Normalization Corrections and Conditions of Validity for Integral Representations of Bessel Type Equations

An integral representation of a singular differential equation must satisfy the equation and the prescribed endpoint traces. We revisit a weighted Cauchy formula for the Bessel operator and a representation for the Euler–Poisson–Darboux equation. Evaluation of the beta integral fixes the normalization required by the weighted initial velocity; a constant-data test shows that a coefficient printed in a 2018 article and in the first author’s thesis does not have this normalization. For a half-interval Poisson operator, integration by parts also produces a boundary term proportional to the derivative of the seed at the origin. Using smooth, compactly supported data, we show that the unrestricted interpretation of a published two-Bessel representation can fail on an open set. We then prove a sufficient version of the formula for even seed functions. Supporting results identify a Fourier uniqueness class, establish the classical parameter-shift identities on even analytic functions, and specify the range on which a fractional-integral inverse applies. These findings concern the normalization and domains of known representations; they do not assert a new transmutation method or priority over earlier work.