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Date: 7-7-2021
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Let be a real vector space (e.g., the real continuous functions on a closed interval , two-dimensional Euclidean space , the twice differentiable real functions on , etc.). Then is a real subspace of if is a subset of and, for every , and (the reals), and . Let be a homogeneous system of linear equations in , ..., . Then the subset of which consists of all solutions of the system is a subspace of .
More generally, let be a field with , where is prime, and let denote the -dimensional vector space over . The number of -D linear subspaces of is
(1) |
where this is the q-binomial coefficient (Aigner 1979, Exton 1983). The asymptotic limit is
(2) |
where
(3) |
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(4) |
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(5) |
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(6) |
(Finch 2003), where is a Jacobi theta function and is a q-Pochhammer symbol. The case gives the q-analog of the Wallis formula.
REFERENCES:
Aigner, M. Combinatorial Theory. New York: Springer-Verlag, 1979.
Exton, H. q-Hypergeometric Functions and Applications. New York: Halstead Press, 1983.
Finch, S. R. "Lengyel's Constant." Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 316-321, 2003.
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