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Fine's Equation

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Fine's equation is the following q-series identity for |q|<1

product_(n=1)^(infty)((1-q^(2n))(1-q^(3n))(1-q^(8n))(1-q^(12n)))/((1-q^n)(1-q^(24n)))=((q^2)_infty(q^3)_infty(q^8)_infty(q^(12))_infty)/((q)_infty(q^(24))_infty)
(1)
=1+sum_(k=1)^(infty)(q^k(q^(4k)+1)(q^(6k)+1))/(q^(12k)+1)
(2)
=1+sum_(n=1)^(infty)E_(1,5,7,11)(n;24)q^n
(3)
=1+sum_(n=1)^(infty)(sum_(d|n)(-6/d))q^n,
(4)

where (q)_infty is a q-Pochhammer symbol, E_(1,5,7,11)(n;24) is the number of divisors of n that are congruent to 1, 5, 7, and 11 (mod 24) minus the number of divisors of n congruent to -1, -5, -7, and -11 (mod 24), and (-6/d) is a Kronecker symbol.

The coefficients of the first few powers of q starting with n=0, 1, ... are 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 2, 1, ... (OEIS A000377).

See also

q-Series

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References

Andrews, G. E. (Ed.). Percy Alexander MacMahon: Collected Papers, Vol. 2. Cambridge, MA: MIT Press, p. 260, 1986.Andrews, G. E. "Nathan Fine, 1916-1994." Not. Amer. Math. Soc. 42, 678-679, 1995.Sloane, N. J. A. Sequence A000377 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Fine's Equation

Cite this as:

Weisstein, Eric W. "Fine's Equation." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/FinesEquation.html

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