Burning Dragon Claw Fist, originally "Moeru Ryusoken," is one of the most formidable martial arts techniques used by Sabo in the One Piece series. This is a combination of Dragon Claw Fist with Sabo's style and powers of the Flame-Flame Fruit that he received from his sworn brother Ace.
Ryūsoken is a Dragon Claw Fist martial art of the hands, in which the user changes his/her hands into the shape of a dragon's claw. This style is one of the basics within Sabo's fighting technique, which he learned during his time with the Revolutionary Army. By the addition of powers from the Flame-Flame Fruit, this already formidable art raises to new heights of destructive capability.
One of the most powerful techniques within the Burning Dragon Claw Fist arsenal would be the "Moeru Ryusoken: Kaen Ryou," which literally means "Burning Dragon Claw Fist: Fire Flame Dragon King." In Viz Manga and the Funimation dub, this attack is referred to as "Burning Dragon Claw Fist: Flame Dragon King," while in some English translations, it is "Blazing Dragon King.". This variation in naming with different localizations is what makes Japanese attack names nuanced in translation.
The technique is executed by having Sabo form claws with his fists, coat them with Armament Haki, and set it alight with the powers of the Flame-Flame Fruit. This basically combines Haki with marital skill and Devil Fruit ability into one devastating attack that can easily overwhelm powerful opponents. As much as Sabo can use Armament Haki together with the powers of the Flame-Flame Fruit, he is advanced in terms of combat and can therefore synergize a lot of fighting techniques.
The Burning Dragon Claw Fist style that Sabo has developed reflects his flexibility and fighting genius. He took the powers of the Flame-Flame Fruit and mixed them with his old techniques from Ryusoken. He therefore came up with a fighting style that would really respect Ace but propel his fighting skills to the next level. This fusion of techniques increases not only the offensive abilities of Sabo but is also a symbolically continued legacy of Ace's.
Most outstanding, though, is how much the Burning Dragon Claw Fist techniques come across visually within the anime series. When Sabo attacks with his different techniques, his hands are visibly always enfolded within intense flames that usually take on the visual appearance of fiery dragon claws. It not only adds to making such powerful attacks but also enhances this dragon theme within the Ryusoken martial arts style.
In the greater scheme of things, Burning Dragon Claw Fist emphasizes a certain way of considering power development within the One Piece world at large. While most characters specialize in one or the other, between Devil Fruit and the more traditionalist path of martial arts, Sabo's style reveals that there are ways in which these miscellaneous power systems of the series can complement each other. It is this concept of fighting that thusly ties into the general theme of the series: constant growth and adaptation, whereby characters find new ways to make themselves stronger before ever-more powerful enemies.
Having eaten the Flame-Flame Fruit, Sabo has proved the effectiveness of Burning Dragon Claw Fist in several key battles so far. Amongst such techniques was that during the Dressrosa Arc, Sabo could take down some high-powered enemies using his combined powers, portraying the potential of destruction associated with the power. Specifically, one attack of his, Fire Flame Dragon King, has been claimed to be one of his strongest moves, capable of shifting the balance in favor of the situation at hand.
Another interesting thing to note is that the development of Burning Dragon Claw Fist style gives rise to the speculation of fans about the real nature and origin of the Ryusoken technique. Some of these theories are: probably the Dragon Claw Fist has some connections with the Uruti, ancient history in the world of One Piece, or with the mysterious figure of Dragon, Luffy's father and leader of the Revolutionary Army. While these are mere theories, they do show the degree of interest fans have in the details of fighting styles and how the series works.