Action of higher derivations on prime rings with involution

Abstract In this paper, we monitor the behavior of a prime ring with involution of second kind. Such rings are studied through the prism of higher derivations satisfying certain differential identities. Precisely, we prove that for a higher derivation <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>D</m:mi> <m:mo>=</m:mo> <m:msub> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>d</m:mi> <m:mi>i</m:mi> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mrow> <m:mi>i</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mi>ℕ</m:mi> <m:mo>∪</m:mo> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:mn>0</m:mn> <m:mo stretchy="false">}</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:msub> </m:mrow> </m:math> {D=(d_{i})_{i\in\mathbb{N}\cup\{{0}\}}} , if we are able to establish the identity, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:msub> <m:mi>d</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">[</m:mo> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:msup> <m:mi>x</m:mi> <m:mo>*</m:mo> </m:msup> <m:mo stretchy="false">]</m:mo> </m:mrow> </m:mrow> <m:mo>∈</m:mo> <m:mrow> <m:mi>Z</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>R</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> {d_{n}[x,x^{*}]\in Z(R)} for a single positive integer n , then the structure exhibits certain interesting properties. Some similar looking results are also presented.