# §10.6 Recurrence Relations and Derivatives

## §10.6(i) Recurrence Relations

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

 10.6.1 $\displaystyle\mathop{\mathscr{C}_{\nu-1}\/}\nolimits\!\left(z\right)+\mathop{% \mathscr{C}_{\nu+1}\/}\nolimits\!\left(z\right)$ $\displaystyle=(2\nu/z)\mathop{\mathscr{C}_{\nu}\/}\nolimits\!\left(z\right),$ $\displaystyle\mathop{\mathscr{C}_{\nu-1}\/}\nolimits\!\left(z\right)-\mathop{% \mathscr{C}_{\nu+1}\/}\nolimits\!\left(z\right)$ $\displaystyle=2\mathop{\mathscr{C}_{\nu}\/}\nolimits'\!\left(z\right).$ Symbols: $\mathop{\mathscr{C}_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{z}\right)$: 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)
 10.6.2 $\displaystyle\mathop{\mathscr{C}_{\nu}\/}\nolimits'\!\left(z\right)$ $\displaystyle=\mathop{\mathscr{C}_{\nu-1}\/}\nolimits\!\left(z\right)-(\nu/z)% \mathop{\mathscr{C}_{\nu}\/}\nolimits\!\left(z\right),$ $\displaystyle\mathop{\mathscr{C}_{\nu}\/}\nolimits'\!\left(z\right)$ $\displaystyle=-\mathop{\mathscr{C}_{\nu+1}\/}\nolimits\!\left(z\right)+(\nu/z)% \mathop{\mathscr{C}_{\nu}\/}\nolimits\!\left(z\right).$ Symbols: $\mathop{\mathscr{C}_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{z}\right)$: 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)
 10.6.3 $\displaystyle\mathop{J_{0}\/}\nolimits'\!\left(z\right)$ $\displaystyle=-\mathop{J_{1}\/}\nolimits\!\left(z\right),$ $\displaystyle\hskip 10.0pt\mathop{Y_{0}\/}\nolimits'\!\left(z\right)$ $\displaystyle=-\mathop{Y_{1}\/}\nolimits\!\left(z\right),$ $\displaystyle\mathop{{H^{(1)}_{0}}\/}\nolimits'\!\left(z\right)$ $\displaystyle=-\mathop{{H^{(1)}_{1}}\/}\nolimits\!\left(z\right),$ $\displaystyle\hskip 10.0pt\mathop{{H^{(2)}_{0}}\/}\nolimits'\!\left(z\right)$ $\displaystyle=-\mathop{{H^{(2)}_{1}}\/}\nolimits\!\left(z\right).$

If $f_{\nu}(z)=z^{p}\mathop{\mathscr{C}_{\nu}\/}\nolimits\!\left(\lambda z^{q}\right)$, 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)
 10.6.5 $\displaystyle zf_{\nu}^{\prime}(z)$ $\displaystyle=\lambda qz^{q}f_{\nu-1}(z)+(p-\nu q)f_{\nu}(z),$ $\displaystyle zf_{\nu}^{\prime}(z)$ $\displaystyle=-\lambda qz^{q}f_{\nu+1}(z)+(p+\nu q)f_{\nu}(z).$ 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(ii) Derivatives

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

 10.6.6 $\displaystyle\left(\frac{1}{z}\frac{\mathrm{d}}{\mathrm{d}z}\right)^{k}\left(z% ^{\nu}\mathop{\mathscr{C}_{\nu}\/}\nolimits\!\left(z\right)\right)$ $\displaystyle=z^{\nu-k}\mathop{\mathscr{C}_{\nu-k}\/}\nolimits\!\left(z\right),$ $\displaystyle\left(\frac{1}{z}\frac{\mathrm{d}}{\mathrm{d}z}\right)^{k}(z^{-% \nu}\mathop{\mathscr{C}_{\nu}\/}\nolimits\!\left(z\right))$ $\displaystyle=(-1)^{k}z^{-\nu-k}\mathop{\mathscr{C}_{\nu+k}\/}\nolimits\!\left% (z\right).$
 10.6.7 ${\mathop{\mathscr{C}_{\nu}\/}\nolimits^{(k)}}\!\left(z\right)=\frac{1}{2^{k}}% \sum_{n=0}^{k}(-1)^{n}\binom{k}{n}\mathop{\mathscr{C}_{\nu-k+2n}\/}\nolimits\!% \left(z\right).$

## §10.6(iii) Cross-Products

Let

 10.6.8 $\displaystyle p_{\nu}$ $\displaystyle=\mathop{J_{\nu}\/}\nolimits\!\left(a\right)\mathop{Y_{\nu}\/}% \nolimits\!\left(b\right)-\mathop{J_{\nu}\/}\nolimits\!\left(b\right)\mathop{Y% _{\nu}\/}\nolimits\!\left(a\right),$ $\displaystyle q_{\nu}$ $\displaystyle=\mathop{J_{\nu}\/}\nolimits\!\left(a\right)\mathop{Y_{\nu}\/}% \nolimits'\!\left(b\right)-\mathop{J_{\nu}\/}\nolimits'\!\left(b\right)\mathop% {Y_{\nu}\/}\nolimits\!\left(a\right),$ $\displaystyle r_{\nu}$ $\displaystyle=\mathop{J_{\nu}\/}\nolimits'\!\left(a\right)\mathop{Y_{\nu}\/}% \nolimits\!\left(b\right)-\mathop{J_{\nu}\/}\nolimits\!\left(b\right)\mathop{Y% _{\nu}\/}\nolimits'\!\left(a\right),$ $\displaystyle s_{\nu}$ $\displaystyle=\mathop{J_{\nu}\/}\nolimits'\!\left(a\right)\mathop{Y_{\nu}\/}% \nolimits'\!\left(b\right)-\mathop{J_{\nu}\/}\nolimits'\!\left(b\right)\mathop% {Y_{\nu}\/}\nolimits'\!\left(a\right),$ Defines: $p_{\nu}$: cross-product (locally), $q_{\nu}$: cross-product (locally), $r_{\nu}$: cross-product (locally) and $s_{\nu}$: cross-product (locally) Symbols: $\mathop{J_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{z}\right)$: Bessel function of the first kind, $\mathop{Y_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{z}\right)$: Bessel function of the second kind 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)

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)

and

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