# §10.65 Power Series

## §10.65(i) $\mathop{\mathrm{ber}_{\nu}\/}\nolimits x$ and $\mathop{\mathrm{bei}_{\nu}\/}\nolimits x$

 10.65.1 $\displaystyle\mathop{\mathrm{ber}_{\nu}\/}\nolimits x$ $\displaystyle=(\tfrac{1}{2}x)^{\nu}\sum_{k=0}^{\infty}\frac{\mathop{\cos\/}% \nolimits\!\left(\frac{3}{4}\nu\pi+\frac{1}{2}k\pi\right)}{k!\mathop{\Gamma\/}% \nolimits\!\left(\nu+k+1\right)}(\tfrac{1}{4}x^{2})^{k},$ $\displaystyle\mathop{\mathrm{bei}_{\nu}\/}\nolimits x$ $\displaystyle=(\tfrac{1}{2}x)^{\nu}\sum_{k=0}^{\infty}\frac{\mathop{\sin\/}% \nolimits\!\left(\frac{3}{4}\nu\pi+\frac{1}{2}k\pi\right)}{k!\mathop{\Gamma\/}% \nolimits\!\left(\nu+k+1\right)}(\tfrac{1}{4}x^{2})^{k}.$
 10.65.2 $\displaystyle\mathop{\mathrm{ber}\/}\nolimits x$ $\displaystyle=1-\frac{(\frac{1}{4}x^{2})^{2}}{(2!)^{2}}+\frac{(\frac{1}{4}x^{2% })^{4}}{(4!)^{2}}-\cdots,$ $\displaystyle\mathop{\mathrm{bei}\/}\nolimits x$ $\displaystyle=\tfrac{1}{4}x^{2}-\frac{(\frac{1}{4}x^{2})^{3}}{(3!)^{2}}+\frac{% (\frac{1}{4}x^{2})^{5}}{(5!)^{2}}-\cdots.$

## §10.65(ii) $\mathop{\mathrm{ker}_{\nu}\/}\nolimits x$ and $\mathop{\mathrm{kei}_{\nu}\/}\nolimits x$

When $\nu$ is not an integer combine (10.65.1) with (10.61.6). Also, with $\mathop{\psi\/}\nolimits\!\left(x\right)=\mathop{\Gamma\/}\nolimits'\!\left(x% \right)/\mathop{\Gamma\/}\nolimits\!\left(x\right)$,

 10.65.3 $\displaystyle\mathop{\mathrm{ker}_{n}\/}\nolimits x$ $\displaystyle=\tfrac{1}{2}(\tfrac{1}{2}x)^{-n}\sum_{k=0}^{n-1}\frac{(n-k-1)!}{% k!}\mathop{\cos\/}\nolimits\!\left(\tfrac{3}{4}n\pi+\tfrac{1}{2}k\pi\right)(% \tfrac{1}{4}x^{2})^{k}-\mathop{\ln\/}\nolimits\!\left(\tfrac{1}{2}x\right)% \mathop{\mathrm{ber}_{n}\/}\nolimits x+\tfrac{1}{4}\pi\mathop{\mathrm{bei}_{n}% \/}\nolimits x+\tfrac{1}{2}(\tfrac{1}{2}x)^{n}\sum_{k=0}^{\infty}\frac{\mathop% {\psi\/}\nolimits\!\left(k+1\right)+\mathop{\psi\/}\nolimits\!\left(n+k+1% \right)}{k!(n+k)!}\mathop{\cos\/}\nolimits\!\left(\tfrac{3}{4}n\pi+\tfrac{1}{2% }k\pi\right)(\tfrac{1}{4}x^{2})^{k},$ 10.65.4 $\displaystyle\mathop{\mathrm{kei}_{n}\/}\nolimits x$ $\displaystyle=-\tfrac{1}{2}(\tfrac{1}{2}x)^{-n}\sum_{k=0}^{n-1}\frac{(n-k-1)!}% {k!}\mathop{\sin\/}\nolimits\!\left(\tfrac{3}{4}n\pi+\tfrac{1}{2}k\pi\right)(% \tfrac{1}{4}x^{2})^{k}-\mathop{\ln\/}\nolimits\!\left(\tfrac{1}{2}x\right)% \mathop{\mathrm{bei}_{n}\/}\nolimits x-\tfrac{1}{4}\pi\mathop{\mathrm{ber}_{n}% \/}\nolimits x+\tfrac{1}{2}(\tfrac{1}{2}x)^{n}\sum_{k=0}^{\infty}\frac{\mathop% {\psi\/}\nolimits\!\left(k+1\right)+\mathop{\psi\/}\nolimits\!\left(n+k+1% \right)}{k!(n+k)!}\mathop{\sin\/}\nolimits\!\left(\tfrac{3}{4}n\pi+\tfrac{1}{2% }k\pi\right)(\tfrac{1}{4}x^{2})^{k}.$
 10.65.5 $\displaystyle\mathop{\mathrm{ker}\/}\nolimits x$ $\displaystyle=-\mathop{\ln\/}\nolimits\!\left(\tfrac{1}{2}x\right)\mathop{% \mathrm{ber}\/}\nolimits x+\tfrac{1}{4}\pi\mathop{\mathrm{bei}\/}\nolimits x+% \sum_{k=0}^{\infty}(-1)^{k}\frac{\mathop{\psi\/}\nolimits\!\left(2k+1\right)}{% ((2k)!)^{2}}(\tfrac{1}{4}x^{2})^{2k},$ $\displaystyle\mathop{\mathrm{kei}\/}\nolimits x$ $\displaystyle=-\mathop{\ln\/}\nolimits\!\left(\tfrac{1}{2}x\right)\mathop{% \mathrm{bei}\/}\nolimits x-\tfrac{1}{4}\pi\mathop{\mathrm{ber}\/}\nolimits x+% \sum_{k=0}^{\infty}(-1)^{k}\frac{\mathop{\psi\/}\nolimits\!\left(2k+2\right)}{% ((2k+1)!)^{2}}(\tfrac{1}{4}x^{2})^{2k+1}.$

