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For positive integer $D$, define $j(D)=\sum_{n=1}^\infty \frac{1}{(n^D)!}$.

For $D \le 6$, sage finds closed form in terms of hypergeometric functions at algrebraic arguments and fails to find closed form for $D>6$.

According to sage $j(2)= \,_1F_4\left(\begin{matrix} 1 \\ -i + 1,i + 1,-i \, \sqrt{2} + 1,i \, \sqrt{2} + 1 \end{matrix} ; 1 \right)$

The length of the latex sage's closed form of $j(6)$ is 2534 characters long.

Sage's results in machine readable form are available

Wolfram Alpha can't solve $D>1$ and chatGPT "believes" there isn't simple closed form for $j(2)$.

Q1 Are sage's results correct?

Q2 Is there closed form for $D>6$?

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    $\begingroup$ All these results cannot be true since $(n^2)!$ grows too fast to be expressible by a finite product of pochammer symbols, so no hypergeometric evaluation is possible. In addition, since all the series converge extremely fast, you can immediately check that the result for $D=2$ is numerically wrong. $\endgroup$
    – Henri Cohen
    Commented Feb 12 at 11:10
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    $\begingroup$ for $d=2$ your hypergeometric function evaluates to 1.17227, while the correct answer is 1.04167 $\endgroup$
    – Carlo Beenakker
    Commented Feb 12 at 11:13
  • $\begingroup$ The bug comes from Maxima's simplify_sum command used by Sage, see groups.google.com/g/sage-devel/c/E-JooEu5QTo/m/uR3RCXP8AQAJ $\endgroup$
    – Dima Pasechnik
    Commented Feb 14 at 19:42

1 Answer 1

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The bug is in Maxima's simplify_sum - which is used by SageMath

(%i4) load("simplify_sum");
(%o4) "/usr/share/maxima/5.45.1/share/solve_rec/simplify_sum.mac"

(%i5) sum(1/factorial(n^2), n, 1, inf), simpsum;
(%o5) 'sum(1/(n^2)!,n,1,inf)

(%i6) simplify_sum(%);
1/'product(n^2+%,%,1,2*n+1) non-rational term ratio to nusum
1/'product(n^2+%,%,1,2*n+1) non-rational term ratio to nusum
(%o6) %f[1,4]([1],[1-%i,%i+1,1-sqrt(2)*%i,sqrt(2)*%i+1],1)
```
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