# An infinite product. Is it the value of a special function?

Let E(k) be the Euler numbers. Then $$\prod _{k=1}^{\infty }{ \exp\left({\frac {4-{E} \left( 2\,k \right) }{4\,k \left( 4\,x+3 \right) ^{2\,k}}} \right)} = \frac {\Gamma \left( x+1/2 \right) \left( x+3/4 \right) ^{ 3/2}}{\Gamma \left( x+2 \right) }.$$ Whenever the infinite product converges (for example for $x\ge 2$) I suspect the equation to hold, but was not able to give a proof. Any information related to this product is welcome.

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