# §28.23 Expansions in Series of Bessel Functions

We use the following notations:

 28.23.1 $\displaystyle{\cal C}_{\mu}^{(1)}$ $\displaystyle=J_{\mu},$ $\displaystyle{\cal C}_{\mu}^{(2)}$ $\displaystyle=Y_{\mu},$ $\displaystyle{\cal C}_{\mu}^{(3)}$ $\displaystyle={H^{(1)}_{\mu}},$ $\displaystyle{\cal C}_{\mu}^{(4)}$ $\displaystyle={H^{(2)}_{\mu}};$ ⓘ Defines: $\mathcal{C}_{\mu}^{(j)}$: cylinder functions (locally) Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$: Bessel function of the first kind, $Y_{\NVar{\nu}}\left(\NVar{z}\right)$: Bessel function of the second kind, ${H^{(1)}_{\NVar{\nu}}}\left(\NVar{z}\right)$: Bessel function of the third kind (or Hankel function), ${H^{(2)}_{\NVar{\nu}}}\left(\NVar{z}\right)$: Bessel function of the third kind (or Hankel function) and $j$: integer A&S Ref: 20.4.7 (in different notation) Referenced by: §28.28(ii) Permalink: http://dlmf.nist.gov/28.23.E1 Encodings: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png See also: Annotations for §28.23 and Ch.28

compare §10.2(ii). For the coefficients $c^{\nu}_{n}(q)$ see §28.14. For $A_{n}^{m}(q)$ and $B_{n}^{m}(q)$ see §28.4.

 28.23.2 $\displaystyle\mathrm{me}_{\nu}\left(0,h^{2}\right){\mathrm{M}^{(j)}_{\nu}}% \left(z,h\right)$ $\displaystyle=\sum_{n=-\infty}^{\infty}(-1)^{n}c_{2n}^{\nu}(h^{2}){\cal C}_{% \nu+2n}^{(j)}(2h\cosh z),$ 28.23.3 $\displaystyle\mathrm{me}_{\nu}'\left(0,h^{2}\right){\mathrm{M}^{(j)}_{\nu}}% \left(z,h\right)$ $\displaystyle=\mathrm{i}\tanh z\sum_{n=-\infty}^{\infty}(-1)^{n}(\nu+2n)c_{2n}% ^{\nu}(h^{2}){\cal C}_{\nu+2n}^{(j)}(2h\cosh z),$

valid for all $z$ when $j=1$, and for $\Re z>0$ and $|\cosh z|>1$ when $j=2,3,4$.

 28.23.4 $\displaystyle\mathrm{me}_{\nu}\left(\tfrac{1}{2}\pi,h^{2}\right){\mathrm{M}^{(% j)}_{\nu}}\left(z,h\right)$ $\displaystyle=e^{\mathrm{i}\nu\ifrac{\pi}{2}}\sum_{n=-\infty}^{\infty}c_{2n}^{% \nu}(h^{2}){\cal C}_{\nu+2n}^{(j)}(2h\sinh z),$ 28.23.5 $\displaystyle\mathrm{me}_{\nu}'\left(\tfrac{1}{2}\pi,h^{2}\right){\mathrm{M}^{% (j)}_{\nu}}\left(z,h\right)$ $\displaystyle=\mathrm{i}e^{\mathrm{i}\nu\ifrac{\pi}{2}}\coth z\sum_{n=-\infty}% ^{\infty}(\nu+2n)c_{2n}^{\nu}(h^{2}){\cal C}_{\nu+2n}^{(j)}(2h\sinh z),$

valid for all $z$ when $j=1$, and for $\Re z>0$ and $|\sinh z|>1$ when $j=2,3,4$.

