5
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I have the following term which I would like to express correctly in Mathematica:

$$ \prod_{j=1,j\neq i}^m(\rho_i-\rho_j) $$

Can you please help?

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  • 1
    $\begingroup$ The question is unclearly formulated. Is it a product by $j$ or by $i,j$? $\endgroup$ – user64494 Jun 3 at 6:36
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Long form:

Product[(ρ[i] - ρ[j]), {i, 1, m}, {j, i + 1, m}] Product[(ρ[i] - ρ[j]), {i, 1, m}, {j, 1, i - 1}]

By observing that each factor occurs twice (up to sign):

Product[ - (ρ[i] - ρ[j])^2, {i, 1, m}, {j, i + 1, m}]

By counting the signs:

(-1)^(m (m - 1)/2) Product[(ρ[i] - ρ[j])^2, {i, 1, m}, {j, i + 1, m}]
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  • $\begingroup$ Thank you so much. You mean without the ^2 cous its not in the question, but it helped. $\endgroup$ – Y.L Jun 2 at 20:51
  • $\begingroup$ Yes the square is in the question: every term occurs twice (with opposite signs). There should be a $(-1)^{m(m-1)/2}$ factor too, to account for these signs. $\endgroup$ – Roman Jun 2 at 20:59
  • $\begingroup$ @Roman Huh, I completely forgot about the sign. Thanks for pointing this out! $\endgroup$ – Henrik Schumacher Jun 2 at 21:01
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Perhaps

ClearAll[ prod]
prod[m_, i_] := Product[ρ[i] - ρ[j], {j, DeleteCases[Range[m], i]}]

prod[5, 2]

(-ρ[1] + ρ[2]) (ρ[2] - ρ[3]) (ρ[2] - ρ[4]) (ρ[2] - ρ[5])

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