scholarly journals Families of orthogonal Laurent polynomials, hyperelliptic lie algebras and elliptic integrals

2016 ◽  
Vol 27 (11) ◽  
pp. 899-919
Author(s):  
Ben Cox ◽  
Mee Seong Im
Keyword(s):  
Lie Algebras ◽  
Elliptic Integrals ◽  
Laurent Polynomials ◽  
Orthogonal Laurent Polynomials

scholarly journals Combinatorial Interpretations for Rank-Two Cluster Algebras of Affine Type

10.37236/933 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Gregg Musiker ◽  
James Propp
Keyword(s):  
Lie Algebras ◽  
Cluster Algebras ◽  
Laurent Polynomials ◽  
Finite Dimensional ◽  
Combinatorial Interpretations ◽  
Rank 2 ◽  
The Individual ◽  
Affine Type ◽  
Positive Integers ◽  
Two Parameter

Fomin and Zelevinsky show that a certain two-parameter family of rational recurrence relations, here called the $(b,c)$ family, possesses the Laurentness property: for all $b,c$, each term of the $(b,c)$ sequence can be expressed as a Laurent polynomial in the two initial terms. In the case where the positive integers $b,c$ satisfy $bc < 4$, the recurrence is related to the root systems of finite-dimensional rank $2$ Lie algebras; when $bc>4$, the recurrence is related to Kac-Moody rank $2$ Lie algebras of general type. Here we investigate the borderline cases $bc=4$, corresponding to Kac-Moody Lie algebras of affine type. In these cases, we show that the Laurent polynomials arising from the recurence can be viewed as generating functions that enumerate the perfect matchings of certain graphs. By providing combinatorial interpretations of the individual coefficients of these Laurent polynomials, we establish their positivity.

scholarly journals Orthogonal Laurent polynomials

1986 ◽  
Vol 89 (1) ◽  
pp. 17-36 ◽  
Author(s):  
E. Hendriksen ◽  
H. van Rossum
Keyword(s):  
Laurent Polynomials ◽  
Orthogonal Laurent Polynomials

scholarly journals Sequences of orthogonal Laurent polynomials, bi-orthogonality and quadrature formulas on the unit circle

2007 ◽  
Vol 206 (2) ◽  
pp. 950-966 ◽  
Author(s):  
Ruymán Cruz-Barroso ◽  
Leyla Daruis ◽  
Pablo González-Vera ◽  
Olav NjÅstad
Keyword(s):  
Unit Circle ◽  
Quadrature Formulas ◽  
Laurent Polynomials ◽  
Orthogonal Laurent Polynomials
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