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For recursively generated shifts, we provide definitive answers to twooutstanding problems in the theory of unilateral weighted shifts: theSubnormality Problem ({\bf SP}) (related to the Aluthge transform) and theSquare Root Problem ({\bf SRP}) (which deals with Berger measures of subnormalshifts). We use the Mellin Transform and the theory of exponential polynomialsto establish that ({\bf SP}) and ({\bf SRP}) are equivalent if and only if anatural functional equation holds for the canonically associated Mellintransform. For $p$--atomic measures with $p \le 6$, our main result provides anew and simple proof of the above-mentioned equivalence. Subsequently, weobtain an example of a $7$--atomic measure for which the equivalence fails.This provides a negative answer to a problem posed by G.R. Exner in 2009, andto a recent conjecture formulated by R.E. Curto et al in 2019.

The Square Root Problem and Subnormal Aluthge Transforms of Recursively Generated Weighted Shifts