suppose we have the following two sequences


$$\alpha\_k = (k-1)\left(1-\frac {1}{1+(k+1)l}\right) \quad , k \geq 2$$
$$\beta\_k = (k-1)\left(1+\frac {1}{1+(k-1)l}\right) \quad , k \geq 2$$


where $l$ is a positive constant
and define the sequence $c\_k$ recursively by:
$$c\_2 = - 1/\beta\_2 $$
$$c\_3 = 0 $$
$$c\_{k+1} = \frac{\alpha\_{k-1}}{\beta\_{k+1}}c\_{k-1} \quad , k \geq 3 $$


it is not hard to see that this would give 
$$c\_2 = - 1/\beta\_2$$
$$c\_{2k} = -\frac{\alpha\_2}{\beta\_2 }\cdot\frac{\alpha\_4}{\beta\_4 }\cdot\cdot\cdot \frac{\alpha\_{2k-2}}{\beta\_{2k-2} }\cdot \frac{1}
{\beta\_{2k}} \quad , k \geq 2$$


$$c\_{2k+1} = 0 \quad , k \geq 1 $$


apparently we must have $d\_k = c\_{2k} \sim k^{-(1+1/l)}$ but I have no idea how to show this. can anyone shed some light on this?


also a similar question for the recursion
$$c\_2 = - 1/\beta\_2 $$
$$c\_3 = \frac{\gamma\_2}{\beta\_3}c\_2 $$
$$c\_{k+1} = \frac{\alpha\_{k-1}}{\beta\_{k+1}}c\_{k-1} + \frac{\gamma\_k}{\beta\_{k+1}}c\_k \quad , k \geq 3 $$


where 
$$\gamma\_k = \sigma \frac{l(k^3-k)}{1+kl} \quad , k \geq 2$$
$\sigma$ being also a positive constant 


How would the asymptotics look like in this case?



Asymptotics of a recursion
For the first sequence, $$d\_k=\frac{\alpha\_{2k-2}}{\beta\_{2k}}d\_{k-1}=\frac{k-3/2}{k+1/2-1/\ell}d\_{k-1}$$
so one has 
$$d\_k=C\frac{\Gamma(k-1/2)}{\Gamma(k+3/2-1/\ell)}\ ,$$
the constant $C$ being determined by the initial condition $d\_1$, namely
$$C=d\_1\frac{\Gamma(5/2- 1/\ell)}{\Gamma(1/2)}\ . $$ Recall that $\Gamma(x+a)=x^a\Gamma(x)(1+o(1))$ as $x\to+\infty$, so
$$d\_k=Ck^{-1-1/\ell}(1+o(1)). $$


For the second sequence, $$c\_{k+1}=\frac{(k-1)(k+1)}{k+2/\ell }c\_k+\frac{k-2}{k+2/\ell}c\_{k-1}$$
which implies $c\_{k-1}/c\_k=O(1/k)$; if we plug this in the recursion again we have
$$c\_{k+1}=\frac{(k-1)(k+1)}{k+2/\ell }(1+O(1/k^2))\ , $$
whence $$c\_k= A\frac{\Gamma(k+1)\Gamma(k-1)}{\Gamma(k+2/\ell) }(1+o(1))\ ,$$ because the infinite product of $1+O(1/k^2)$ is convergent. By the Stirling formula $$c\_k= B k^{k-1/2+2/\ell}e^{-k} (1+o(1))$$ for a certain constant $B$. 


