# §10.63 Recurrence Relations and Derivatives

## §10.63(i) $\mathop{\mathrm{ber}_{\nu}\/}\nolimits x$, $\mathop{\mathrm{bei}_{\nu}\/}\nolimits x$, $\mathop{\mathrm{ker}_{\nu}\/}\nolimits x$, $\mathop{\mathrm{kei}_{\nu}\/}\nolimits x$

Let $f_{\nu}(x)$, $g_{\nu}(x)$ denote any one of the ordered pairs:

 10.63.1 $\mathop{\mathrm{ber}_{\nu}\/}\nolimits x,\mathop{\mathrm{bei}_{\nu}\/}% \nolimits x;$ $\mathop{\mathrm{bei}_{\nu}\/}\nolimits x,-\mathop{\mathrm{ber}_{\nu}\/}% \nolimits x;$ $\mathop{\mathrm{ker}_{\nu}\/}\nolimits x,\mathop{\mathrm{kei}_{\nu}\/}% \nolimits x;$ $\mathop{\mathrm{kei}_{\nu}\/}\nolimits x,-\mathop{\mathrm{ker}_{\nu}\/}% \nolimits x.$

Then

 10.63.2 $\displaystyle f_{\nu-1}(x)+f_{\nu+1}(x)$ $\displaystyle=-(\nu\sqrt{2}/x)\left(f_{\nu}(x)-g_{\nu}(x)\right),$ $\displaystyle f_{\nu+1}(x)+g_{\nu+1}(x)-f_{\nu-1}(x)-g_{\nu-1}(x)$ $\displaystyle=2\sqrt{2}f_{\nu}^{\prime}(x),$ $\displaystyle f_{\nu}^{\prime}(x)$ $\displaystyle=-(1/\sqrt{2})\left(f_{\nu-1}(x)+g_{\nu-1}(x)\right)-(\nu/x)f_{% \nu}(x),$ $\displaystyle f_{\nu}^{\prime}(x)$ $\displaystyle=(1/\sqrt{2})\left(f_{\nu+1}(x)+g_{\nu+1}(x)\right)+(\nu/x)f_{\nu% }(x).$ Symbols: $x$: real variable, $\nu$: complex parameter, $f_{\nu}$ a Kelvin function and $g_{\nu}$ a Kelvin function A&S Ref: 9.9.14 Referenced by: §10.71(i) Permalink: http://dlmf.nist.gov/10.63.E2 Encodings: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png See also: Annotations for 10.63(i)
 10.63.3 $\displaystyle\sqrt{2}\mathop{\mathrm{ber}\/}\nolimits'x$ $\displaystyle=\mathop{\mathrm{ber}_{1}\/}\nolimits x+\mathop{\mathrm{bei}_{1}% \/}\nolimits x,$ $\displaystyle\sqrt{2}\mathop{\mathrm{bei}\/}\nolimits'x$ $\displaystyle=-\mathop{\mathrm{ber}_{1}\/}\nolimits x+\mathop{\mathrm{bei}_{1}% \/}\nolimits x.$ Symbols: $\mathop{\mathrm{bei}_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{x}\right)$: Kelvin function, $\mathop{\mathrm{ber}_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{x}\right)$: Kelvin function and $x$: real variable A&S Ref: 9.9.16 Permalink: http://dlmf.nist.gov/10.63.E3 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 10.63(i)
 10.63.4 $\displaystyle\sqrt{2}\mathop{\mathrm{ker}\/}\nolimits'x$ $\displaystyle=\mathop{\mathrm{ker}_{1}\/}\nolimits x+\mathop{\mathrm{kei}_{1}% \/}\nolimits x,$ $\displaystyle\sqrt{2}\mathop{\mathrm{kei}\/}\nolimits'x$ $\displaystyle=-\mathop{\mathrm{ker}_{1}\/}\nolimits x+\mathop{\mathrm{kei}_{1}% \/}\nolimits x.$ Symbols: $\mathop{\mathrm{kei}_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{x}\right)$: Kelvin function, $\mathop{\mathrm{ker}_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{x}\right)$: Kelvin function and $x$: real variable A&S Ref: 9.9.17 Permalink: http://dlmf.nist.gov/10.63.E4 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 10.63(i)

