# §10.29 Recurrence Relations and Derivatives

## §10.29(i) Recurrence Relations

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

 10.29.1 $\displaystyle\mathop{\mathscr{Z}_{\nu-1}\/}\nolimits\!\left(z\right)-\mathop{% \mathscr{Z}_{\nu+1}\/}\nolimits\!\left(z\right)$ $\displaystyle=(2\nu/z)\mathop{\mathscr{Z}_{\nu}\/}\nolimits\!\left(z\right),$ $\displaystyle\mathop{\mathscr{Z}_{\nu-1}\/}\nolimits\!\left(z\right)+\mathop{% \mathscr{Z}_{\nu+1}\/}\nolimits\!\left(z\right)$ $\displaystyle=2\!\mathop{\mathscr{Z}_{\nu}\/}\nolimits'\!\left(z\right).$ Symbols: $\mathop{\mathscr{Z}_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{z}\right)$: 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)
 10.29.2 $\displaystyle\mathop{\mathscr{Z}_{\nu}\/}\nolimits'\!\left(z\right)$ $\displaystyle=\mathop{\mathscr{Z}_{\nu-1}\/}\nolimits\!\left(z\right)-(\nu/z)% \mathop{\mathscr{Z}_{\nu}\/}\nolimits\!\left(z\right),$ $\displaystyle\mathop{\mathscr{Z}_{\nu}\/}\nolimits'\!\left(z\right)$ $\displaystyle=\mathop{\mathscr{Z}_{\nu+1}\/}\nolimits\!\left(z\right)+(\nu/z)% \mathop{\mathscr{Z}_{\nu}\/}\nolimits\!\left(z\right).$ Symbols: $\mathop{\mathscr{Z}_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{z}\right)$: 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)
 10.29.3 $\displaystyle\mathop{I_{0}\/}\nolimits'\!\left(z\right)$ $\displaystyle=\mathop{I_{1}\/}\nolimits\!\left(z\right),$ $\displaystyle\mathop{K_{0}\/}\nolimits'\!\left(z\right)$ $\displaystyle=-\mathop{K_{1}\/}\nolimits\!\left(z\right).$ Symbols: $\mathop{I_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{z}\right)$: modified Bessel function of the first kind, $\mathop{K_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{z}\right)$: modified Bessel function of the second kind and $z$: complex variable A&S Ref: 9.6.27 Permalink: http://dlmf.nist.gov/10.29.E3 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 10.29(i)

## §10.29(ii) Derivatives

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

 10.29.4 $\displaystyle\left(\frac{1}{z}\frac{\mathrm{d}}{\mathrm{d}z}\right)^{k}(z^{\nu% }\mathop{\mathscr{Z}_{\nu}\/}\nolimits\!\left(z\right))$ $\displaystyle=z^{\nu-k}\mathop{\mathscr{Z}_{\nu-k}\/}\nolimits\!\left(z\right),$ $\displaystyle\left(\frac{1}{z}\frac{\mathrm{d}}{\mathrm{d}z}\right)^{k}(z^{-% \nu}\mathop{\mathscr{Z}_{\nu}\/}\nolimits\!\left(z\right))$ $\displaystyle=z^{-\nu-k}\mathop{\mathscr{Z}_{\nu+k}\/}\nolimits\!\left(z\right).$
 10.29.5 ${\mathop{\mathscr{Z}_{\nu}\/}\nolimits^{(k)}}\!\left(z\right)=\frac{1}{2^{k}}% \left(\mathop{\mathscr{Z}_{\nu-k}\/}\nolimits\!\left(z\right)+\binom{k}{1}% \mathop{\mathscr{Z}_{\nu-k+2}\/}\nolimits\!\left(z\right)+\binom{k}{2}\mathop{% \mathscr{Z}_{\nu-k+4}\/}\nolimits\!\left(z\right)+\cdots+\mathop{\mathscr{Z}_{% \nu+k}\/}\nolimits\!\left(z\right)\right).$ Symbols: $\binom{\NVar{m}}{\NVar{n}}$: binomial coefficient, $\mathop{\mathscr{Z}_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{z}\right)$: 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)