# §30.5 Functions of the Second Kind

Other solutions of (30.2.1) with $\mu=m$, $\lambda=\mathop{\lambda^{m}_{n}\/}\nolimits\!\left(\gamma^{2}\right)$, and $z=x$ are

 30.5.1 $\mathop{\mathsf{Qs}^{m}_{n}\/}\nolimits\!\left(x,\gamma^{2}\right),$ $n=m,m+1,m+2,\dots$.

They satisfy

 30.5.2 $\mathop{\mathsf{Qs}^{m}_{n}\/}\nolimits\!\left(-x,\gamma^{2}\right)=(-1)^{n-m+% 1}\mathop{\mathsf{Qs}^{m}_{n}\/}\nolimits\!\left(x,\gamma^{2}\right),$

and

 30.5.3 $\mathop{\mathsf{Qs}^{m}_{n}\/}\nolimits\!\left(x,0\right)=\mathop{\mathsf{Q}^{% m}_{n}\/}\nolimits\!\left(x\right);$

compare §14.3(i). Also,

 30.5.4 $\mathop{\mathscr{W}\/}\nolimits\left\{\mathop{\mathsf{Ps}^{m}_{n}\/}\nolimits% \!\left(x,\gamma^{2}\right),\mathop{\mathsf{Qs}^{m}_{n}\/}\nolimits\!\left(x,% \gamma^{2}\right)\right\}=\frac{(n+m)!}{(1-x^{2})(n-m)!}A_{n}^{m}(\gamma^{2})A% _{n}^{-m}(\gamma^{2})\quad(\neq 0),$

with $A_{n}^{\pm m}(\gamma^{2})$ as in (30.11.4).

For further properties see Meixner and Schäfke (1954) and §30.8(ii).