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 $i,j$? $\endgroup$– user64494Commented Jun 3, 2019 at 6:36
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2 Answers
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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}]
answered Jun 2, 2019 at 20:42
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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.LCommented Jun 2, 2019 at 20:51
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$\begingroup$ 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. $\endgroup$– RomanCommented Jun 2, 2019 at 20:59
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$\begingroup$ @Roman Huh, I completely forgot about the sign. Thanks for pointing this out! $\endgroup$ Commented Jun 2, 2019 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])
