ISSN:
1572-9036
Keywords:
10 B05
;
20C35
;
33A30
;
81-47
;
81G30
;
Racah coefficients
;
Clebsch-Gordan coefficients
;
weight 2 zeros
;
Pell equations
;
Diophantine equations
;
orbit classification of zeros
;
symbolic formula manipulation
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract The interface between Racah coefficients and mathematics is reviewed and several unsolved problems pointed out. The specific goal of this investigation is to determine zeros of these coefficients. The general polynomial is given whose set of zeros contains all nontrivial zeros of Racah (6j) coefficients [this polynomial is also given for the Wigner-Clebsch-Gordan (3j) coefficients]. Zeros of weight 1 3j- and 6j-coefficients are known to be related to the solutions of classic Diophantine equations. Here it is shown how solutions of the quadratic Diophantine equation known as Pell's equation are related to weight 2 zeros of 3j- and 6j-coefficients. This relation involves transformations of quadratic forms over the integers, the orbit classification of zeros of Pell's equation, and an algorithm for determining numerically the fundamental solutions of Pell's equation. The symbol manipulation program MACSYMA was used extensively to effect various factorings and transformations and to give a proof.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF00047095
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