The r-bi\(^{q}\)nomial coefficient and some properties

In this paper we give a new generalization of the binomial coefficient: the r-bi \(^{q}\) ${}^q$ nomial coefficient \(\left( {\begin{array}{c}L\\ k\end{array}}\right) _{q,r}\) ${\left(\begin{array}{c}L\\ k\end{array}\right)}_{q,r}$ defined as the coefficient of \(x^{k}\) $x^k$ in the expansion of \(\begin{aligned} \left( 1+x+\cdots +x^{q}\right) ^{L}\left( 1+2x+\cdots +qx^{q-1}\right) ^{r}. \end{aligned}\) $$\begin{array}{c}\hfill {\left(1,+,x,+,\cdots ,+,x^q\right)}^L{\left(1,+,2,x,+,\cdots ,+,q,x^{q-1}\right)}^r.\end{array}$$ We establish a connection between these coefficients and the partial r-Bell polynomials and we provide many combinatorial properties of these newcoefficients.