Calculation of $6j$-symbols for the Lie algebra $\mathfrak{gl}_n$

An explicit description of the multiplicity space that describes occurrences of irreducible representations in a splitting of a tensor product of two irreducible representations of $\mathfrak{gl}_n$ is given. Using this description an explicit formula for an arbitrary $6j$-symbol for the algebra $\mathfrak{gl}_n$ is derived. The $6j$-symbol is expressed through a value of a generalized hypergeometric function.