Can anyone help me evaluate $f(3)$ with $f(x):=\prod_{k=2}^\infty (1+\frac{2}{x^2-(8k-3)x+4k(4k-3)})$? Thanks in advance.
2026-04-05 18:34:51.1775414091
Help required in limits
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Hint: Substituting $x=3$ in the given term $$1+\frac{2}{x^2-(8k-3)x+4k(4k-3)}$$ and factorizing we get $$2\,{\frac { \left( k-1 \right) \left( 4\,k-5 \right) }{ \left( 4\,k-3 \right) \left( 2\,k-3 \right) }} $$ and the infinite product is given by $$1/4\,{\frac {{\pi}^{3/2}\sqrt {2}}{ \left( \Gamma \left( 3/4 \right) \right) ^{2}}} $$