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On asymptotics for lacunary partition functions. (English) Zbl 1540.11131

Summary: We give asymptotic analysis of power series associated with lacunary partition functions. New partition theoretic interpretations of some basic hypergeometric series are offered as examples.

MSC:

11N37 Asymptotic results on arithmetic functions
30B10 Power series (including lacunary series) in one complex variable

References:

[1] Blomer, V.—Granville, A.: Estimates for representation numbers of quadratic forms, Duke Math. J. 135(2) (2006), 261-302. · Zbl 1135.11020
[2] Gasper, G.—Rahman, M.: Basic Hypergeometric Series, Encyclopedia Math. Appl. 35, Cambridge Univ. Press, Cambridge, 1990. · Zbl 0695.33001
[3] Hardy, G. H.—Littlewood, J. E.: Tauberian theorems concerning power series and Dirichlet’s series whose coefficients are positive, Proc. London Math. Soc. 13 (1914), 174-191. · JFM 45.0389.02
[4] Hecke, E.: Über einen neuen Zusammenhang zwischen elliptischen Modulfunktionen und indefiniten quadratischen Formen, Mathematische Werke, Vandenhoeck und Ruprecht, Göttingen, 1959, pp. 418-427.
[5] Lovejoy, J.: Lacunary partition functions, Math. Res. Lett. 9 (2002), 191-198. · Zbl 1007.11060
[6] Lovejoy, J.: A Bailey lattice, Proc. Amer. Math. Soc. 132 (2004), 1507-1516. · Zbl 1050.33011
[7] Odoni, R.: Representations of algebraic integers by binary quadratic forms and norm forms from full modules of extension fields, J. Number Theory 10 (1978), 324-333 · Zbl 0411.12010
[8] Patkowski, A. E.: Partitions related to positive definite binary quadratic forms, Integers 19 (2019), Art. A25. · Zbl 1457.11140
[9] Rogers, L. J.: Second memoir on the expansion of certain infinite products, Proc. London Math. Soc. 25 (1894), 318-343.
[10] Slater, L. J.: A new proof of Rogers transformations of infinite series, Proc. London Math. Soc. (2) 53 (1951), 460-475. · Zbl 0044.06102
[11] Titchmarsh, E. C.: The Theory of the Riemann Zeta Function, 2nd edition, Oxford University Press, 1986. · Zbl 0601.10026
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