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How to write the hypergeometric function ${}_2F_0$ in Mathematica?

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    $\begingroup$ HypergeometricPFQ[{a, b}, {}, z] is the straight answer, but depending on what you're doing, you may be better off with formulating things in terms of Tricomi's function, HypergeometricU[]. $\endgroup$ Commented Aug 1, 2022 at 19:44
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    $\begingroup$ @Roman, ${}_2F_0$ is indeed a function defined as the analytic continuation of a certain divergent power series (see e.g. this). $\endgroup$ Commented Aug 1, 2022 at 21:09
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    $\begingroup$ @Hans: the correct statement is that if any of a and b in HypergeometricPFQ[{a, b}, {}, z] are nonpositive integers, then you have a polynomial result (since all the Pochhammer symbols after a certain point zero out). $\endgroup$ Commented Aug 1, 2022 at 21:13
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    $\begingroup$ Some comments appeared while I was typing... HypergeometricPFQ[{a, b}, {}, z] is divergent (just like all $_pF_q$ with $p-q\geq2$), but can be expanded into a power series around $z=0$. Only if $a$ or $b$ is a negative integer, the series stops when the Pochhammer symbol hits zero, leading to polynomials. Note that the divergent series can be regularized, and Sum[Pochhammer[a, k] Pochhammer[b, k] z^k/k!, {k, 0, ∞}, Regularization -> "Borel"] gives (-(1/z))^a HypergeometricU[a, 1 + a - b, -(1/z)] $\endgroup$
    – Fred Hucht
    Commented Aug 1, 2022 at 21:16
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    $\begingroup$ ...and the last expression in @Fred's comment is exactly why I said it might be better to reformulate in terms of HypergeometricU[]. $\endgroup$ Commented Aug 1, 2022 at 21:19

1 Answer 1

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Based on the many comments, it appears that the correct function is

HypergeometricPFQ[{a, b}, {}, z] 

You can confirm the correspondence to the traditional form

HypergeometricPFQ[{a, b}, {}, z] // TraditionalForm

This outputs something like $$\, _2F_0(a,b;;z)$$

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