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Andrews-Gordon Identity


The Andrews-Gordon identity (Andrews 1974) is the analytic counterpart of Gordon's combinatorial generalization of the Rogers-Ramanujan identities (Gordon 1961). It has a number of important applications in mathematical physics (Fulman 1999).

The identity states

 sum_(n_1,...,n_(k-1)>=0)(x^(N_1^2+...+N_(k-1)^2+N_i+...+N_(k-1)))/((x)_(n_1)...(x)_(n_(k-1)))=product_(r=1; r!=0,+/-i (mod 2k+1))1/(1-x^r),

where 1<=i<=k, k>=2, x is complex with |x|<1, and N_j=n_j+...+n_(k-1) (Andrews 1974; Andrews 1984, p. 111; Fulman 1999).

There are also a more general combinatorial theorems which include the Andrews-Gordon identity, Andrews's analytic generalization of the Göllnitz-Gordon identities, Gordon's partition theorem, and Schur's partition theorem as special cases. However, the statements of these theorems are quite complicated.


See also

Göllnitz-Gordon Identities, Gordon's Partition Theorem, Rogers-Ramanujan Identities, Schur's Partition Theorem

Portions of this entry contributed by Andrew Sills

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References

Andrews, G. E. "A Generalization of the Classical Partition Theorems." Trans. Amer. Math. Soc. 145, 205-221, 1969.Andrews, G. E. On the General Rogers-Ramanujan Theorem. Providence, RI: Amer. Math. Soc., 1974.Andrews, G. E. Encyclopedia of Mathematics and Its Applications, Vol. 2: The Theory of Partitions. Cambridge, England: Cambridge University Press, 1984.Fulman, J. "The Rogers-Ramanujan Identities, The Finite General Linear Groups, and the Hall-Littlewood Polynomials." Proc. Amer. Math. Soc. 128, 17-25, 1999.Gordon, B. "A Combinatorial Generalization of the Rogers-Ramanujan Identities." Amer. J. Math. 83, 393-399, 1961.Mc Laughlin, J.; Sills, A. V.; and Zimmer, P. "Dynamic Survey DS15: Rogers-Ramanujan-Slater Type Identities." Electronic J. Combinatorics, DS15, 1-59, May 31, 2008. http://www.combinatorics.org/Surveys/ds15.pdf.

Referenced on Wolfram|Alpha

Andrews-Gordon Identity

Cite this as:

Sills, Andrew and Weisstein, Eric W. "Andrews-Gordon Identity." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Andrews-GordonIdentity.html

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