In this paper, traditional single-hidden layer feedforward network (SLFN) is extended to novel generalized SLFN (GSLFN) by employing polynomial functions of inputs as output weights connecting randomly generated hidden units with corresponding output nodes. The significant contributions of this paper are as follows: 1) a primal GSLFN (P-GSLFN) is implemented using randomly generated hidden nodes and polynomial output weights whereby the regression matrix is augmented by full or partial input variables and only polynomial coefficients are to be estimated; 2) a simplified GSLFN (S-GSLFN) is realized by decomposing the polynomial output weights of the P-GSLFN into randomly generated polynomial nodes and tunable output weights; 3) both P- and S-GSLFN are able to achieve universal approximation if the output weights are tuned by ridge regression estimators; and 4) by virtue of the developed batch and online sequential ridge ELM (BR-ELM and OSR-ELM) learning algorithms, high performance of the proposed GSLFNs in terms of generalization and learning speed is guaranteed. Comprehensive simulation studies and comparisons with standard SLFNs are carried out on real-world regression benchmark data sets. Simulation results demonstrate that the innovative GSLFNs using BR-ELM and OSR-ELM are superior to standard SLFNs in terms of accuracy, training speed, and structure compactness.