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 for the Lie algebra $\mathfrak{gl}_3$. The symbol is expressed as the ratio of values of hypergeometric functions with $\pm 1$ substituted for all arguments. Finding a $3j$-symbol is essentially equivalent to the determination of an arbitrary Clebsch–Gordan coefficient for $\mathfrak{gl}_3$. The coefficients are important in the quark theory of quantum mechanics.