×

Three-term relations for Hardy sums. (English) Zbl 0604.10003

The authors prove a general summation formula by extending to three dimensions the well-known idea of Eisenstein for counting the lattice points in a rectangle. From this formula they deduce a three-term relation for polynomials. Then, using this relation, they give elementary proofs of all the three-term and mixed three-term relations for Hardy sums given by L. A. Goldberg in his 1981 thesis (University of Illinois, Urbana).
Reviewer: K.S.Williams

MSC:

11B39 Fibonacci and Lucas numbers and polynomials and generalizations
11L99 Exponential sums and character sums
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Apostol, T. M.; Vu, T. H., Elementary proofs of Berndt’s reciprocity laws, Pacific J. Math., 98, 17-23 (1982) · Zbl 0479.10010
[2] Berndt, B. C., A new proof of the reciprocity theorem for Dedekind sums, Elem. Math., 29, 93-94 (1974) · Zbl 0283.10010
[3] Berndt, B. C., Dedekind sums and a paper of Hardy, J. London Math. Soc., 13, 2, 129-136 (1976) · Zbl 0319.10006
[4] Berndt, B. C., Reciprocity theorems for Dedekind sums and generalizations, Adv. in Math., 23, 285-316 (1977) · Zbl 0342.10014
[5] Berndt, B. C., Analytic Eisenstein series, theta-functions, and series relations in the spirit of Ramanujan, J. Reine Angew. Math., 303/304, 332-365 (1978) · Zbl 0384.10011
[6] Berndt, B. C.; Dieter, U., Sums involving the greatest integer function and Riemann-Stieltjes integration, J. Reine Angew. Math., 337, 208-220 (1982) · Zbl 0487.10002
[7] Berndt, B. C.; Goldberg, L. A., Analytic properties of arithmetic sums arising in the theory of the classical theta-functions, SIAM J. Math. Anal., 15, 143-150 (1984) · Zbl 0537.10006
[8] Carlitz, L., Some sums containing the greatest integer function, Rev. Roumaine Math. Pures Appl., 20, 521-530 (1975) · Zbl 0307.10003
[9] Carlitz, L., Some polynomials associated with Dedekind sums, Acta Math. Sci. Hungar., 26, 311-319 (1975) · Zbl 0317.10017
[10] B. Davis and R. Sitaramachandrarao; B. Davis and R. Sitaramachandrarao
[11] Dedekind, R., Erläuterungen zu der Riemannschen Fragmenten über die Grenzfalle der Elliptischen Funktionen, Gesammalte Math. Werke 1, 159-173 (1930), Braunschweig
[12] Dieter, U., Beziehungen zwischen Dedekindschen Summen, Abh. Math. Sem. Univ. Hamburg, 21, 109-125 (1957) · Zbl 0078.07002
[13] Dieter, U., Cotangent sums, a further generalization of Dedekind sums, J. Number Theory, 18, 289-305 (1984) · Zbl 0537.10005
[14] Goldberg, L. A., Transformations of Theta-Functions and Analogues of Dedekind Sums, (thesis (1981), Univ. of Illinois: Univ. of Illinois Urbana)
[15] Hardy, G. H., (Collected Papers, Vol. IV (1969), Oxford Univ. Press (Clarendon): Oxford Univ. Press (Clarendon) London/New York), 362-392
[16] Kuipers, L., Aufgabe 866, Elem. Math., 36, 143 (1981)
[17] Rademacher, H., Generalization of the reciprocity formula for Dedekind sums, Duke Math. J., 21, 391-397 (1954) · Zbl 0057.03801
[18] Rademacher, H.; Grosswald, E., (Dedekind Sums (1972), Math. Assoc. Amer: Math. Assoc. Amer Washington, D.C), Carus Mathematical Monograph No. 16 · Zbl 0251.10020
[19] R. SitaramachandraraoActa Arith.; R. SitaramachandraraoActa Arith. · Zbl 0635.10002
[20] Udrescu, V. St, A remark on a result of L. Carlitz, Rev. Roumaine Math. Pures Appl., 20, 605-608 (1975) · Zbl 0307.10004
[21] Whittaker, E. T.; Watson, G. N., (A Course of Modern Analysis (1962), Cambridge Univ. Press: Cambridge Univ. Press Cambridge) · Zbl 0105.26901
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.