# §10.6 Recurrence Relations and Derivatives

## §10.6(i) Recurrence Relations

With $\mathscr{C}_{\nu}\left(z\right)$ defined as in §10.2(ii),

 10.6.1 $\displaystyle\mathscr{C}_{\nu-1}\left(z\right)+\mathscr{C}_{\nu+1}\left(z\right)$ $\displaystyle=(2\nu/z)\mathscr{C}_{\nu}\left(z\right),$ $\displaystyle\mathscr{C}_{\nu-1}\left(z\right)-\mathscr{C}_{\nu+1}\left(z\right)$ $\displaystyle=2\mathscr{C}_{\nu}'\left(z\right).$ ⓘ Symbols: $\mathscr{C}_{\NVar{\nu}}\left(\NVar{z}\right)$: cylinder function, $z$: complex variable and $\nu$: complex parameter A&S Ref: 9.1.27 Referenced by: §10.51(i), §10.6(i), §10.6(ii), §10.63(i), §10.74(iv) Permalink: http://dlmf.nist.gov/10.6.E1 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 10.6(i), 10.6 and 10
 10.6.2 $\displaystyle\mathscr{C}_{\nu}'\left(z\right)$ $\displaystyle=\mathscr{C}_{\nu-1}\left(z\right)-(\nu/z)\mathscr{C}_{\nu}\left(% z\right),$ $\displaystyle\mathscr{C}_{\nu}'\left(z\right)$ $\displaystyle=-\mathscr{C}_{\nu+1}\left(z\right)+(\nu/z)\mathscr{C}_{\nu}\left% (z\right).$ ⓘ Symbols: $\mathscr{C}_{\NVar{\nu}}\left(\NVar{z}\right)$: cylinder function, $z$: complex variable and $\nu$: complex parameter A&S Ref: 9.1.27 Referenced by: §10.21(ii), §10.22(i), §10.22(ii), §10.5, §10.51(i), §10.6(i), §10.63(i), §10.63(ii), §10.74(vi) Permalink: http://dlmf.nist.gov/10.6.E2 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 10.6(i), 10.6 and 10
 10.6.3 $\displaystyle J_{0}'\left(z\right)$ $\displaystyle=-J_{1}\left(z\right),$ $\displaystyle\hskip 10.0ptY_{0}'\left(z\right)$ $\displaystyle=-Y_{1}\left(z\right),$ $\displaystyle{H^{(1)}_{0}}'\left(z\right)$ $\displaystyle=-{H^{(1)}_{1}}\left(z\right),$ $\displaystyle\hskip 10.0pt{H^{(2)}_{0}}'\left(z\right)$ $\displaystyle=-{H^{(2)}_{1}}\left(z\right).$

If $f_{\nu}(z)=z^{p}\mathscr{C}_{\nu}\left(\lambda z^{q}\right)$, where $p,q$, and $\lambda$ ($\neq 0$) are real or complex constants, then

 10.6.4 $\displaystyle f_{\nu-1}(z)+f_{\nu+1}(z)$ $\displaystyle=(2\nu/\lambda)z^{-q}f_{\nu}(z),$ $\displaystyle(p+\nu q)f_{\nu-1}(z)+(p-\nu q)f_{\nu+1}(z)$ $\displaystyle=(2\nu/\lambda)z^{1-q}f_{\nu}^{\prime}(z).$ ⓘ Symbols: $z$: complex variable, $\nu$: complex parameter and $f_{\nu}(z)$ A&S Ref: 9.1.29 Referenced by: §10.6(i) Permalink: http://dlmf.nist.gov/10.6.E4 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 10.6(i), 10.6 and 10
 10.6.5 $\displaystyle zf_{\nu}^{\prime}(z)$ $\displaystyle=\lambda qz^{q}f_{\nu-1}(z)+(p-\nu q)f_{\nu}(z),$ $\displaystyle zf_{\nu}^{\prime}(z)$ $\displaystyle=-\lambda qz^{q}f_{\nu+1}(z)+(p+\nu q)f_{\nu}(z).$ ⓘ Symbols: $z$: complex variable, $\nu$: complex parameter and $f_{\nu}(z)$ A&S Ref: 9.1.29 Referenced by: §10.6(i) Permalink: http://dlmf.nist.gov/10.6.E5 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 10.6(i), 10.6 and 10

