Overlaps of deformed and non-deformed harmonic oscillator basis states

•Analytic results were derived for the first time between deformed and non-deformed harmonic oscillator basis states.•The overlaps are independent of the harmonic oscillator frequency in a given space, deformed or non-deformed.•Useful for studying coupled oscillators, nuclear shell model, coherent state theory, deformed effective field theory and damped oscillators.