# §28.22 Connection Formulas

## §28.22(i) Integer $\nu$

 28.22.1 $\displaystyle\mathop{{\mathrm{Mc}^{(1)}_{m}}\/}\nolimits\!\left(z,h\right)$ $\displaystyle=\sqrt{\dfrac{2}{\pi}}\dfrac{1}{g_{\mathit{e},m}(h)\mathop{% \mathrm{ce}_{m}\/}\nolimits\!\left(0,h^{2}\right)}\mathop{\mathrm{Ce}_{m}\/}% \nolimits\!\left(z,h^{2}\right),$ 28.22.2 $\displaystyle\mathop{{\mathrm{Ms}^{(1)}_{m}}\/}\nolimits\!\left(z,h\right)$ $\displaystyle=\sqrt{\dfrac{2}{\pi}}\frac{1}{g_{\mathit{o},m}(h)\mathop{\mathrm% {se}_{m}\/}\nolimits'\!\left(0,h^{2}\right)}\mathop{\mathrm{Se}_{m}\/}% \nolimits\!\left(z,h^{2}\right),$ 28.22.3 $\displaystyle\mathop{{\mathrm{Mc}^{(2)}_{m}}\/}\nolimits\!\left(z,h\right)$ $\displaystyle=\sqrt{\frac{2}{\pi}}\dfrac{1}{g_{\mathit{e},m}(h)\mathop{\mathrm% {ce}_{m}\/}\nolimits\!\left(0,h^{2}\right)}\*\left(-f_{\mathit{e},m}(h)\mathop% {\mathrm{Ce}_{m}\/}\nolimits\!\left(z,h^{2}\right)+\dfrac{2}{\pi C_{m}(h^{2})}% \mathop{\mathrm{Fe}_{m}\/}\nolimits\!\left(z,h^{2}\right)\right),$ 28.22.4 $\displaystyle\mathop{{\mathrm{Ms}^{(2)}_{m}}\/}\nolimits\!\left(z,h\right)$ $\displaystyle=\sqrt{\frac{2}{\pi}}\dfrac{1}{g_{\mathit{o},m}(h)\mathop{\mathrm% {se}_{m}\/}\nolimits'\!\left(0,h^{2}\right)}\*\left(-f_{\mathit{o},m}(h)% \mathop{\mathrm{Se}_{m}\/}\nolimits\!\left(z,h^{2}\right)-\dfrac{2}{\pi S_{m}(% h^{2})}\mathop{\mathrm{Ge}_{m}\/}\nolimits\!\left(z,h^{2}\right)\right).$

The joining factors in the above formulas are given by

 28.22.5 $\displaystyle g_{\mathit{e},2m}(h)$ $\displaystyle=(-1)^{m}\sqrt{\dfrac{2}{\pi}}\dfrac{\mathop{\mathrm{ce}_{2m}\/}% \nolimits\!\left(\frac{1}{2}\pi,h^{2}\right)}{A_{0}^{2m}(h^{2})},$ 28.22.6 $\displaystyle g_{\mathit{e},2m+1}(h)$ $\displaystyle=(-1)^{m+1}\sqrt{\frac{2}{\pi}}\dfrac{\mathop{\mathrm{ce}_{2m+1}% \/}\nolimits'\!\left(\frac{1}{2}\pi,h^{2}\right)}{hA_{1}^{2m+1}(h^{2})},$ 28.22.7 $\displaystyle g_{\mathit{o},2m+1}(h)$ $\displaystyle=(-1)^{m}\sqrt{\dfrac{2}{\pi}}\dfrac{\mathop{\mathrm{se}_{2m+1}\/% }\nolimits\!\left(\frac{1}{2}\pi,h^{2}\right)}{hB_{1}^{2m+1}(h^{2})},$ 28.22.8 $\displaystyle g_{\mathit{o},2m+2}(h)$ $\displaystyle=(-1)^{m+1}\sqrt{\dfrac{2}{\pi}}\dfrac{\mathop{\mathrm{se}_{2m+2}% \/}\nolimits'\!\left(\frac{1}{2}\pi,h^{2}\right)}{h^{2}B_{2}^{2m+2}(h^{2})},$ 28.22.9 $\displaystyle f_{\mathit{e},m}(h)$ $\displaystyle=-\sqrt{\ifrac{\pi}{2}}g_{\mathit{e},m}(h)\mathop{{\mathrm{Mc}^{(% 2)}_{m}}\/}\nolimits\!\left(0,h\right),$ 28.22.10 $\displaystyle f_{\mathit{o},m}(h)$ $\displaystyle=-\sqrt{\ifrac{\pi}{2}}g_{\mathit{o},m}(h)\mathop{{\mathrm{Ms}^{(% 2)}_{m}}\/}\nolimits'\!\left(0,h\right),$

