How to show that $$ \frac{4^n}{n^{3/2}\sqrt \pi} $$ has not the form $p_1(n) \lambda_1^n + \ldots p_i(n) \lambda_i^{n}$ for some polynomials $p_i(n)$ and numbers $\lambda_i$?
2026-05-15 19:05:50.1778871950
How to show that $\frac{4^n}{n^{3/2}\sqrt \pi}$ could not be expressed as $\sum_i^m p_i(n)\lambda_i^n$
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The Laplace transform of $p(x) K^x$ is a rational function whose singularities are poles.
The Laplace transform of $\frac{4^x}{x^{3/2}}$ has a branch point (of the $C\sqrt{s-2\log 2}$ kind) at $s=2\log 2$, hence $\frac{4^x}{x^{3/2}}$ and $\sum p_m(x) K_m^x$ cannot be the same function.