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For the first sequence, $$d_k=\frac{\alpha_{2k-2}}{\beta_{2k}}d_{k-1}=\frac{k-3/2}{k-1/2-1/\ell}d_{k-1}$$ so one has $$d_k=C\frac{\Gamma(k-1/2)}{\Gamma(k+1/2-1/\ell)}\ ,$$ the constant $C$ being determined by the initial condition $d_1$. Recall that $\Gamma(x+a)=x^a\Gamma(x)(1+o(1))$ as $x\to+\infty$, so $$d_k=Cx^{-1+1/\ell}(1+o(1)). $$