, J. Combin. Theory Ser. A 53 (1990), no. 2, 165–182. MR 1041444, DOI 10.1016/0097-3165(90)90056-3
Philippe Flajolet and Michèle Soria, General combinatorial schemas: Gaussian limit distributions and exponential tails, Discrete Math. 114 (1993), no. 1-3, 159–180. Combinatorics and algorithms (Jerusalem, 1988). MR 1217750, DOI 10.1016/0012-365X(93)90364-Y
Danièle Gardy, Some results on the asymptotic behaviour of coefficients of large powers of functions, Discrete Math. 139 (1995), no. 1-3, 189–217. Formal power series and algebraic combinatorics (Montreal, PQ, 1992). MR 1336840, DOI 10.1016/0012-365X(94)00133-4
I. P. Goulden and D. M. Jackson, Combinatorial enumeration, Wiley-Interscience Series in Discrete Mathematics, John Wiley & Sons, Inc., New York, 1983. With a foreword by Gian-Carlo Rota; A Wiley-Interscience Publication. MR 702512
Zhicheng Gao and L. Bruce Richmond, Central and local limit theorems applied to asymptotic enumeration. IV. Multivariate generating functions, J. Comput. Appl. Math. 41 (1992), no. 1-2, 177–186. Asymptotic methods in analysis and combinatorics. MR 1181718, DOI 10.1016/0377-0427(92)90247-U
L. H. Harper, Stirling behavior is asymptotically normal, Ann. Math. Statist. 38 (1967), 410–414. MR 211432, DOI 10.1214/aoms/1177698956
W. K. Hayman, A generalisation of Stirling’s formula, J. Reine Angew. Math. 196 (1956), 67–95. MR 80749, DOI 10.1515/crll.1956.196.67
Bernard Harris and Lowell Schoenfeld, Asymptotic expansions for the coefficients of analytic functions, Illinois J. Math. 12 (1968), 264–277. MR 224801
Hsien-Kuei Hwang, Large deviations for combinatorial distributions. I. Central limit theorems, Ann. Appl. Probab. 6 (1996), no. 1, 297–319. MR 1389841, DOI 10.1214/aoap/1034968075
Hsien-Kuei Hwang, On convergence rates in the central limit theorems for combinatorial structures, European J. Combin. 19 (1998), no. 3, 329–343. MR 1621021, DOI 10.1006/eujc.1997.0179
Dan Richardson, Bruno Salvy, John Shackell, and Joris Van der Hoeven. Asymptotic expansions of exp-log functions. In Y. N. Lakshman, editor, ISSAC’96, pages 309–313. ACM Press, 1996. Proceedings of the 1996 International Symposium on Symbolic and Algebraic Computation. July 24–26, 1996. Zürich, Switzerland.
B. Salvy. Examples of automatic asymptotic expansions. SIGSAM Bulletin, 25(2):4–17, April 1991.
John Shackell, Inverses of Hardy $L$-functions, Bull. London Math. Soc. 25 (1993), no. 2, 150–156. MR 1204067, DOI 10.1112/blms/25.2.150
Bruno Salvy and John Shackell, Symbolic asymptotics: multiseries of inverse functions, J. Symbolic Comput. 27 (1999), no. 6, 543–563. MR 1701094, DOI 10.1006/jsco.1999.0281