In the case when $\nu$ is an integer

 28.23.6 $\displaystyle{\mathrm{Mc}^{(j)}_{2m}}\left(z,h\right)$ $\displaystyle=(-1)^{m}\left(\mathrm{ce}_{2m}\left(0,h^{2}\right)\right)^{-1}% \sum_{\ell=0}^{\infty}(-1)^{\ell}A_{2\ell}^{2m}(h^{2}){\cal C}_{2\ell}^{(j)}(2% h\cosh z),$ 28.23.7 $\displaystyle{\mathrm{Mc}^{(j)}_{2m}}\left(z,h\right)$ $\displaystyle=(-1)^{m}\left(\mathrm{ce}_{2m}\left(\tfrac{1}{2}\pi,h^{2}\right)% \right)^{-1}\sum_{\ell=0}^{\infty}A_{2\ell}^{2m}(h^{2}){\cal C}_{2\ell}^{(j)}(% 2h\sinh z),$
 28.23.8 $\displaystyle{\mathrm{Mc}^{(j)}_{2m+1}}\left(z,h\right)$ $\displaystyle=(-1)^{m}\left(\mathrm{ce}_{2m+1}\left(0,h^{2}\right)\right)^{-1}% \sum_{\ell=0}^{\infty}(-1)^{\ell}A_{2\ell+1}^{2m+1}(h^{2}){\cal C}_{2\ell+1}^{% (j)}(2h\cosh z),$ 28.23.9 $\displaystyle{\mathrm{Mc}^{(j)}_{2m+1}}\left(z,h\right)$ $\displaystyle=(-1)^{m+1}\left(\mathrm{ce}_{2m+1}'\left(\tfrac{1}{2}\pi,h^{2}% \right)\right)^{-1}\coth z\sum_{\ell=0}^{\infty}(2\ell+1)A_{2\ell+1}^{2m+1}(h^% {2}){\cal C}_{2\ell+1}^{(j)}(2h\sinh z),$ 28.23.10 $\displaystyle{\mathrm{Ms}^{(j)}_{2m+1}}\left(z,h\right)$ $\displaystyle=(-1)^{m}\left(\mathrm{se}_{2m+1}'\left(0,h^{2}\right)\right)^{-1% }\tanh z\sum_{\ell=0}^{\infty}(-1)^{\ell}(2\ell+1)B_{2\ell+1}^{2m+1}(h^{2}){% \cal C}_{2\ell+1}^{(j)}(2h\cosh z),$ 28.23.11 $\displaystyle{\mathrm{Ms}^{(j)}_{2m+1}}\left(z,h\right)$ $\displaystyle=(-1)^{m}\left(\mathrm{se}_{2m+1}\left(\tfrac{1}{2}\pi,h^{2}% \right)\right)^{-1}\sum_{\ell=0}^{\infty}B_{2\ell+1}^{2m+1}(h^{2}){\cal C}_{2% \ell+1}^{(j)}(2h\sinh z),$ 28.23.12 $\displaystyle{\mathrm{Ms}^{(j)}_{2m+2}}\left(z,h\right)$ $\displaystyle=(-1)^{m}\left(\mathrm{se}_{2m+2}'\left(0,h^{2}\right)\right)^{-1% }\tanh z\sum_{\ell=0}^{\infty}(-1)^{\ell}(2\ell+2)B_{2\ell+2}^{2m+2}(h^{2}){% \cal C}_{2\ell+2}^{(j)}(2h\cosh z),$ 28.23.13 $\displaystyle{\mathrm{Ms}^{(j)}_{2m+2}}\left(z,h\right)$ $\displaystyle=(-1)^{m+1}\left(\mathrm{se}_{2m+2}'\left(\tfrac{1}{2}\pi,h^{2}% \right)\right)^{-1}\coth z\sum_{\ell=0}^{\infty}(2\ell+2)B_{2\ell+2}^{2m+2}(h^% {2}){\cal C}_{2\ell+2}^{(j)}(2h\sinh z).$

When $j=1$, each of the series (28.23.6)–(28.23.13) converges for all $z$. When $j=2,3,4$ the series in the even-numbered equations converge for $\Re z>0$ and $|\cosh z|>1$, and the series in the odd-numbered equations converge for $\Re z>0$ and $|\sinh z|>1$.

For proofs and generalizations, see Meixner and Schäfke (1954, §§2.62 and 2.64).