# §10.63 Recurrence Relations and Derivatives

## §10.63(i) $\operatorname{ber}_{\nu}x$, $\operatorname{bei}_{\nu}x$, $\operatorname{ker}_{\nu}x$, $\operatorname{kei}_{\nu}x$

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

 10.63.1 $\operatorname{ber}_{\nu}x,\operatorname{bei}_{\nu}x;$ $\operatorname{bei}_{\nu}x,-\operatorname{ber}_{\nu}x;$ $\operatorname{ker}_{\nu}x,\operatorname{kei}_{\nu}x;$ $\operatorname{kei}_{\nu}x,-\operatorname{ker}_{\nu}x.$

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 and 10
 10.63.3 $\displaystyle\sqrt{2}\operatorname{ber}'x$ $\displaystyle=\operatorname{ber}_{1}x+\operatorname{bei}_{1}x,$ $\displaystyle\sqrt{2}\operatorname{bei}'x$ $\displaystyle=-\operatorname{ber}_{1}x+\operatorname{bei}_{1}x.$ ⓘ Symbols: $\operatorname{bei}_{\NVar{\nu}}\left(\NVar{x}\right)$: Kelvin function, $\operatorname{ber}_{\NVar{\nu}}\left(\NVar{x}\right)$: 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 and 10
 10.63.4 $\displaystyle\sqrt{2}\operatorname{ker}'x$ $\displaystyle=\operatorname{ker}_{1}x+\operatorname{kei}_{1}x,$ $\displaystyle\sqrt{2}\operatorname{kei}'x$ $\displaystyle=-\operatorname{ker}_{1}x+\operatorname{kei}_{1}x.$ ⓘ Symbols: $\operatorname{kei}_{\NVar{\nu}}\left(\NVar{x}\right)$: Kelvin function, $\operatorname{ker}_{\NVar{\nu}}\left(\NVar{x}\right)$: 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 and 10

## §10.63(ii) Cross-Products

Let

 10.63.5 $\displaystyle p_{\nu}$ $\displaystyle={\operatorname{ber}_{\nu}^{2}}x+{\operatorname{bei}_{\nu}^{2}}x,$ $\displaystyle q_{\nu}$ $\displaystyle=\operatorname{ber}_{\nu}x\operatorname{bei}_{\nu}'x-% \operatorname{ber}_{\nu}'x\operatorname{bei}_{\nu}x,$ $\displaystyle r_{\nu}$ $\displaystyle=\operatorname{ber}_{\nu}x\operatorname{ber}_{\nu}'x+% \operatorname{bei}_{\nu}x\operatorname{bei}_{\nu}'x,$ $\displaystyle s_{\nu}$ $\displaystyle=\left(\operatorname{ber}_{\nu}'x\right)^{2}+\left(\operatorname{% bei}_{\nu}'x\right)^{2}.$ ⓘ Defines: $p_{\nu}$: cross-product (locally), $q_{\nu}$: cross-product (locally), $r_{\nu}$: cross-product (locally) and $s_{\nu}$: cross-product (locally) Symbols: $\operatorname{bei}_{\NVar{\nu}}\left(\NVar{x}\right)$: Kelvin function, $\operatorname{ber}_{\NVar{\nu}}\left(\NVar{x}\right)$: 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), 10.63 and 10

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), 10.63 and 10

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), 10.63 and 10

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