Formulas for Calculating the $ 3j $-Symbols of the Representations of the Lie Algebra $ {\mathfrak{gl}}_{3} $ for the Gelfand–Tsetlin Bases

We give a simple explicit formula for an arbitrary $ 3j -symbol the Lie algebra {\mathfrak{gl}}_{3} . The symbol is expressed as ratio of values hypergeometric functions with \pm 1 substituted all arguments. Finding essentially equivalent to determination Clebsch–Gordan coefficient coefficients are important in quark theory quantum mechanics.