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Mathematica 14.0 on Windows produces

Sum[x^(9 + 6 k)/(9 + 6 k)!, {k, 0, Infinity}]

1/6 x^3 (-1 + HypergeometricPFQ[{}, {2/3, 5/6, 7/6, 4/3, 3/2}, x^6/46656])

The above is an analytical expression (see Wiki for the definitions), whereas Maple 2024 produces a closed-form expression for it $$ \frac{x^{9} \left(-\frac{60480}{x^{6}}+\frac{60480 \,{\mathrm e}^{x}}{x^{9}}-\frac{120960 \,{\mathrm e}^{\frac{x}{2}} \cos \left(\frac{\sqrt{3}\, x}{2}\right)}{x^{9}}+\frac{120960 \,{\mathrm e}^{-\frac{x}{2}} \cos \left(\frac{\sqrt{3}\, x}{2}\right)}{x^{9}}-\frac{60480 \,{\mathrm e}^{-x}}{x^{9}}\right)}{362880}. $$

The question arises: how to obtain that closed-form expression with Mathematica?

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1 Answer 1

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Maple:

s:=sum(x^(9 + 6*k)/(9 + 6*k)!, k=0..infinity)
simplify(convert(s,trig))

enter image description here

Mathematica V 14:

s = Sum[x^(9 + 6  k)/(9 + 6  k)!, {k, 0, Infinity}];
FunctionExpand[s];
ExpToTrig[%]

enter image description here

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    $\begingroup$ Don't hurry. The Maple result is obtained by s := evalc(sum(x^(6*k + 9)/(6*k + 9)!, k = 0 .. infinity)). The same in Mathematica is obtained by s = Sum[x^(9 + 6 k)/(9 + 6 k)!, {k, 0, Infinity}]; ComplexExpand[FunctionExpand[s]]. Series multisection is used by Maple. $\endgroup$
    – user64494
    Commented May 28 at 8:52
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    $\begingroup$ Don't hurry. I was in middle of making coeffee so I was in a bit of a rush and did not have time to try all other options. $\endgroup$
    – Nasser
    Commented May 28 at 10:27
  • $\begingroup$ Are the two expressions equivalent? The last term seems different... $\endgroup$
    – mattiav27
    Commented May 28 at 12:35
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    $\begingroup$ +1 s // FunctionExpand // FullSimplify gives a cleaner form, i.e., 1/6 (-x^3 - 4 Cos[(Sqrt[3] x)/2] Sinh[x/2] + 2 Sinh[x]) $\endgroup$
    – Bob Hanlon
    Commented May 28 at 20:00

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