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Starting from fixed-order perturbation theory (FOPT) we derive expressions for the heavy-flavour components of the deep-inelastic structure functions (F i,H (x, Q 2 , m 2 H ), i = 2, L; H = c, b, t) in the variable-flavour number scheme (VFNS). These expressions are valid in all orders of perturbation theory. This derivation establishes a relation between the parton densities 1 supported by the Foundation for Fundamental Research on Matter (FOM) 2 on leave from ITP, SUNY at Stony Brook, New York 11794-3840, USA 1 parametrized at n f and n f + 1 light flavours. The consequences for the existing parametrizations of the parton densities are discussed. Further we show that in charm electroproduction the exact and asymptotic expressions for the heavy-quark coefficient functions yield identical results for F 2,c (x, Q 2 , m 2 c ) when Q 2 ≥ 20 (GeV/c) 2 . We also study the differences between the FOPT and the VFNS descriptions for F 2,c (x, Q 2 , m 2 c ). It turns out that the charm structure function in the VFNS is larger than the one obtained in FOPT over the whole Q 2 -range. Furthermore inspection of the perturbation series reveals that the higher order corrections in the VFNS are smaller than those present in FOPT for Q 2 ≥ 10 (GeV/c) 2 . Therefore the VFNS gives a better prediction for the charm structure function at large Q 2 -values than FOPT.2 c [10]. Although it was shown in NLO [11] that these logarithms lead to rather stable charm structure functions F i,c (x, Q 2 , m 2 c ) with respect to variations in the factorization and renormalization scales µ, their size still warrants some special treatment (for a discussion of the scale dependence of the charm con-4

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