How to compute multivariate Bessel expansions

Abstract

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