# §28.24 Expansions in Series of Cross-Products of Bessel Functions or Modified Bessel Functions

Throughout this section $\varepsilon_{0}=2$ and $\varepsilon_{s}=1$, $s=1,2,3,\dotsc$.

With ${\cal C}_{\mu}^{(j)}$, $c^{\nu}_{n}(q)$, $A_{n}^{m}(q)$, and $B_{n}^{m}(q)$ as in §28.23,

 28.24.1 $c_{2n}^{\nu}(h^{2}){\mathrm{M}^{(j)}_{\nu}}\left(z,h\right)=\sum_{\ell=-\infty% }^{\infty}(-1)^{\ell}c_{2\ell}^{\nu}(h^{2})J_{\ell-n}\left(he^{-z}\right)% \mathcal{C}_{\nu+n+\ell}^{(j)}(he^{z}),$

where $j=1,2,3,4$ and $n\in\mathbb{Z}$.

In the case when $\nu$ is an integer,

 28.24.2 $\displaystyle\varepsilon_{s}{\mathrm{Mc}^{(j)}_{2m}}\left(z,h\right)$ $\displaystyle=(-1)^{m}\sum_{\ell=0}^{\infty}(-1)^{\ell}\frac{A_{2\ell}^{2m}(h^% {2})}{A_{2s}^{2m}(h^{2})}\left(J_{\ell-s}\left(he^{-z}\right){\cal C}_{\ell+s}% ^{(j)}(he^{z})+J_{\ell+s}\left(he^{-z}\right){\cal C}_{\ell-s}^{(j)}(he^{z})% \right),$ 28.24.3 $\displaystyle{\mathrm{Mc}^{(j)}_{2m+1}}\left(z,h\right)$ $\displaystyle=(-1)^{m}\sum_{\ell=0}^{\infty}(-1)^{\ell}\frac{A_{2\ell+1}^{2m+1% }(h^{2})}{A_{2s+1}^{2m+1}(h^{2})}\left(J_{\ell-s}\left(he^{-z}\right){\cal C}_% {\ell+s+1}^{(j)}(he^{z})+J_{\ell+s+1}\left(he^{-z}\right){\cal C}_{\ell-s}^{(j% )}(he^{z})\right),$
 28.24.4 $\displaystyle{\mathrm{Ms}^{(j)}_{2m+1}}\left(z,h\right)$ $\displaystyle=(-1)^{m}\sum_{\ell=0}^{\infty}(-1)^{\ell}\frac{B_{2\ell+1}^{2m+1% }(h^{2})}{B_{2s+1}^{2m+1}(h^{2})}\left(J_{\ell-s}\left(he^{-z}\right){\cal C}_% {\ell+s+1}^{(j)}(he^{z})-J_{\ell+s+1}\left(he^{-z}\right){\cal C}_{\ell-s}^{(j% )}(he^{z})\right),$ 28.24.5 $\displaystyle{\mathrm{Ms}^{(j)}_{2m+2}}\left(z,h\right)$ $\displaystyle=(-1)^{m}\sum_{\ell=0}^{\infty}(-1)^{\ell}\frac{B_{2\ell+2}^{2m+2% }(h^{2})}{B_{2s+2}^{2m+2}(h^{2})}\left(J_{\ell-s}\left(he^{-z}\right){\cal C}_% {\ell+s+2}^{(j)}(he^{z})-J_{\ell+s+2}\left(he^{-z}\right){\cal C}_{\ell-s}^{(j% )}(he^{z})\right),$

where $j=1,2,3,4$, and $s=0,1,2,\dots$.

Also, with $I_{n}$ and $K_{n}$ denoting the modified Bessel functions (§10.25(ii)), and again with $s=0,1,2,\dots$,

