General recurrence formulas for evaluating molecular integrals over contracted Cartesian Gaussian functions are derived by introducing auxiliary contracted hyper-Gaussian (ACH) functions. By using a contracted Gaussian function, this ACH represents an extension of the Gaussian function named derivative of Fourier-kernel multiplied Gaussian [J. Chem. Phys. 94, 3790 (1991)]. The ACH is reducible to contracted Cartesian Gaussian functions, contracted modified Hermite Gaussian functions, and to contracted Gaussian functions multiplied by phase factors, or the so-called GIAO, and is also reducible to various spatial operators necessary for ab initio molecular orbital calculations. In our formulation, all molecular integrals are expressed in terms of ACH. Therefore, the formulations have wide applicability for calculating various kinds of molecular integrals in ab initio calculations. Recursive calculations based on our formulation do not depend on the number of contraction terms, because the contraction step is completed at the evaluation of the initial integrals. Therefore, we expect that more efficient recursive calculations will be accomplished by using our formulas for evaluating molecular integrals over contracted Gaussian functions.