Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

If $$A = \sum_{n\geq 0} a_nx^n$$ $$B = \sum_{n\geq 0} b_nx^n$$ and $$xA = B+x$$.

If I now want to express $a_n$ using $b_n$ how does the $x$-term in the RHS of $xA = B+x$ come into play? since Is $a_i = b_{i+1}$ for all $i \neq 1$ and $a_1 = b_2 +1$ $x$ has the generating function $0,1,0,0,0,0....$?

share|improve this question

1 Answer 1

up vote 2 down vote accepted

Almost: your indexing is off by $1$.

From $xA=B+x$ you have $$x\sum_{n\ge 0}a_nx^n=x+\sum_{n\ge 0}b_nx^n\;,$$ so $$\sum_{n\ge 0}a_nx^{n+1}=b_0+(b_1+1)x+\sum_{n\ge 2}b_nx^n\;.$$ Equivalently, $$a_0x+\sum_{n\ge 2}a_{n-1}x^n=b_0+(b_1+1)x+\sum_{n\ge 2}b_nx^n\;,$$ and now it’s easy to equate coefficients: $b_0=0,a_0=b_1+1$, and $a_n=b_{n+1}$ for $n\ge 1$.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.