# §28.23 Expansions in Series of Bessel Functions

We use the following notations:

 28.23.1 $\displaystyle{\cal C}_{\mu}^{(1)}$ $\displaystyle=\mathop{J_{\mu}\/}\nolimits,$ $\displaystyle{\cal C}_{\mu}^{(2)}$ $\displaystyle=\mathop{Y_{\mu}\/}\nolimits,$ $\displaystyle{\cal C}_{\mu}^{(3)}$ $\displaystyle=\mathop{{H^{(1)}_{\mu}}\/}\nolimits,$ $\displaystyle{\cal C}_{\mu}^{(4)}$ $\displaystyle=\mathop{{H^{(2)}_{\mu}}\/}\nolimits;$ Defines: $\mathcal{C}_{\mu}^{(j)}$: cylinder functions (locally) Symbols: $\mathop{J_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{z}\right)$: Bessel function of the first kind, $\mathop{Y_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{z}\right)$: Bessel function of the second kind, $\mathop{{H^{(1)}_{\NVar{\nu}}}\/}\nolimits\!\left(\NVar{z}\right)$: Bessel function of the third kind (or Hankel function), $\mathop{{H^{(2)}_{\NVar{\nu}}}\/}\nolimits\!\left(\NVar{z}\right)$: 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

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

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

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

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

In the case when $\nu$ is an integer

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

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 $|\mathop{\cosh\/}\nolimits z|>1$, and the series in the odd-numbered equations converge for $\Re{z}>0$ and $|\mathop{\sinh\/}\nolimits z|>1$.

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