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if I'm not mistaken $\partial H(\lambda,x)/\partial x\rightarrow 2^{\lambda} \lambda x^{\lambda-1}$ for $x\rightarrow \infty$, so the $a_2$ term vanishes for any $\lambda<1$; so if the condition you are looking for exists, it can only involve $\lambda$, and then the condition seems to be $\lambda=-2,-4,-6,\ldots$.