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The Gamma function has the property that $\Gamma(z+1)=\Gamma(z)z$, so this expression:

( Sqrt[Gamma[1 + l - m]] Sqrt[Gamma[2 + l + m]] )/(Sqrt[ Gamma[l - m]] Sqrt[Gamma[1 + l + m]])

should simplify to something like $\sqrt{(l-m)(l+m+1)}$ (if I am not mistaken), but FullSimplify with the Assumption that l and m are integers does not simplify it.

How can I simplify that expression?

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    $\begingroup$ It seems that you play with rules of calculating angular momentum in quantum mechanics. Anyway it might be interesting if you add a context where from your expression come out. $\endgroup$
    – Artes
    Commented Aug 8, 2020 at 19:41
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    $\begingroup$ @Artes, indeed, it almost looks like a ratio of the prefactors for spherical harmonics. $\endgroup$
    – J. M.'s missing motivation ♦
    Commented Aug 9, 2020 at 1:48

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For non-positive integers Gamma takes ComplexInfinity value, and so

FullSimplify[( Sqrt[Gamma[1 + l - m]] Sqrt[Gamma[2 + l + m]] )/(
               Sqrt[ Gamma[l - m]] Sqrt[Gamma[1 + l + m]]),
              (l|m) \[Element] Integers && l >= m>= 0]

  Sqrt[(l - m)(1 + l + m)]
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  • $\begingroup$ A simpler command FullSimplify[a, Assumptions -> l - m > 0 && l + m > -1] works here. $\endgroup$
    – user64494
    Commented Aug 8, 2020 at 9:44
  • $\begingroup$ @user64494 Mind your strong inequality, while the weak also is true. Nevertheless you should know what l and m means in quantum mechanics and why they are integer. $\endgroup$
    – Artes
    Commented Aug 8, 2020 at 9:55
  • $\begingroup$ @ Artes, you are not right: my assumptions are weaker than tours. A simpler example is x>0 is weaker than x\[Element] Integers &&x>0. Hope I am clear, however, don't hesitate to ask for further explanation in need. $\endgroup$
    – user64494
    Commented Aug 8, 2020 at 9:59
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    $\begingroup$ @user64494 They are not, you exclude the case l == m == 0 and since you don't understand my first comment, better back to school. $\endgroup$
    – Artes
    Commented Aug 8, 2020 at 10:06
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    $\begingroup$ @user64494 My arguments are emotional sometimes, this is not bad. You comments however are pointless. If you have something to add you should provide another answer or upvote this one. Since you've done neither this nor that I guess you are ill. My expectation is that you shouldn't litter up this site with invaluable comments. Nethertheless If they are, just provide another answer and persuade others that you have something constructive to say. You could not persuade me, perhaps you will be successful with others. $\endgroup$
    – Artes
    Commented Aug 8, 2020 at 19:22

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