Asymptotics of number of possibilities that $V$ dice rolls add up to $n V$ for $V\to\infty$ If we roll an $(s+1)$-sided dice $V$ times (assuming the sides are labelled by $0$, $1$ up to $s$) the number of possibilities to get exactly $N$ is well-known to be given by (see derivation here)
\begin{align}
d(V,N,s)=\sum_j (-1)^j\binom{V}{t}\binom{V+n-j(s+1)-1}{V-1}\,.
\end{align}
We also know that the resulting probability distribution
\begin{align}
p(V,N,s)=(s+1)^{-V}d(V,N,s)
\end{align}
approaches a normal distribution centered at $\mu=\frac{s V}{2}$ with standard deviation $\sigma=\sqrt{\frac{V s (2+s)}{12}}$.
However, I would like to find the asymptotics of $d(V,n V,s)$ as we take $V\to\infty$, where $n$ is a fixed fraction $n\in[0,s]$ and $s$ a fixed integer (of course!). I know that the Gaussian approximation is only exact right at the center $N=\frac{s V}{2}$, i.e.,
\begin{align}
d(V,\frac{sV}{2},s)\sim \sqrt{\frac{6}{s(2+s)\pi V}}\,e^{\log(1+s)V}\quad\text{as}\quad V\to\infty\,.
\end{align}
I'm struggling to compute the asymptotics for other choices of $n$, as I do not know how to deal with the sum. I expect a structure alike to
\begin{align}
d(V,nV,s)\sim \frac{\alpha(s,n)}{\sqrt{V}}\,e^{f(s,n) V}\quad\text{as}\quad V\to\infty\
\end{align}
with $f(s,\frac{s}{2})=\log(1+s)$, but I don't know how to find $f$ for other values of $n$.
 A: I'm not sure if this helps, but in L. Comtet, Advanced Combinatorics (1974) the following formula is provided:
$$ d(V,N,s) = \binom{V,s}{N}=\frac{2}{\pi} \int_{0}^{\pi/2}  \left( \frac{\sin (s+1)\, \phi}{\sin \phi} \right)^V \cos [ (V s-2N) \phi ] \, d\phi$$
Edit: see also
A Note on Extended Binomial Coefficients, Thorsten Neuschel
A: Partial update:
This is not a full solution, but based on the helpful comments I wanted to give an update. What I am looking for is clearly the asymptotic behavior of the extended binomial coefficients (also known as polynomial coefficients - not to be confused with the $q$-binomial coefficients). While the $q$-binomial coefficients usually have the $q$ as lower subscript, the extended binomial coefficients either have the $q$ separated by a comma or as superscript.
There appears to be a slight difference in convention, but I followed the convention where
$$ \sum^\infty_{k=0}\binom{n,q}{k}x^k=(1+x+x^2+\dots+x^q)^n.$$
There is also the convention where the sum only goes up to $x^{q-1}$, so everything in this case $q\to q+1$ (with respect to my convention).
The asymptotics of these coefficients has been studied, but to my knowledge the case considered by me has not been solved in generality (but rather only for $n=q/2$, which I had already found). Recent references include:

*

*JIYOU LI: ASYMPTOTIC ESTIMATE FOR THE POLYNOMIAL COEFFICIENTS

*Steffen Eger: Stirling's approximation for central extended binomial coefficients

*Thorsten Neuschel: A Note on Extended Binomial Coefficients
I'm still trying to figure out if the integral representation suggested by leonbloy provides some help, but using a Gaussian approximation only allowed me to rederive the result for $n=q/2$. As this concerns integration, I started this separate question.
I also found an interesting reference in the appendix of this book, where the trinomial case ($s=2$) is solved. The relevant asymptotics is given in Theorem D.4 and the form of the function $f(n,s=2)$ is explained in Corollary D.6.
