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$q$-analogue triangular numbers and distance geometry

Author(s): Kenneth B. Stolarsky
Journal: Proc. Amer. Math. Soc. 125 (1997), 35-39.
MSC (1991): Primary 05A19, 05A30, 51K05; Secondary 11B65
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Abstract: The so-called ``$q$-identities'' play a major role in classical combinatorics. Most of them can be viewed as arising somehow in the context of hypergeometric series. Here we present a ``sum of squares'' identity involving $q$-analogues of the triangular numbers that, by contrast, arises in the context of distance geometry.

References:

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Andrews, G. E., q-Series: their development and application in analysis, number theory, combinatorics, physics, and computer algebra, Regional Conference Series in Math. 66, Amer. Math. Soc. (1986). MR 88b:11063

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Blumenthal, L. M., Theory and Applications of Distance Geometry, Chelsea, New York, 1970. MR 42:3678

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Cleary, J., Morris, S. A., and Yost, D., Numerical geometry-numbers for shapes, Amer. Math. Monthly 93 (1986), 260-275. MR 87h:51043

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Croft, H. T., Falconer, K. J., and Guy, Richard K., Unsolved Problems in Geometry, Springer-Verlag, New York, 1991. MR 92c:52001

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Gross, O., The rendezvous value of a metric space, Advances in Game Theory, Ann. Math. Studies, vol. 52, Princeton University, Princeton, 1964, pp. 49-53. MR 28:5841

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Stolarsky, K. B., Sums of distances between points on a sphere, Proc. Amer. Math. Soc. 35 (1972), 547-549. MR 46:2555

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Additional Information:

Kenneth B. Stolarsky
Affiliation: Department of Mathematics, University of Illinois, 1409 W. Green St., Urbana, Illinois 61801
Email: stolarsk@math.uiuc.edu

DOI: 10.1090/S0002-9939-97-03823-9
PII: S 0002-9939(97)03823-9
Keywords: Distance geometry, $q$-identity, $q$-analogue triangular numbers, triangular numbers
Received by editor(s): June 29, 1995
Communicated by: Jeffry N. Kahn
Copyright of article: Copyright 1997, American Mathematical Society

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