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In my MSE question, "Conjectured connection between $e$ and $\pi$ in a semidisk", the answer included

$$\prod_{k=1}^\infty\left[1-\big((k+1)^{1/3}-k^{1/3}\big)^3\right]\approx 0.96454\ldots.$$

Does this infinite product have a closed form?

By "closed form", I mean an expression in terms of known constants. I realize that this is still subjective, but I have found that certain kinds of geometrical constructions yield infinite products with closed forms (example1, example2, example3). The infinite product in this question also comes from one of these kinds of geometrical constructions, so I wonder if it also has a closed form.

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    $\begingroup$ Do you want an answer beyond "obviously not"? Proving that something doesn't have a closed form is close to impossible. I don't even know how to prove that $\sum_{k=1}^{\infty} 1/k^5$ doesn't have a closed form as a rational number. $\endgroup$
    – user491858
    Commented May 16 at 22:38
  • $\begingroup$ @user491858 If no one here provides a closed form, I'll assume that it probably doesn't have a closed form. $\endgroup$
    – Dan
    Commented May 16 at 22:39
  • $\begingroup$ @user491858 Isn't there a whole theory that is able to say, e.g., $\int e^{-x^2}dx$ does not have a "closed form"? $\endgroup$
    – mathworker21
    Commented May 17 at 0:10
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    $\begingroup$ @mathworker21 It is very much a different (and easier) question to say that a function does not have a closed form. Proving that $\log(x)$ is a transcendental function over $\mathbf{Q}(x)$ is easy. Proving that $\log(-1) = \pi i$ is transcendental is harder. $\endgroup$
    – user491858
    Commented May 17 at 0:34
  • $\begingroup$ @Dandif you check the OEIS? $\endgroup$
    – Max Lonysa Muller
    Commented May 17 at 9:10

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