I have a question concerning the series:
$$c_n:= \sum_{k=0}^{\lfloor \frac{n}{2} \rfloor} \left(-a\right)^k \frac{1}{k!} \frac{b^{n-2k}}{\left(n-2k\right)!} ~ ~ ~ ~; ~ ~ ~ ~ a,b>0 ~ ~ ~ ; ~ ~ n\in\mathbb{N }$$
I could show, that $c_n$ converges to $0$ for $n$ to infinity, but I need a bound for the speed of decay (e.g. $\vert c_n \vert \leq 1/2^n$). If we destinguish between odd and even $n$ and apply wolfram alpha we get:
$$ c_n = \frac{4^n a^n ~ U(-n,1/2,b^2/4a)}{(2n)!} ~~~~\text{ ( n even )}$$ $$ c_n = \frac{4^n a^n ~ b ~ U(-n,3/2,b^2/4a)}{(2n+1)!} ~~~~\text{ ( n odd)}$$
Here $U$ denotes the confluent hypergeometric function. I know the definition of the confluent hypergeometric function as a series and I know that it is one of the solutions of Kummer's differential equation. Unfortunately I know nothing else about this topic. I look it up in the handbook of Abramowitz and Stegun and got the impression that there is a very big theory behind hypergeometric functions.
Is there maybe any elementary way to determine a good bound for the decay rate? The whole field of mathematics around the confluent hypergeometric function seems to be advanced and I guess it would take very long to get some basic knowledge. To be honest: all my standard textbooks do not cover this topic. By the way: is there any readable introduction to the field ?
With best regards, Mat
This is only a draft/sketch, which must be expanded at some stages. I think for your problem you should use formula 13.5.16 Abramowitz and Stegun to estimate the U values: $$ U(-n,1/2,b^2/4a) \sim \Gamma(\frac{1}{4} + n + \frac{1}{4})\Big(c_{even}(a,b)+O(1)\Big)$$ $$ U(-n,3/2,b^2/4a) \sim \Gamma(\frac{3}{4} + n + \frac{1}{4})\Big(c_{odd}(a,b)+O(1)\Big)$$ where $c_{odd}, c_{even}$ do not depend on $n$, but the $O(1)$ includes a $\cos(f\sqrt{n}+g)$ term. Plugging these estimates in your expressions for the $c_n$, Maple gets the asymptotic expressions $$ c_n = \frac{4^n a^n ~ U(-n,1/2,b^2/4a)}{(2n)!} \sim \frac{\sqrt{2}}{4 n}\left(\frac{ea}{n^{3/2}}\right)^n c'_{even}(a,b)~~~~\text{ ( n even )}$$ $$ c_n = \frac{4^n a^n ~ b ~ U(-n,3/2,b^2/4a)}{(2n+1)!} \sim \frac{\sqrt{2}}{4 n}\left(\frac{ea}{n}\right)^n c'_{odd}(a,b)~~~~\text{ ( n odd)}$$ which shows the decay rate for the $c_n$. But note: If had not made severe errors, the rate is much slower than $2^{-n}$.