Digit Sums Generalizing Binomial Coefficients

We define four different generalizations of binomial coefficients, using the digit sum of a string with digits in {0, 1, 2,..., g}, for positive integers g. We identify one generalization with the extended binomial coefficients, and we express every other generalization in terms of the extended binomial coefficients. We also express every generalization in terms of the binomial coefficients and find the explicit formula for each generalization.