What is the expression for an oversampled signal?
I go with the answer of Vasiliy and proofed it.

$$y\left[m\right] = x\left[{\Big\lfloor{\frac{m}{\beta}}\Big\rfloor}\right]$$

As I really can't think while writing TeX and you may need this before the end of the month, I wrote on a tablet.

The solution is to describe the input signal x in the time domain. Then the output signal y is derived from x using natural sampling. Based on the knowledge about the clock domains (


synchronized at time zero and
receiver clock equal or faster then transmitter clock
both clocks are exactly periodically


), the expression can be simplified.



Edit: The clock relation expression
$$n\left[m\right] = {\Big\lfloor{\frac{m}{\beta}}\Big\rfloor}$$
is not really obvious, but can be expressed very well using system theory. The idea is to describe the clock cycle count of the TX clock (TX clock value) as an integer value signal over time. This signal can be expressed using the floor function.

Then this signal is sampled using natural sampling at the positions of the RX clock events. In this way the counter value n of the slower TX clock can be expressed for all RX clock event positions mT1. Because the positions only depend on m, n can be expressed as a function of m.