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The generating function method that we had developing has variousapplications in physics and not only interress undergraduate students but alsophysicists. We solve simply difficult problems or unsolved commonly used inquantum, nuclear and group theory textbooks. We find simply: the generatingfunction of the harmonic oscillator, the Feynman propagators of the oscillatorand the oscillator in uniform magnetic field. We derive the invariants of SU(2)and the expressions of 3-j,6-j symbols. We find also the octonions or Hurwitzquadratic transformations. We show that the cross-product exist only in E3 andE7. We determine the {p} representation of hydrogen atom in three andn-dimensions. We generalize the Cramer's rule for the calculation of therotational spectrum of the nucleus. We find the expression of the Hamiltonianin terms of quasi-bosons for study the collective vibration. We determine thebasis and the expressions of 3-j symbols of SU (3) and SU(n).We find theSchr\"odinger equation from Hamilton-Jacobi formalism. We present theseapplications in independent chapters.

Generating function method and its applications to Quantum, Nuclear and the Classical Groups