signed mixtures [X’ed]

Following a question on X validated, the hypoexponential distribution, I came across (for the second time) a realistic example of a mixture (of exponentials) whose density wrote as a signed mixture, i.e. involving both negative and positive weights (with sum still equal to one). Namely,

representing the density of a sum of d Exponential variates. The above is only well-defined when all rates differ, while a more generic definition involving matrix exponentiation exists. But the case when (only) two rates are equal can rather straightforwardly be derived by a direct application of L’Hospital rule, which my friend George considered as the number one calculus rule!