The generic multiplicity-induced-dominancy property from retarded to neutral delay-differential equations: When delay-systems characteristics meet the zeros of Kummer functions

The paper proves that, under the coefficient constraints ensuring maximal multiplicity m+n+1 at s0 (Theorem 3.1), the root s0 is dominant: strictly dominant for retarded (m<n) and non-strictly dominant with all remaining roots on the vertical line Re s = s0 for neutral (m=n), with the spectrum explicitly characterized by the set Ξn (Theorem 3.3). The argument proceeds via a translation/dilation z = τ(s−s0), an exact factorization of the translated characteristic function through a Kummer confluent hypergeometric function (equation (3.7)), and localization of its zeros (Proposition 2.10 and Lemmas 3.5–3.6), yielding Re z<0 for m<n and Re z=0 with the formula (3.6) for m=n (see the change-of-variables Lemma 2.2, the coefficient identities (3.1), the Kummer factorization (3.7), and the spectrum formula (3.6) in the neutral case: , , , , and the statement of Theorem 3.3 in ). The candidate solution reaches the same conclusions via a different route: it reduces to an entire-function model H_{n,m}(z)=e^{z}P_{n,m}(z)−Q_{n,m}(z), asserts a strict modulus gap |P_{n,m}(iζ)|>|Q_{n,m}(iζ)| for ζ≠0 (m<n) using total-positivity/Gram-form ideas, and then applies Rouché on large rectangles to exclude zeros in Re z≥0; for m=n it derives the explicit imaginary-axis equation tan(ζ/2)=ζ B_n(ζ)/A_n(ζ), i.e., Ξn, and excludes off-axis zeros by an argument-principle/max-modulus argument—matching the paper’s (3.6). Minor issues: the candidate’s normalization includes an extraneous factor e^{s0τ} in the rescaled identity; while irrelevant for the zero set, the paper’s exact translated identity has no such factor (cf. (2.2)–(2.4) and (3.2) yielding ∆̃(z)=(−1)^n n![P_{n,m}(z)−e^{−z}Q_{n,m}(z)], see , ). The core conclusions and spectral description agree. Therefore, both are correct, with substantially different proofs.