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This paper contains a general theory for asymptotic expansions of type (*)f(x)=a_1 phi_1(x)+...+a_n phi_n(x)+o(phi_n(x)), x tends to x_0, n>=3, where theasymptotic scale phi_1(x)>>phi_2(x)>>...>>phi_n(x), x tends to x_0, is assumedto be an extended complete Chebyshev system on a one-sided neighborhood of x_0."Factorizational theory" refers to proofs being based on various types offactorizations of a differential operator associated to (phi_1,...,phi_n),hence we preliminarly collect various results concerning the concept ofChebyshev asymptotic scale, associated disconjugate operators and canonicalfactorizations. Another guiding thread of our theory is the property of formaldifferentiation and we aim at characterizing some n-tuples of asymptoticexpansions formed by (*) and (n-1) expansions obtained by formal applicationsof suitable linear differential operators of orders 1,2,...,n-1. Our secondpreliminary step will be that of discovering that the class of the operatorsnaturally associated to "canonical" factorizations seems to be the mostmeaningful to be used in a context of formal differentiation. This gives riseto conjectures whose proofs build an analytic theory of finite asymptoticexpansions in the real domain which, though not elementary, parallels thefamiliar results about Taylor's formula. One of the results states that to eachscale of the type under consideration it remains associated an important classof functions (namely that of generalized convex functions) enjoying theproperty that the expansion (*), if valid, is automatically formallydifferentiable (n-1) times in two special senses.

Analytic theory of finite asymptotic expansions in the real domain. Part II: the factorizational theory for Chebyshev asymptotic scales