# §28.16 Asymptotic Expansions for Large $q$

Let $s=2m+1$, $m=0,1,2,\dots$, and $\nu$ be fixed with $m<\nu. Then as $h(=\sqrt{q})\to+\infty$

 28.16.1 $\mathop{\lambda_{\nu}\/}\nolimits\!\left(h^{2}\right)\sim-2h^{2}+2sh-\dfrac{1}% {8}(s^{2}+1)-\dfrac{1}{2^{7}h}(s^{3}+3s)-\dfrac{1}{2^{12}h^{2}}(5s^{4}+34s^{2}% +9)-\dfrac{1}{2^{17}h^{3}}(33s^{5}+410s^{3}+405s)-\dfrac{1}{2^{20}h^{4}}(63s^{% 6}+1260s^{4}+2943s^{2}+486)-\dfrac{1}{2^{25}h^{5}}(527s^{7}+15617s^{5}+69001s^% {3}+41607s)+\cdots.$

For graphical interpretation, see Figures 28.13.1 and 28.13.2.