estimating parameter using fitdistr function

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enter image description hereI have this original pdf... and I'm trying to estimate the parameters required in both dgamma and dnbinomial. my professor implied It'd be better if I use the fitdistr function according to the paper I'm working on. but the thing is Idk how to use it especially this it's a new distribution so I don't know how to call it within the fitdistr function. I tried the optim function and it provided me with results but it didn't provide me with the correct answer since we've done the paper work calculation and it is not what is required with the paper can you help me please?? thank you

I'd really appreciate your help

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