# §28.22 Connection Formulas

## §28.22(i) Integer $\nu$

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

The joining factors in the above formulas are given by

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

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

## §28.22(ii) Noninteger $\nu$

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

Here $\operatorname{me}_{\nu}\left(0,h^{2}\right)$ $(\neq 0)$ is given by (28.14.1) with $z=0$, and ${\operatorname{M}^{(1)}_{\nu}}\left(0,h\right)$ 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 ${\operatorname{M}^{(2)}_{\nu}}\left(z,h\right)=\cot\left(\nu\pi\right){% \operatorname{M}^{(1)}_{\nu}}\left(z,h\right)-\frac{1}{\sin\left(\nu\pi\right)% }{\operatorname{M}^{(1)}_{-\nu}}\left(z,h\right).$