## §10.63(ii) Cross-Products

Let

 10.63.5 $\displaystyle p_{\nu}$ $\displaystyle={\mathop{\mathrm{ber}_{\nu}\/}\nolimits^{2}}x+{\mathop{\mathrm{% bei}_{\nu}\/}\nolimits^{2}}x,$ $\displaystyle q_{\nu}$ $\displaystyle=\mathop{\mathrm{ber}_{\nu}\/}\nolimits x\mathop{\mathrm{bei}_{% \nu}\/}\nolimits'x-\mathop{\mathrm{ber}_{\nu}\/}\nolimits'x\mathop{\mathrm{bei% }_{\nu}\/}\nolimits x,$ $\displaystyle r_{\nu}$ $\displaystyle=\mathop{\mathrm{ber}_{\nu}\/}\nolimits x\mathop{\mathrm{ber}_{% \nu}\/}\nolimits'x+\mathop{\mathrm{bei}_{\nu}\/}\nolimits x\mathop{\mathrm{bei% }_{\nu}\/}\nolimits'x,$ $\displaystyle s_{\nu}$ $\displaystyle=\left(\mathop{\mathrm{ber}_{\nu}\/}\nolimits'x\right)^{2}+\left(% \mathop{\mathrm{bei}_{\nu}\/}\nolimits'x\right)^{2}.$ Defines: $p_{\nu}$: cross-product (locally), $q_{\nu}$: cross-product (locally), $r_{\nu}$: cross-product (locally) and $s_{\nu}$: cross-product (locally) Symbols: $\mathop{\mathrm{bei}_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{x}\right)$: Kelvin function, $\mathop{\mathrm{ber}_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{x}\right)$: Kelvin function, $x$: real variable and $\nu$: complex parameter A&S Ref: 9.9.18 Referenced by: §10.63(ii), §10.63(ii) Permalink: http://dlmf.nist.gov/10.63.E5 Encodings: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png See also: Annotations for 10.63(ii)

Then

 10.63.6 $\displaystyle p_{\nu+1}$ $\displaystyle=p_{\nu-1}-(4\nu/x)r_{\nu},$ $\displaystyle q_{\nu+1}$ $\displaystyle=-(\nu/x)p_{\nu}+r_{\nu}=-q_{\nu-1}+2r_{\nu},$ $\displaystyle r_{\nu+1}$ $\displaystyle=-((\nu+1)/x)p_{\nu+1}+q_{\nu},$ $\displaystyle s_{\nu}$ $\displaystyle=\tfrac{1}{2}p_{\nu+1}+\tfrac{1}{2}p_{\nu-1}-(\nu^{2}/x^{2})p_{% \nu},$ Symbols: $x$: real variable, $\nu$: complex parameter, $p_{\nu}$: cross-product, $q_{\nu}$: cross-product, $r_{\nu}$: cross-product and $s_{\nu}$: cross-product A&S Ref: 9.9.19 Referenced by: §10.63(ii) Permalink: http://dlmf.nist.gov/10.63.E6 Encodings: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png See also: Annotations for 10.63(ii)

and

 10.63.7 $p_{\nu}s_{\nu}=r_{\nu}^{2}+q_{\nu}^{2}.$ Symbols: $\nu$: complex parameter, $p_{\nu}$: cross-product, $q_{\nu}$: cross-product, $r_{\nu}$: cross-product and $s_{\nu}$: cross-product A&S Ref: 9.9.20 Referenced by: §10.63(ii) Permalink: http://dlmf.nist.gov/10.63.E7 Encodings: TeX, pMML, png See also: Annotations for 10.63(ii)

Equations (10.63.6) and (10.63.7) also hold when the symbols $\mathop{\mathrm{ber}\/}\nolimits$ and $\mathop{\mathrm{bei}\/}\nolimits$ in (10.63.5) are replaced throughout by $\mathop{\mathrm{ker}\/}\nolimits$ and $\mathop{\mathrm{kei}\/}\nolimits$, respectively.