# §10.29 Recurrence Relations and Derivatives

## §10.29(i) Recurrence Relations

With $\mathscr{Z}_{\nu}\left(z\right)$ defined as in §10.25(ii),

 10.29.1 $\displaystyle\mathscr{Z}_{\nu-1}\left(z\right)-\mathscr{Z}_{\nu+1}\left(z\right)$ $\displaystyle=(2\nu/z)\mathscr{Z}_{\nu}\left(z\right),$ $\displaystyle\mathscr{Z}_{\nu-1}\left(z\right)+\mathscr{Z}_{\nu+1}\left(z\right)$ $\displaystyle=2\mathscr{Z}_{\nu}'\left(z\right).$ ⓘ Symbols: $\mathscr{Z}_{\NVar{\nu}}\left(\NVar{z}\right)$: modified cylinder function, $z$: complex variable and $\nu$: complex parameter A&S Ref: 9.6.26 Referenced by: §10.29(ii), §10.51(ii), §7.6(ii) Permalink: http://dlmf.nist.gov/10.29.E1 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.29(i), §10.29 and Ch.10
 10.29.2 $\displaystyle\mathscr{Z}_{\nu}'\left(z\right)$ $\displaystyle=\mathscr{Z}_{\nu-1}\left(z\right)-(\nu/z)\mathscr{Z}_{\nu}\left(% z\right),$ $\displaystyle\mathscr{Z}_{\nu}'\left(z\right)$ $\displaystyle=\mathscr{Z}_{\nu+1}\left(z\right)+(\nu/z)\mathscr{Z}_{\nu}\left(% z\right).$ ⓘ Symbols: $\mathscr{Z}_{\NVar{\nu}}\left(\NVar{z}\right)$: modified cylinder function, $z$: complex variable and $\nu$: complex parameter A&S Ref: 9.6.26 Referenced by: §10.28, §10.43(i), §10.51(ii) Permalink: http://dlmf.nist.gov/10.29.E2 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.29(i), §10.29 and Ch.10
 10.29.3 $\displaystyle I_{0}'\left(z\right)$ $\displaystyle=I_{1}\left(z\right),$ $\displaystyle K_{0}'\left(z\right)$ $\displaystyle=-K_{1}\left(z\right).$ ⓘ Symbols: $I_{\NVar{\nu}}\left(\NVar{z}\right)$: modified Bessel function of the first kind, $K_{\NVar{\nu}}\left(\NVar{z}\right)$: modified Bessel function of the second kind and $z$: complex variable A&S Ref: 9.6.27 Referenced by: §10.29(i), Erratum (V1.1.3) for Subsections 10.6(i), 10.29(i) Permalink: http://dlmf.nist.gov/10.29.E3 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.29(i), §10.29 and Ch.10

For results on modified quotients of the form $\ifrac{z\mathscr{Z}_{\nu\pm 1}\left(z\right)}{\mathscr{Z}_{\nu}\left(z\right)}$ see Onoe (1955) and Onoe (1956).

## §10.29(ii) Derivatives

For $k=0,1,2,\dotsc$,

 10.29.4 $\displaystyle\left(\frac{1}{z}\frac{\mathrm{d}}{\mathrm{d}z}\right)^{k}(z^{\nu% }\mathscr{Z}_{\nu}\left(z\right))$ $\displaystyle=z^{\nu-k}\mathscr{Z}_{\nu-k}\left(z\right),$ $\displaystyle\left(\frac{1}{z}\frac{\mathrm{d}}{\mathrm{d}z}\right)^{k}(z^{-% \nu}\mathscr{Z}_{\nu}\left(z\right))$ $\displaystyle=z^{-\nu-k}\mathscr{Z}_{\nu+k}\left(z\right).$
 10.29.5 ${\mathscr{Z}_{\nu}}^{(k)}\left(z\right)=\frac{1}{2^{k}}\left(\mathscr{Z}_{\nu-% k}\left(z\right)+\genfrac{(}{)}{0.0pt}{}{k}{1}\mathscr{Z}_{\nu-k+2}\left(z% \right)+\genfrac{(}{)}{0.0pt}{}{k}{2}\mathscr{Z}_{\nu-k+4}\left(z\right)+% \cdots+\mathscr{Z}_{\nu+k}\left(z\right)\right).$ ⓘ Symbols: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$: binomial coefficient, $\mathscr{Z}_{\NVar{\nu}}\left(\NVar{z}\right)$: modified cylinder function, $k$: nonnegative integer, $z$: complex variable and $\nu$: complex parameter A&S Ref: 9.6.29 Referenced by: §10.29(ii) Permalink: http://dlmf.nist.gov/10.29.E5 Encodings: TeX, pMML, png See also: Annotations for §10.29(ii), §10.29 and Ch.10