How to compute multivariate Bessel expansions

We develop a constructive method for computing explicitly multivariate Bessel expansions of the type \(\begin{aligned} \sum _{m\ge 1} \alpha _m \prod _{i=1}^k \frac{J_{\mu _i}(\zeta _m x_i)}{(\zeta _m x_i)^{\mu _i}}, \end{aligned}\) $$\begin{array}{c}\hfill \sum _{m\ge 1}{\alpha }_m\prod _{i=1}^k\frac{J_{{\mu }_i}({\zeta }_m{x}_i)}{{({\zeta }_m{x}_i)}^{{\mu }_i}},\end{array}$$ assuming that for a particular value \(\eta \) $\eta$ a closed expression for the single-variable Bessel expansion \(\begin{aligned} \sum _{m\ge 1}\alpha _m \frac{J_{\eta }(\zeta _m x)}{(\zeta _m x)^\eta } \end{aligned}\) $$\begin{array}{c}\hfill \sum _{m\ge 1}{\alpha }_m\frac{J_{\eta }({\zeta }_{m}x)}{{({\zeta }_{m}x)}^{\eta }}\end{array}$$ as a power series of \(x^{2j}\) $x^{2j}$ , \(j\in \mathbb {N}\) $j\in \mathbb{N}$ , is known. Using the method we compute in a closed form a bunch of examples of multivariate Bessel expansions.