Structural properties of  s -regular functions in the split quaternionic setting

In this paper we study a slice-based notion of regularity for the split quaternionic algebra <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi mathvariant="double-struck">S</m:mi> </m:math> S , whose natural quadratic form has signature (2,2). For each hyperbolic unit <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi>J</m:mi> <m:mo>∈</m:mo> <m:mi mathvariant="script">T</m:mi> </m:math> J∈ T (with J 2 = 1) we consider the slice <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msub> <m:mrow> <m:mi>L</m:mi> </m:mrow> <m:mrow> <m:mi>J</m:mi> </m:mrow> </m:msub> <m:mo>=</m:mo> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo>+</m:mo> <m:mi>J</m:mi> <m:mi>y</m:mi> <m:mspace width="0.17em"/> <m:mo>:</m:mo> <m:mspace width="0.17em"/> <m:mspace width="0.3333em"/> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>y</m:mi> <m:mo>∈</m:mo> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mo stretchy="false">}</m:mo> </m:mrow> </m:math> LJ=+Jy : x,y∈ R\ and define left s -regularity by the vanishing of the hyperbolic Cauchy–Riemann operator <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msubsup> <m:mrow> <m:mi>D</m:mi> </m:mrow> <m:mrow> <m:mi>J</m:mi> </m:mrow> <m:mrow> <m:mo>*</m:mo> </m:mrow> </m:msubsup> </m:math> DJ^ on L J . We establish structural results that clarify how s -regularity behaves under orthogonal choices of units and how it interacts with power series on slices. In particular, we prove a splitting principle (Theorem 9) and a rigorous Abel-type boundary limit theorem for slice power series (Theorem 6), where non-tangential (Stolz-type) approach regions and an idempotent decomposition replace norm-based arguments that fail in the indefinite setting. We also provide a precise relation between the original M -approach condition and Stolz regions (Proposition 6.1) and discuss the wave/ultra-hyperbolic character of the induced second-order operators. A comparison with quaternionic slice-regularity and a brief Schur-analysis outlook position the present theory within the broader landscape of hypercomplex analysis.