I'm not sure, but I think that the function
cannot be written through more elementary functions. Am I right? I am asking this as a non-mathematician.
Or in other way: Can above expression be treated as coefficient of $z^m$ in the product
$$ (1+xz)^n\,(1+z)^m.$$ say? (This has been the one answer on the related question Binomial series with two binomial coefficents)
I hope that I do not break any rule with posting like this.....