The Royal Rekka is a multi-part move not unlike Base Goku's old Dragon Fist.
It has a starter attack (a gut punch, as well as a multi-punch EX version) and from that, various paths you can take. Let's break it down:
STARTERS
LP has the fastest start-up but the weakest hitstun on impact. Just enough hitstun to link into the MP Path, not the LP or SP ones.
MP has a medium start-up and a decent hitstun on impact, allowing you to link into a Light attack.
SP has the slowest start-up but the longest hitstun on impact, allowing you to link into a Light or Medium attack.
The Starter attack's range is the same for all versions. For all versions you can go into a Path both on impact, on block and on whiff.
The EX version has the start-up time and hitstun of the MP version and gains quite some range, traveling forward at least 2/5 of the screen. It also does more damage by automatically following into a couple more punches. You can cancel out of the final punch into a Path quicker than any of the Normal versions, almost immediately upon impact.
However you cannot go into a Path if the move whiffs. When it's blocked, Vegeta will keep throwing out the additional punches. The hitstun is inbetween that of the LP and MP versions.
PATHS. The stronger the Starter leading up to it, the stronger the Path.
LP hits low. Due to its medium-long start-up speed it does NOT combo off of the LP Starter. Doing it out of MP Starter does enough hitstun for you to follow with a Light Attack. Out of a SP Starter, Mediums will also connect.
MP hits mid. Doing it out of MP Starter does enough hitstun for you to follow with a Light Attack. Out of a SP Starter which does even more hitstun, Mediums will also connect.
SP hits high. Due to its long start-up speed it does NOT combo off of the LP Starter. It also has the longest recovery time. Doing it out of SP Starter causes enough hitstun for you to follow with a Light Attack. Not Mediums.
EX creates a small multi-hitting Atomic Bomb, enabling Vegeta to retreat while leaving behind a present for the opponent.
ENDERS. As with the Path, the stronger the Starter leading up to it is, the stronger the Ender will be.
LP hits low and pop-trips the enemy (similar to N/M Vegeta's Dashing Slide Kick). Out of a Strong Starter, it will flip KO the opponent up for even better juggling options.
MP hits mid and knocks the enemy away. Acting as a combo ender. However in the corner, you can catch p2 with a st/LK/cr.LK. Doing it out of an LP Path does a simple knock-away, while out of a MP Path results in a Twist KO. Out of SP Path it also does a Twist KO but makes p2 fly up and away higher and when done close to the wall, p2 will wallsplat against it, allowing for new follow-up options.
SP hits high and launches p2, enabling you to go Persuit and aircombos. Out of a LP Starter it does quite a small launch but it still allows for a persuit into aircombo. Out of a MP Starter it does the common launcher, and out of an SP Starter it will Flip KO the opponent high into the air.
EX Ender is the same Atomic Bomb as the EX Path.