Trying to calculate derivatives of spherical harmonics

Question

I am trying to confirm what is shown on this website: http://functions.wolfram.com/Polynomials/SphericalHarmonicY/20/01/01/0001/

But when I type

D[SphericalHarmonicY[n, m, ϑ, φ], ϑ] 

Mathematica gives me this expression:

m Cot[ϑ] SphericalHarmonicY[n, m, ϑ, φ] + (
E^(-I φ) Sqrt[Gamma[1 - m + n]] Sqrt[Gamma[2 + m + n]]
SphericalHarmonicY[n, 1 + m, ϑ, φ])/(
Sqrt[Gamma[-m + n]] Sqrt[Gamma[1 + m + n]])

I tried to check if that expression is equal to what the website says it should be, by placing each expression in between

===

but it says False. Using just two equal signs doesn't give me any evaluation.

How can Mathematica give a different answer to the one on the website? I would really like to know what the answer to this partial derivative is and would prefer to use one that doesn't involve Gamma's, hence my desire to reproduce the expression given in the website.

Any help would be much appreciated!

Answer 1

The Gamma function has a property that

$ \Gamma(x+1)=x*\Gamma(x) $

So in the result that Mathematica returns

$ \sqrt{\frac{\Gamma(n-m+1)\Gamma(n+m+2)}{\Gamma(n-m)\Gamma(n+m+1)}} = \sqrt{\frac{\Gamma(n-m+1)}{\Gamma(n-m)}}\times\sqrt{\frac{\Gamma(n+m+2)}{\Gamma(n+m+1)}} = \sqrt{(n-m)(n+m+1)}$

which is exactly the result shown in the website.