Combining gamma functions

Hi I have customer revenue data that I am modelling with gamma functions (coding in python)
I want to combine my gamma functions to look at spending in a whole year - so I was hoping someone could confirm the theory on this:

combining two Gamma functions is the convolution of the two functions?

I believe Gamma(a,2b) = 2Gamma(a,b)

if I combine Gamma(a,2b) and Gamma(c,3b) is this equivalent to (2+3)Gamma(a+c,b)?

Is this correct in adding values from several distributions, or is there a more complicated way of doing this correctly?

Thanks for you answers!