I managed to do the first part, stuck in the part circled. Any help will be appreciated, thanks.
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Use the Rodrigues formula plus the following identity:
Does that give you some ideas?
Use your generating function
Then differentiating wrt x gives
so using the generating function again gives
Now, shifting indicies gives
since the first term of the series on the left is zero. Equating coefficients of u gives your result.
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