## §10.65(iii) Cross-Products and Sums of Squares

 10.65.6 ${\mathop{\mathrm{ber}_{\nu}\/}\nolimits^{2}}x+{\mathop{\mathrm{bei}_{\nu}\/}% \nolimits^{2}}x=(\tfrac{1}{2}x)^{2\nu}\sum_{k=0}^{\infty}\frac{1}{\mathop{% \Gamma\/}\nolimits\!\left(\nu+k+1\right)\mathop{\Gamma\/}\nolimits\!\left(\nu+% 2k+1\right)}\frac{(\frac{1}{4}x^{2})^{2k}}{k!},$
 10.65.7 $\mathop{\mathrm{ber}_{\nu}\/}\nolimits x\mathop{\mathrm{bei}_{\nu}\/}\nolimits% 'x-\mathop{\mathrm{ber}_{\nu}\/}\nolimits'x\mathop{\mathrm{bei}_{\nu}\/}% \nolimits x=(\tfrac{1}{2}x)^{2\nu+1}\sum_{k=0}^{\infty}\frac{1}{\mathop{\Gamma% \/}\nolimits\!\left(\nu+k+1\right)\mathop{\Gamma\/}\nolimits\!\left(\nu+2k+2% \right)}\frac{(\frac{1}{4}x^{2})^{2k}}{k!},$
 10.65.8 $\mathop{\mathrm{ber}_{\nu}\/}\nolimits x\mathop{\mathrm{ber}_{\nu}\/}\nolimits% 'x+\mathop{\mathrm{bei}_{\nu}\/}\nolimits x\mathop{\mathrm{bei}_{\nu}\/}% \nolimits'x=\tfrac{1}{2}(\tfrac{1}{2}x)^{2\nu-1}\sum_{k=0}^{\infty}\frac{1}{% \mathop{\Gamma\/}\nolimits\!\left(\nu+k+1\right)\mathop{\Gamma\/}\nolimits\!% \left(\nu+2k\right)}\frac{(\frac{1}{4}x^{2})^{2k}}{k!},$
 10.65.9 $\left(\mathop{\mathrm{ber}_{\nu}\/}\nolimits'x\right)^{2}+\left(\mathop{% \mathrm{bei}_{\nu}\/}\nolimits'x\right)^{2}=(\tfrac{1}{2}x)^{2\nu-2}\sum_{k=0}% ^{\infty}\frac{2k^{2}+2\nu k+\frac{1}{4}\nu^{2}}{\mathop{\Gamma\/}\nolimits\!% \left(\nu+k+1\right)\mathop{\Gamma\/}\nolimits\!\left(\nu+2k+1\right)}\frac{(% \frac{1}{4}x^{2})^{2k}}{k!}.$

## §10.65(iv) Compendia

For further power series summable in terms of Kelvin functions and their derivatives see Hansen (1975).