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Logarithmic and complex constant term identities. (English) Zbl 1290.05152
Bailey, David H. (ed.) et al., Computational and analytical mathematics. In Honor of Jonathan Borwein’s 60th birthday. Selected papers based on the presentations at the workshop, also known as JonFest, Simon Fraser University, BC, Canada, May 16–20, 2011. New York, NY: Springer (ISBN 978-1-4614-7620-7/hbk; 978-1-4614-7621-4/ebook). Springer Proceedings in Mathematics & Statistics 50, 219-250 (2013).
The authors explore generalizations of the Macdonald-Morris constant term identity \[ CT\left[\prod_{i=1}^{n} (1-x_{i})^{a}(1-x_{i}^{-1})^{b} \prod_{1 \leq i \neq j \leq n} \left(1 - \frac{x_{i}}{x_{j}} \right)^{k}\right] = \prod_{i=0}^{n-1} \frac{(a+b+ik)!((i+1)k!)}{(a+ik)!(b+ik)!k!}, \] related to results (some still conjectural) of D. Adamović and A. Milas [J. Algebra 344, No. 1, 313–332 (2011; Zbl 1244.17015); Int. Math. Res. Not. 2010, No. 20, 3896–3934 (2010; Zbl 1210.17031)].
The authors give a conjectured complex version of the Macdonald-Morris identity, where the parameter \(k\) is replaced by a complex parameter \(u\), and prove the conjecture in the case \(n=3\). They also show that this conjecture implies a “logarithmic” version of this identity.
The paper also includes a discussion of related identities in Type \(G_{2}\). (The Macdonald-Morris identity is a generalization of an identity originally conjectured by Macdonald, which can be phrased in terms of root systems. The results for \(G_{2}\) here are a generalization of that identity, while the Macdonald-Morris identity is a stronger generalization specific to Type A.) The authors also include a discussion of why such elegant identities do not seem to occur in the other classical types.
For the entire collection see [Zbl 1276.00018].

05E05 Symmetric functions and generalizations
05A05 Permutations, words, matrices
05A10 Factorials, binomial coefficients, combinatorial functions
05A19 Combinatorial identities, bijective combinatorics
33C20 Generalized hypergeometric series, \({}_pF_q\)
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