## §10.6(ii) Derivatives

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

 10.6.6 $\displaystyle\left(\frac{1}{z}\frac{\mathrm{d}}{\mathrm{d}z}\right)^{k}\left(z% ^{\nu}\mathscr{C}_{\nu}\left(z\right)\right)$ $\displaystyle=z^{\nu-k}\mathscr{C}_{\nu-k}\left(z\right),$ $\displaystyle\left(\frac{1}{z}\frac{\mathrm{d}}{\mathrm{d}z}\right)^{k}(z^{-% \nu}\mathscr{C}_{\nu}\left(z\right))$ $\displaystyle=(-1)^{k}z^{-\nu-k}\mathscr{C}_{\nu+k}\left(z\right).$
 10.6.7 ${\mathscr{C}_{\nu}^{(k)}}\left(z\right)=\frac{1}{2^{k}}\sum_{n=0}^{k}(-1)^{n}% \genfrac{(}{)}{0.0pt}{}{k}{n}\mathscr{C}_{\nu-k+2n}\left(z\right).$ ⓘ Symbols: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$: binomial coefficient, $\mathscr{C}_{\NVar{\nu}}\left(\NVar{z}\right)$: cylinder function, $n$: integer, $k$: nonnegative integer, $z$: complex variable and $\nu$: complex parameter A&S Ref: 9.1.31 Referenced by: §10.6(ii) Permalink: http://dlmf.nist.gov/10.6.E7 Encodings: TeX, pMML, png See also: Annotations for 10.6(ii), 10.6 and 10

## §10.6(iii) Cross-Products

Let

 10.6.8 $\displaystyle p_{\nu}$ $\displaystyle=J_{\nu}\left(a\right)Y_{\nu}\left(b\right)-J_{\nu}\left(b\right)% Y_{\nu}\left(a\right),$ $\displaystyle q_{\nu}$ $\displaystyle=J_{\nu}\left(a\right)Y_{\nu}'\left(b\right)-J_{\nu}'\left(b% \right)Y_{\nu}\left(a\right),$ $\displaystyle r_{\nu}$ $\displaystyle=J_{\nu}'\left(a\right)Y_{\nu}\left(b\right)-J_{\nu}\left(b\right% )Y_{\nu}'\left(a\right),$ $\displaystyle s_{\nu}$ $\displaystyle=J_{\nu}'\left(a\right)Y_{\nu}'\left(b\right)-J_{\nu}'\left(b% \right)Y_{\nu}'\left(a\right),$ ⓘ Defines: $p_{\nu}$: cross-product (locally), $q_{\nu}$: cross-product (locally), $r_{\nu}$: cross-product (locally) and $s_{\nu}$: cross-product (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 and $\nu$: complex parameter A&S Ref: 9.1.32 Permalink: http://dlmf.nist.gov/10.6.E8 Encodings: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png See also: Annotations for 10.6(iii), 10.6 and 10

where $a$ and $b$ are independent of $\nu$. Then

 10.6.9 $\displaystyle p_{\nu+1}-p_{\nu-1}$ $\displaystyle=-\frac{2\nu}{a}q_{\nu}-\frac{2\nu}{b}r_{\nu},$ $\displaystyle q_{\nu+1}+r_{\nu}$ $\displaystyle=\frac{\nu}{a}p_{\nu}-\frac{\nu+1}{b}p_{\nu+1},$ $\displaystyle r_{\nu+1}+q_{\nu}$ $\displaystyle=\frac{\nu}{b}p_{\nu}-\frac{\nu+1}{a}p_{\nu+1},$ $\displaystyle s_{\nu}$ $\displaystyle=\tfrac{1}{2}p_{\nu+1}+\tfrac{1}{2}p_{\nu-1}-\frac{\nu^{2}}{ab}p_% {\nu},$ ⓘ Symbols: $\nu$: complex parameter, $p_{\nu}$: cross-product, $q_{\nu}$: cross-product, $r_{\nu}$: cross-product and $s_{\nu}$: cross-product A&S Ref: 9.1.33 Permalink: http://dlmf.nist.gov/10.6.E9 Encodings: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png See also: Annotations for 10.6(iii), 10.6 and 10

and

 10.6.10 $p_{\nu}s_{\nu}-q_{\nu}r_{\nu}=4/(\pi^{2}ab).$