\[w(n)\sim\frac{\sqrt{\pi}e^{\frac{1}{2}z}z^{\frac{1}{4}(4a-2b+1)}}{\Gamma\left(%
a\right)\Gamma\left(a+1-b\right)}n^{\frac{1}{4}(4a-2b-3)}e^{-2\sqrt{nz}},\]