Expressing one generating function like combination of another generating functions.

Question

Let $A (t), B (t)$, and $C (t)$ - generating functions for sequences $a_0, a_1, a_2,\dots; b_0, b_1, b_2,\dots and\ c_0, c_1, c_2,\dots$ Express $C (t)$ through $A (t)$ and $B (t)$, if $c_n=\sum_{j+4k<=n}a_jb_k$ . I tried to build series for $c_n$ and then express from it series for $a_n$ and $b_n$ but without success. Please help me. Thank you in advance.

Answer 1

If it's not easy to see the connection between $C(x)=\sum_{n=0}^{\infty}c_nx^n$ with $A(x)=\sum_{j=0}^{\infty}a_jx^j$ and $B(x)=\sum_{k=0}^{\infty}c_kx^k$ we could look at the coefficients $c_n$ for small $n$ and check if we detect some pattern or regularity.

Assuming $c_n$ is given by \begin{align*} c_n=\sum_{{j+4k=n}\atop{j,k\geq 0}}a_jb_k\qquad n\geq 0 \end{align*}

we see:

\begin{align*} c_0=a_0b_0\qquad&c_4=a_4b_0+a_0b_1&c_8=a_8b_0+a_4b_1+a_0b_2\\ c_1=a_1b_0\qquad&c_5=a_5b_0+a_1b_1&\ldots\qquad\qquad\qquad\qquad\\ c_2=a_2b_0\qquad&c_6=a_6b_0+a_2b_1&\\ c_3=a_3b_0\qquad&c_7=a_7b_0+a_3b_1&\\ \end{align*}

We observe the index of $b_n$ is increasing by one whenever the index of $c_n$ is increased by $4$. So let's start with:

\begin{align*} B(x^4)=b_0+b_1x^4+b_2x^8+\ldots \end{align*}

Multiplication of $A(x)$ and $B(x^4)$ results in:

\begin{align*} A(x)B(x^4)&=\left(\sum_{j=0}^{\infty}a_jx^j\right)\left(\sum_{k=0}^{\infty}b_kx^{4k}\right)\\ &=\sum_{n=0}^{\infty}\sum_{j+4k=n}a_jx^jb_kx^{4k}\\ &=\sum_{n=0}^{\infty}\sum_{j+4k=n}a_jb_kx^{n}\\ &=\sum_{n=0}^{\infty}c_nx^n\\ &=C(x) \end{align*}

and we see that no more adaptations are necessary and we're already finished.