Express generating function g(x) using f(x)

Question

Let f(x) be generating function of the sequence {a_n} and g(x) for the sequence {b_n}. In following cases, express g(x) using f(x):

a) b₃ = 3; b_n = a_n; n≠3

b) b₁ = 1; b3 = 3; b_n = 2a_n, n≠1.3

c) b₁ = 1; b_n = 2a_n + 5, n≠1

I searched literally whole internet a I could not find any similar problem. Can you please help me?

Answer 1

Applying the definitions, you have $$f(x)=a_0+a_1x+a_2x^2+a_3x^3+a_4x^4+\dotsb$$ and $$g(x)=a_0+a_1x+a_2x^2+3x^3+a_4x^4+\dotsb$$ from which you obtain $$ g(x)=f(x)+(3-a_3)x^3 $$ noting that $$ f^{(3)}(0)=6{a_3} $$ you can express $a_3$ in terms of $f$ and close the first question.

The others seem similar.