The series resistor is to limit the current sourced by the Ardiuno port pin into the input capacitance of the FET. For PWM, you want the FET to switch on and off as fast as possible, so you want the smallest resistor appropriate. A 220 ohm resistor will limit the current at 5V to 22 mA which is the maximum comfortably rated for the mega328 (or most similar) chip.
The parallel resistor is merely to pull the gate to ground when nothing is otherwise controlling the voltage, and the gate has a very high input impedance (many megohms) so this resistor can be quite a high value. 47k is really quite low and will have no effect on the PWM driving. A 4k7 would be OK if the series resistor was 220 ohms, but slightly reduces the drive voltage (by one twentieth), but 4k7 against 2k2 would prevent the circuit from working as it would reduce the gate drive voltage by one third.
You could in fact, put the 4k7 resistor not from the gate to ground, but from the Arduino port to ground if those are the only resistors to hand, but the values I carefully specified would be much preferred.