Thanks Dr. Yuval. how I work on each one? I means if it's possible, add for (1) or (2) some methods to work on it.

@LogicLove Added more explanations on how to implement (1). Implementing (2) is for you to complete.

thanks Dr. +1. I Works on these to understand it. thanks Dr.

Dear Dr. I'm sorry for wasting your time. I try more links and do with myself but the second one is very hard. I read this article researchgate.net/profile/Varmo_Vene/publication/… but I couldn't get it. would you please provide hint for second one? I think it's similar to first but couldn't solve it up to yet.

I try more after +15 hours. nice solution is mentioned by you for first type. anyway thanks. maybe the second one is completely different, thanks again.

so because I didn't see any example I couldent understand it. It's very hard. I have a three proof on my note about it, but really your teaching is so interesting, I get first one completely, is it possible complete for type two?

Dr, you are right. I'm a TA and know it. but this complicated one is boring for me for lot's of effort to understand. thanks anyway, I accept it but If you prefer please add some direct hint for me. thanks for your valuable and value to answer me, Dr.

Infact my trouble is how we get show entire sequence is P.R?

I know f(n)=[f(0), f(1),...f(n-1)] n!=0 for f(n)=g(n,f(n)) is P.R. but I didn't understand how take it for type 2/

maybe I think it's solve for $f_2(0)=c$ but not for this mentioned in the type 2.

Dr Would you please explain more for me? How I can overcome to this challenge, specially when comes to entire sequence?

I accept it, please complete the second part for encoding. thanks Dr.

I read another answer that mentioned on another Math links, but I love to your method, please complete it for part 2 after 29 hours :)

Would you please complete the tricky encoding it for type (2) Dr ?

Dr This is not assignment, I see another answer but it's not satisfy me, because you solve the first one very nice and creative, two days I didn't sleep to learn it. I'm a TA on university. Just I want to learn. the purpose of this site is to learning not doing assignment. I try more but I confused. please kindly learn more to me. thanks again Dr. for wasting your time.

Dr. I do lots of search on some books, the topic course of value recursion is not have an enough example. I don't say give me a solution, I say your hint is not enough to reach the solution for me. the other link on MATH mentioned the solution, If I want solution, I didn't try to ask you for getting more hint to reach solution.

Let us continue this discussion in chat.

math site has a solution, but I want to figure out how reach to solution with your method that mentioned here. I failed with your method to reach the second one.

I didn't have enough reputation, so I couldn't chat,

please complete encoding. anyway thanks,