$3j$-symbols for representation of the Lie algebra ${\mathfrak{g}\mathfrak{l}}_3$ in the Gelfand-Tselin base

In the paper a simple explicit formula for an arbitrary $3j$-symbol for the Lie algebra ${\mathfrak{g}\mathfrak{l}}_3$ is given. More precise necessary conditions for non-vanishing of a $3j$-symbol are given, in the case when these conditions hold we give an explicit expression for a $3j$-symbol. It is expressed through a fraction of values of $A$-hypergeometric function when one substitutes $\pm 1$insteadofalli{t}^{\mathrm{\prime }}sarguments\mathrm{.}Theproblemofcalculationofanarbitrary$3j$-symbol is equivalent to the problem of calculation of an arbitrary Clebsh-Gordan coefficient for the algebra ${\mathfrak{g}\mathfrak{l}}_3$. These coefficients play an important role in quantum mechanics in the theory of quarks.