 28.24.6 $\displaystyle\varepsilon_{s}\mathrm{Ie}_{2m}\left(z,h\right)$ $\displaystyle=(-1)^{s}\sum_{\ell=0}^{\infty}(-1)^{\ell}\dfrac{A_{2\ell}^{2m}(h% ^{2})}{A_{2s}^{2m}(h^{2})}\left(I_{\ell-s}\left(he^{-z}\right)I_{\ell+s}\left(% he^{z}\right)+I_{\ell+s}\left(he^{-z}\right)I_{\ell-s}\left(he^{z}\right)% \right),$ 28.24.7 $\displaystyle\mathrm{Io}_{2m+2}\left(z,h\right)$ $\displaystyle=(-1)^{s}\sum_{\ell=0}^{\infty}(-1)^{\ell}\dfrac{B_{2\ell+2}^{2m+% 2}(h^{2})}{B_{2s+2}^{2m+2}(h^{2})}\left(I_{\ell-s}\left(he^{-z}\right)I_{\ell+% s+2}\left(he^{z}\right)-I_{\ell+s+2}\left(he^{-z}\right)I_{\ell-s}\left(he^{z}% \right)\right),$ 28.24.8 $\displaystyle\mathrm{Ie}_{2m+1}\left(z,h\right)$ $\displaystyle=(-1)^{s}\sum_{\ell=0}^{\infty}(-1)^{\ell}\dfrac{B_{2\ell+1}^{2m+% 1}(h^{2})}{B_{2s+1}^{2m+1}(h^{2})}\left(I_{\ell-s}\left(he^{-z}\right)I_{\ell+% s+1}\left(he^{z}\right)+I_{\ell+s+1}\left(he^{-z}\right)I_{\ell-s}\left(he^{z}% \right)\right),$ 28.24.9 $\displaystyle\mathrm{Io}_{2m+1}\left(z,h\right)$ $\displaystyle=(-1)^{s}\sum_{\ell=0}^{\infty}(-1)^{\ell}\frac{A_{2\ell+1}^{2m+1% }(h^{2})}{A_{2s+1}^{2m+1}(h^{2})}\left(I_{\ell-s}\left(he^{-z}\right)I_{\ell+s% +1}\left(he^{z}\right)-I_{\ell+s+1}\left(he^{-z}\right)I_{\ell-s}\left(he^{z}% \right)\right),$ 28.24.10 $\displaystyle\varepsilon_{s}\mathrm{Ke}_{2m}\left(z,h\right)$ $\displaystyle=\sum_{\ell=0}^{\infty}\frac{A_{2\ell}^{2m}(h^{2})}{A_{2s}^{2m}(h% ^{2})}\left(I_{\ell-s}\left(he^{-z}\right)K_{\ell+s}\left(he^{z}\right)+I_{% \ell+s}\left(he^{-z}\right)K_{\ell-s}\left(he^{z}\right)\right),$ 28.24.11 $\displaystyle\mathrm{Ko}_{2m+2}\left(z,h\right)$ $\displaystyle=\sum_{\ell=0}^{\infty}\frac{B_{2\ell+2}^{2m+2}(h^{2})}{B_{2s+2}^% {2m+2}(h^{2})}\left(I_{\ell-s}\left(he^{-z}\right)K_{\ell+s+2}\left(he^{z}% \right)-I_{\ell+s+2}\left(he^{-z}\right)K_{\ell-s}\left(he^{z}\right)\right),$ 28.24.12 $\displaystyle\mathrm{Ke}_{2m+1}\left(z,h\right)$ $\displaystyle=\sum_{\ell=0}^{\infty}\frac{B_{2\ell+1}^{2m+1}(h^{2})}{B_{2s+1}^% {2m+1}(h^{2})}\left(I_{\ell-s}\left(he^{-z}\right)K_{\ell+s+1}\left(he^{z}% \right)-I_{\ell+s+1}\left(he^{-z}\right)K_{\ell-s}\left(he^{z}\right)\right),$ 28.24.13 $\displaystyle\mathrm{Ko}_{2m+1}\left(z,h\right)$ $\displaystyle=\sum_{\ell=0}^{\infty}\frac{A_{2\ell+1}^{2m+1}(h^{2})}{A_{2s+1}^% {2m+1}(h^{2})}\left(I_{\ell-s}\left(he^{-z}\right)K_{\ell+s+1}\left(he^{z}% \right)+I_{\ell+s+1}\left(he^{-z}\right)K_{\ell-s}\left(he^{z}\right)\right).$

The expansions (28.24.1)–(28.24.13) converge absolutely and uniformly on compact sets of the $z$-plane.

For further power series of Mathieu radial functions of integer order for small parameters and improved convergence rate see Larsen et al. (2009).