A Kronecker-type identity and the representations of a number as a sum of three squares. (English) Zbl 1428.11073
Summary: By considering a limiting case of a Kronecker-type identity, we obtain an identity found by both
G. E. Andrews [Trans. Am. Math. Soc. 293, 113–134 (1986; Zbl 0593.10018)] and
R. E. Crandall [Exp. Math. 8, No. 4, 367–379 (1999; Zbl 0949.11062)]. We then use the Andrews-Crandall identity to give a new proof of a formula of Gauss for the representations of a number as a sum of three squares. From the Kronecker-type identity, we also deduce Gauss’ theorem that every positive integer is representable as a sum of three triangular numbers.
MSC:
| 11E25 | Sums of squares and representations by other particular quadratic forms |
| 11B65 | Binomial coefficients; factorials; \(q\)-identities |
| 05A15 | Exact enumeration problems, generating functions |
Online Encyclopedia of Integer Sequences:
Sums of three squares: numbers of the form x^2 + y^2 + z^2.Number of ways of writing n as an unordered sum of at most 3 nonzero triangular numbers.
Theta series of simple cubic lattice; also number of ways of writing a nonnegative integer n as a sum of 3 squares (zero being allowed).
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