where $A_{n}^{m}(h^{2})$, $B_{n}^{m}(h^{2})$ are as in §28.4(i), and $C_{m}(h^{2})$, $S_{m}(h^{2})$ are as in §28.5(i). Furthermore,

 28.22.11 $\displaystyle\mathop{{\mathrm{Mc}^{(2)}_{m}}\/}\nolimits'\!\left(0,h\right)$ $\displaystyle=\sqrt{\ifrac{2}{\pi}}g_{\mathit{e},m}(h),$ $\displaystyle\mathop{{\mathrm{Ms}^{(2)}_{m}}\/}\nolimits\!\left(0,h\right)$ $\displaystyle=-\sqrt{\ifrac{2}{\pi}}g_{\mathit{o},m}(h),$
 28.22.12 $\displaystyle\mathop{\mathrm{fe}_{m}\/}\nolimits'\!\left(0,h^{2}\right)$ $\displaystyle=\tfrac{1}{2}\pi C_{m}(h^{2})\left(g_{\mathit{e},m}(h)\right)^{2}% \mathop{\mathrm{ce}_{m}\/}\nolimits\!\left(0,h^{2}\right),$ $\displaystyle\mathop{\mathrm{ge}_{m}\/}\nolimits\!\left(0,h^{2}\right)$ $\displaystyle=\tfrac{1}{2}\pi S_{m}(h^{2})\left(g_{\mathit{o},m}(h)\right)^{2}% \mathop{\mathrm{se}_{m}\/}\nolimits'\!\left(0,h^{2}\right).$

## §28.22(ii) Noninteger $\nu$

 28.22.13 $\mathop{{\mathrm{M}^{(1)}_{\nu}}\/}\nolimits\!\left(z,h\right)=\frac{\mathop{{% \mathrm{M}^{(1)}_{\nu}}\/}\nolimits\!\left(0,h\right)}{\mathop{\mathrm{me}_{% \nu}\/}\nolimits\!\left(0,h^{2}\right)}\mathop{\mathrm{Me}_{\nu}\/}\nolimits\!% \left(z,h^{2}\right).$

Here $\mathop{\mathrm{me}_{\nu}\/}\nolimits\!\left(0,h^{2}\right)$ $(\neq 0)$ is given by (28.14.1) with $z=0$, and $\mathop{{\mathrm{M}^{(1)}_{\nu}}\/}\nolimits\!\left(0,h\right)$ is given by (28.24.1) with $j=1$, $z=0$, and $n$ chosen so that $|c_{2n}^{\nu}(h^{2})|=\max(|c_{2\ell}^{\nu}(h^{2})|)$, where the maximum is taken over all integers $\ell$.

 28.22.14 $\mathop{{\mathrm{M}^{(2)}_{\nu}}\/}\nolimits\!\left(z,h\right)=\mathop{\cot\/}% \nolimits\!\left(\nu\pi\right)\mathop{{\mathrm{M}^{(1)}_{\nu}}\/}\nolimits\!% \left(z,h\right)-\frac{1}{\mathop{\sin\/}\nolimits\!\left(\nu\pi\right)}% \mathop{{\mathrm{M}^{(1)}_{-\nu}}\/}\nolimits\!\left(z,h\right).$