-analogue triangular numbers

and distance geometry

Author:
Kenneth B. Stolarsky

Journal:
Proc. Amer. Math. Soc. **125** (1997), 35-39

MSC (1991):
Primary 05A19, 05A30, 51K05; Secondary 11B65

DOI:
https://doi.org/10.1090/S0002-9939-97-03823-9

MathSciNet review:
1377009

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Abstract | References | Similar Articles | Additional Information

Abstract: The so-called ``-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 -analogues of the triangular numbers that, by contrast, arises in the context of distance geometry.

**[A]**George E. Andrews,*𝑞-series: their development and application in analysis, number theory, combinatorics, physics, and computer algebra*, CBMS Regional Conference Series in Mathematics, vol. 66, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1986. MR**858826****[B]**Leonard M. Blumenthal,*Theory and applications of distance geometry*, Second edition, Chelsea Publishing Co., New York, 1970. MR**0268781****[C-M-Y]**Joan Cleary, Sidney A. Morris, and David Yost,*Numerical geometry—numbers for shapes*, Amer. Math. Monthly**93**(1986), no. 4, 260–275. MR**835294**, https://doi.org/10.2307/2323675**[C-F-G]**Hallard T. Croft, Kenneth J. Falconer, and Richard K. Guy,*Unsolved problems in geometry*, Problem Books in Mathematics, Springer-Verlag, New York, 1991. Unsolved Problems in Intuitive Mathematics, II. MR**1107516****[G]**O. Gross,*The rendezvous value of metric space*, Advances in game theory, Princeton Univ. Press, Princeton, N.J., 1964, pp. 49–53. MR**0162643****[S]**Kenneth B. Stolarsky,*Sums of distances between points on a sphere*, Proc. Amer. Math. Soc.**35**(1972), 547–549. MR**0303418**, https://doi.org/10.1090/S0002-9939-1972-0303418-3

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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:
https://doi.org/10.1090/S0002-9939-97-03823-9

Keywords:
Distance geometry,
$q$-identity,
$q$-analogue triangular numbers,
triangular numbers

Received by editor(s):
June 29, 1995

Communicated by:
Jeffry N. Kahn

Article copyright:
© Copyright 1997
American Mathematical Society