Multivariate truncated powers [see \textit{W. Dahmen}, SIAM J. Numer. Anal. 17, 179-191 (1980; Zbl 0425.41015)] are distributions which generally can be identified with continuous, sectorwise polynomial functions. Just as in the univariate case B-splines with pairwise distinct knots can be written as linear combinations of these functions. In order to get a similar representation in the case of coalescent knots we define generalized truncated powers by taking distributional limits. Several differentiation and recurrence formulas are proved for generalized truncated powers. These results are used to derive known recurrence relations for B-splines with arbitrary knots.