I am working with the following function for integers $n,m$, defined in $[0,1]$.
$$f_{n,m}(x)= \sum_{i=0}^m \binom{n+i}{i}x^i$$
I know that this can be written as a hypergeometric function, using the binomial series. I am wondering if these kind of polynomials have been studied before... its properties in $[0,1]$, bounds, asymptotics, etc. Any reference will be of help.
On
Ok, I found something that satisfies me, coming from this question:
$$f_{n,m}(x)=\frac{1}{(1-x)^{n+1}}\big(1-(m+1)\tbinom{n+m+1}{n}B_x(n+1,m+1)\big),$$
where $B_x$ is the incomplete Beta function. It was tough to find, probably because of the somewhat misleading title.
Interestingly enough, we have
$$(1-x)^{n+1}f_{n,m}(x) = x^{m+1}f_{m,n}(1-x),$$
following from the fact that the incomplete Beta functions are symmetric and the straightforward $(m+1)\tbinom{n+m+1}{n}=(n+1)\tbinom{n+m+1}{m}$.
These are just a couple of notes.
First, that you can rewrite $f_{n,m}(x)$ as $$ \eqalign{ & f_{\,n,\,m} (x) = \sum\limits_{\left( {0\, \le } \right)\,\,k\, \le \,m} {\left( \matrix{ n + k \cr k \cr} \right)x^{\,k} } = \sum\limits_{\left( {0\, \le } \right)\,\,k\,\left( { \le \,m} \right)} {\left( \matrix{ m - k \cr m - k \cr} \right)\left( \matrix{ n + k \cr k \cr} \right)x^{\,k} } = \cr & = \sum\limits_{\left( {0\, \le } \right)\,\,k\,\left( { \le \,m} \right)} {\left( { - 1} \right)^{\,k} \left( \matrix{ m - k \cr m - k \cr} \right)\left( \matrix{ - n - 1 \cr k \cr} \right)x^{\,k} } = \cr & = \left( { - 1} \right)^{\,m} \sum\limits_{\left( {0\, \le } \right)\,\,k\,\left( { \le \,m} \right)} {\left( \matrix{ - 1 \cr m - k \cr} \right)\left( \matrix{ - n - 1 \cr k \cr} \right)x^{\,k} } \cr} $$ so as to get rid of the summation bounds, which may be of advantage in the algebraic manipulation.
Second , that the formal series on $m$ has a neat and simple form $$ \eqalign{ & F_{\,n} (x,y) = \sum\limits_{0\, \le \,\,m} {f_{\,n,\,m} (x)y^{\,m} } = \sum\limits_{0\, \le \,\,m} {\sum\limits_{\left( {0\, \le } \right)\,\,k\, \le \,m} {\left( \matrix{ n + k \cr k \cr} \right)x^{\,k} y^{\,m} } } = \cr & = \sum\limits_{0\, \le \,\,m} {\sum\limits_{\left( {0\, \le } \right)\,\,k\, \le \,m} {\left( \matrix{ n + k \cr k \cr} \right)x^{\,k} y^{\,m} } } = \sum\limits_{0\, \le \,\,k} {\sum\limits_{k\, \le \,m\,} {\left( \matrix{ n + k \cr k \cr} \right)\left( {yx} \right)^{\,k} y^{\,m - k} } } = \cr & = {1 \over {1 - y}}\sum\limits_{0\, \le \,\,k} {\left( \matrix{ n + k \cr k \cr} \right)\left( {yx} \right)^{\,k} } = {1 \over {1 - y}}\sum\limits_{0\, \le \,\,k} {\left( { - 1} \right)^{\,k} \left( \matrix{ n + k \cr k \cr} \right)\left( { - yx} \right)^{\,k} } = \cr & = {1 \over {1 - y}}\sum\limits_{0\, \le \,\,k} {\left( \matrix{ - n - 1 \cr k \cr} \right)\left( { - yx} \right)^{\,k} } = {1 \over {\left( {1 - y} \right)\left( {1 - xy} \right)^{\,n + 1} }} \cr} $$