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Zagier's Identity

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sum_(n=0)^(infty)[(q)_infty-(q)_n]=g(q)+(q)_inftysum_(k=1)^(infty)(q^k)/(1-q^k)
(1)
=g(q)+(q)_inftyL(q)
(2)
=g(q)+(q)_infty(psi_q(1)+ln(1-q))/(lnq)
(3)
=-q-2q^2-q^3-q^4+2q^5+4q^7+q^8+...
(4)

(OEIS A117586; Andrews 1972, 1998; Knuth and Paterson 1978; Chapman 2000; Zagier 2001), where

g(q)=sum_(n=1)^infty(-1)^n[(3n-1)q^(n(3n-1)/2)+3nq^(n(3n+1)/2)],
(5)

L(q) is a Lambert series, and psi_q(z) is a q-polygamma function.

If

f(x,q)=1+sum_(n=1)^infty(-1)^n[x^(3n-1)q^(n(3n-1)/2)+x^(3n)q^(n(3n+1)/2)],
(6)

then a related identity is given by

f(x,q)=sum_(n=0)^(infty)(x;q)_(n+1)x^n
(7)
=1-x^2q-x^3q^2+x^5q^5+x^6q^7-x^8q^(12)-x^9q^(15)+...
(8)

(Subbarao 1971, Andrews 1972, Andrews 1983, Knuth and Paterson 1978, Chapman 2000, Zagier 2001).

See also

Pentagonal Number Theorem

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References

Andrews, G. E. "Two Theorems of Gauss and Allied Identities Proved Arithmetically." Pacific J. Math. 41, 563-578, 1972.Andrews, G. E. "Euler's Pentagonal Number Theorem." Math. Mag. 56, 279-284, 1983.Andrews, G. E. The Theory of Partitions. Cambridge, England: Cambridge University Press, 1998.Chapman, R. "Franklin's Argument Proves an Identity of Zagier." Electronic J. Combinatorics 7, No. 1, R54, 1-5, 2000. http://www.combinatorics.org/Volume_7/Abstracts/v7i1r54.html.Knuth, D. E. and Paterson, M. S. "Identities from Partition Involutions." Fib. Quart. 16, 198-212, 1978.Sloane, N. J. A. Sequence A117586 in "The On-Line Encyclopedia of Integer Sequences."Subbarao, M. V. "Combinatorial Proofs of Some Identities." Proc. Washington State University Conference on Number Theory. Washington State University, pp. 80-91, 1971.Zagier, D. "Vassiliev Invariants and a Strange Identity Related to the Dedekind Eta-Function." Topology 40, 945-960, 2001.

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Zagier's Identity

Cite this as:

Weisstein, Eric W. "Zagier's Identity." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ZagiersIdentity.html

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