JEE Advanced 2020 Mathematics question

For nonnegative integers and , let

=

For positive integers and , let

g(m, n) =

where for any nonnegative integer ,

f(m, n, p) = .

Then which of the following statements is/are TRUE?

• (A) for all positive integers Correct
• (B) for all positive integers Correct
• (C) for all positive integers
• (D) for all positive integers Correct