Recursive formulas for Morse-oscillator matrix elements of arbitrary powers of 1−exp[−a(r−r e)]

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Abstract
The variable y=1−exp[−a(r−re)] is a natural one to use in connection with a Morse oscillator. A simple formula is derived via the factorization method relating the Morse matrix element 〈0‖ yn ‖0〉 to 〈0‖yn−1 ‖0〉 and 〈0‖yn−2‖ 0〉. Another simple formula is derived, with which all matrix elements 〈v′‖yn ‖v〉 can be calculated recursively, starting from values of 〈0‖yn‖ 0〉.