Closed formulas for the first nontrivial Verlinde series at r=±2

Let $a_r(w)$ and $b_r(w)$ be the Verlinde series associated to the rank parameter $r$. Introduce $t$ by

$$w=\frac{t(2+3t)^3}{16(1+t)^4}.$$

The $r=\pm2$ Verlinde-series conjecture. The first nontrivial series satisfy

$$a_{-2}(w)=\frac{1}{a_2(w)}=\frac{2+3t}{\sqrt{1+t}}\frac{1}{\sqrt{1+t}+\sqrt{1+3t}},$$ $$b_{-2}(w)=b_2(w)=4\sqrt{2+3t}\frac{(1+t)^{1/4}\sqrt{1+3t}}{(\sqrt{1+t}+\sqrt{1+3t})^{5/2}}.$$

These formulas predict the first nontrivial examples beyond the known identities $a_0=a_{\pm1}=b_0=b_{\pm1}=1$. The supplied text presents them as predictions derived using the solution of Lehn's conjecture; the parser marks the candidate as resolved, although the precise resolving result is not identified here.