Find a sequence whose generating function is given.

Question

Chap4 q2) Of which sequence is $U(s)=(1-4pqs^2)^{\frac{-1}{2}}$ the generating function(where $0<p=1-q<1$)?

Solution: So we need to find out a sequence $u_0,u_1,...$ such that $u_0+u_1s+u_2s^2+...=(1-4pqs^2)^{\frac{-1}{2}}$. Given that $\sum_i ar^i=\frac{a}{1-r}$, I feel like I need to make the exponent of $(1-4pqs^2)$ be $-1$

Do you know the generalized binomial series expansion? – Peter Foreman Jul 06 '19 at 23:30 — Peter Foreman, Jul 06 '19 at 23:30
No unfortunately, my knowledge in math is very basic. – user 6663629 Jul 06 '19 at 23:32 — user 6663629, Jul 06 '19 at 23:32
I know the Taylor and McLaurin expansion. – user 6663629 Jul 06 '19 at 23:38 — user 6663629, Jul 06 '19 at 23:38
Oh sounds like a plan. I'll try – user 6663629 Jul 06 '19 at 23:40 — user 6663629, Jul 06 '19 at 23:40

score 1 · Answer 1 · answered Aug 02 '19 at 15:36

1

By the binomial series:

$\begin{align*} (1 + x)^{-1/2} &= \sum_{n \ge 0} \binom{-1/2}{n} x^n \\ &= \sum_{n \ge 0} \frac{(-1)^k}{2^{2 n}} \binom{2 n}{n} x^n \end{align*}$

answered Aug 02 '19 at 15:36

vonbrand

27,812

Find a sequence whose generating function is given.

1 Answers1