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Pell Constant

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The Pell constant is defined by the infinite product expression

P=1-product_(k=0)^(infty)(1-1/(2^(2k+1)))
(1)
=1-(1/2;1/4)_infty
(2)
=0.58057755820489...
(3)

(OEIS A141848), where (a;q)_infty is a q-Pochhammer symbol.

The Pell constant is a transcendental number. This follows from a theorem on modular forms by Nesterenko (1996), as reported by Finch (2015, p. 17).

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References

Finch, S. R. Mathematical Constants. Cambridge, England: Cambridge University Press, p. 119, 2003.Finch, S. R. "Errata and Addenda to Mathematical Constants." Mar. 6, 2015. https://oeis.org/A106533/a106533.pdf.Nesterenko, Yu. V. "Modular Functions and Transcendence Questions." Sb. Math. 187, 1319-1348, 1996. https://doi.org/10.1070/SM1996v187n09ABEH000158.Sloane, N. J. A. Sequence A141848 in "The On-Line Encyclopedia of Integer Sequences."Stevenhagen, P. "The Number of Real Quadratic Fields Having Units of Negative Norm." Experiment. Math. 2, 121-136, 1993.

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Pell Constant

Cite this as:

Weisstein, Eric W. "Pell Constant." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PellConstant.html

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