×

\(q\)-series identities and values of certain \(L\)-functions. (English) Zbl 1005.11048

This paper establishes two general theorems that yield infinitely many \(q\)-series identities obtained by summing the iterated differences between an infinite product and its truncated products. It also shows how such identities are useful in determining the values of certain \(L\)-functions at the negative integers.

MSC:

11P82 Analytic theory of partitions
11M06 \(\zeta (s)\) and \(L(s, \chi)\)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] G. E. Andrews, Ramanujan’s “lost” notebook, V: Euler’s partition identity , Adv. Math. 61 (1986), 156–164. MR 87i:11137 · Zbl 0601.10007
[2] –. –. –. –., “Mock theta functions” in Theta Functions (Brunswick, Maine, 1987), Part 2 , Proc. Sympos. Pure Math. 49 , Amer. Math. Soc., Providence, 1989, 283–298. MR 90h:33005
[3] ——–, The Theory of Partitions , Cambridge Math. Lib., Cambridge Univ. Press, Cambridge, 1998. MR 99c:11126 · Zbl 0996.11002
[4] G. E. Andrews, F. Dyson, and D. Hickerson, Partitions and indefinite quadratic forms , Invent. Math. 91 (1988), 391–407. MR 89f:11071 · Zbl 0642.10012
[5] H. Cohen, \(q\)-identities for Maass waveforms , Invent. Math. 91 (1988), 409–422. MR 89f:11072 · Zbl 0642.10013
[6] P. Erdös, On an elementary proof of some asymptotic formulas in the theory of partitions , Ann. of Math. (2) 43 (1942), 437–450. MR 4:36a JSTOR: · Zbl 0061.07905
[7] N. J. Fine, Basic Hypergeometric Series and Applications , Math. Surveys Monogr. 27 , Amer. Math. Soc., Providence, 1988. MR 91j:33011 · Zbl 0647.05004
[8] G. Gasper and M. Rahman, Basic Hypergeometric Series , Encyclopedia Math. Appl. 35 , Cambridge Univ. Press, Cambridge, 1990. MR 91d:33034 · Zbl 0695.33001
[9] W. J. Leveque, Topics in Number Theory, Vol. 1 , Addison-Wesley, Reading, Mass., 1956. MR 18:283d · Zbl 0070.03804
[10] P. A. MacMahon, Combinatory Analysis , Chelsea, New York, 1960. MR 25:5003 · Zbl 0101.25102
[11] D. Zagier, Vassiliev invariants and a strange identity related to the Dedekind eta-function , to appear in Topology. · Zbl 0989.57009
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.