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On Ramanujan's remarkable summation formula

The author provides a survey of Ramanujan's famous 1ψ1 summation formula and some of its applications. First, he shows that the q-binomial theorem and Jacobi's triple product identity are special cases of the 1ψ1 summation formula. Second, he sketches K. Venkatachaliengar's proof of the 1ψ1 theorem. Third, he shows how classical formulas for r2(n) and r4(n) can be deduced from the 1ψ1 summation formula, where rk(n) denotes the number of representations of the positive integer n as the sum of k squares. Fourth, a limiting case, which includes a famous identity of Jacobi, is discussed. Fifth, certain eta-function identities are shown to be special cases of the 1ψ1 theorem. Sixth, it is demonstrated that some modular equations can be deduced with the help of the 1ψ1 summation formula. Lastly, a q-integral analogue and generalizations are briefly discussed