I'd like to know the asymptotic behaviour as $N\to\infty$ of the following sum


$$ Z\_N(x) := 2^{-N/2} \sum\_{k=0}^{N/2} \frac{N!}{k! (N-2k)!} (N-1)^{-k} (\sqrt{2} x)^{N-2k} $$


in order to compute $p(x):=\lim\_{N\to\infty} \frac{log(Z\_N(x))}{N}$ with $x\geq0$ (I already know this limit exists) .


I found the lower bound $p(x)\geq\log(x)$, that by graphical simulations seems to be very good when $x$ is big enough.


Can you help me to compute $p(x)$?



asymptotic behaviour of a sum
Let $T\_N(x)$ denote the largest term of the sum. Note that 
$$\frac{\log T\_N(x)}{N}\leq \frac{\log (Z\_N(x))}{N}\leq \frac{\log ((N/2)T\_N(x))}{N}=\frac{\log T\_N(x)}{N}+\frac{\log(N/2)}{N}$$
and that the limits of both sides are identical, so by Squeeze theorem $p(x)=\lim\limits\_{N\to\infty} \frac{\log(T\_N(x))}{N}$.
To find the largest term, we4 want to minimize the expression $k!(N-2k)!(N-1)^k(\sqrt{2}x)^{2k-N}$. Note that 
$$\frac{(k+1)!(N-2k-2)!(N-1)^{k+1}(\sqrt{2}x)^{2k+2-N}}{k!(N-2k)!(N-1)^k(\sqrt{2}x)^{2k-N}}=\frac{2x^2(k+1)(N-1)}{(N-2k-1)(N-2k)}$$
and so we continue to make the denominator smaller by increasing $k$ so long as $2x^2(k+1)(N-1)<(N-2k-1)(N-2k)$. Solving for $k$ (and making approximations valid for large $N$) gives 
$$\begin{align}k &< \frac{N x^2+2N-x^2-1-\sqrt{N^2x^4+4N^2x^2-2Nx^4+2Nx^2+x^4-6x^2+1}}{4}\\\\
&< \frac{N x^2+ 2N-\sqrt{x^4+4x^2}N}{4}=\frac{2+x^2-\sqrt{x^4+4x^2}}{4}N\\\\
\end{align}$$
 and so for large $N$ the largest term will be $k=f(x)N$, with $f(x)=\frac{2+x^2-\sqrt{x^4+4x^2}}{4}$. Note that we can assume equality (rather than accuracy to the nearest integer) without hurting our calculations since the limit exists and for any $\epsilon$ we have arbitrarily large $N$ such that this is within $\epsilon$ of an integer. Thus
$$T\_N(x) = 2^{-N/2}\frac{N!}{(f(x)N)!((1-2f(x))N!)}(N-1)^{-f(x)N}(\sqrt{2}x)^{(1-2f(x))N}$$
and so taking logarithms and applying Stirling's approximation ($\log(y!)\approx y\log(y)-y$) we get
$$\begin{align}\log(T\_N(x)) &= -\frac{N}{2}\log(2)+\log(N!)-\log((f(x)N)!)-\log((1-2f(x))N!)\\\\
& \;\;\;\;-f(x)N\log(N-1)+(1-2f(x))N\log(\sqrt{2}x)\\\\
&\approx -\frac{N}{2}\log(2)+N\log(N)-N-f(x)N\log(f(x)N)+f(x)N\\\\
& \;\;\;\;-(1-2f(x))N\log((1-2f(x))N)+(1-2f(x))N-f(x)N\log(N-1)\\\\
& \;\;\;\;+(1-2f(x))N\log(\sqrt{2}x)\\\\
\end{align}$$
and so (note we make the approximation $\log(N-1)\approx \log(N)$)
$$\begin{align}
\frac{\log(T\_N(x))}{N}&\approx -\frac{\log(2)}{2}-f(x)+\log(N)-f(x)\log(f(x)N) \\\\
& \;\;\;\;-(1-2f(x))\log((1-2f(x))N)-f(x)\log(N-1)\\\\
& \;\;\;\;+(1-2f(x))\log(\sqrt{2}x)\\\\
&\approx -\frac{\log(2)}{2}-f(x)+\log(N)-f(x)\log(f(x))-f(x)\log(N) \\\\
& \;\;\;\;-(1-2f(x))\log(1-2f(x))-(1-2f(x))\log(N)-f(x)\log(N-1)\\\\
& \;\;\;\;+(1-2f(x))\log(\sqrt{2}x)\\\\
&\approx -\frac{\log(2)}{2}-f(x)-f(x)\log(f(x))-(1-2f(x))\log(1-2f(x))\\\\
& \;\;\;\;+(1-2f(x))\log(\sqrt{2}x)\\\\
\end{align}$$
thus
$$\begin{align}
p(x) &= \lim\limits\_{N\to\infty} \frac{\log(T\_N(x))}{N}\\\\
&\approx -\frac{\log(2)}{2}-f(x)(1+\log(f(x)))+(1-2f(x))(\log(\sqrt{2}x)-\log(1-2f(x)))\\\\
\end{align}$$
which should be quite close to the actual value due to the accuracy of Stirling's for large values.


