ANNALES POLONICI MATHEMATICI LIX.1 (1994) Diagonal series of rational functions (several variables) by S lawomir Cynk and Piotr Tworzewski (Krak´ ow) Abstract. We give representations of Nash functions in a neighbourhood of a polydisc (torus) in C m as diagonal series of rational functions in a neighbourhood of a polydisc (torus) in C m+1 . 1. Introduction. Let Ω be an open subset of C m . We shall use the following notation: O(Ω) — the space of all holomorphic functions on Ω, N (Ω) — the space of all Nash functions on Ω, R(Ω) — the space of all rational holomorphic functions on Ω. For any compact subset K of C m we denote by O(K) the space of all functions defined on K which have a holomorphic extension to an open neighbourhood of K. In the same way we define N (K) and R(K). We denote by U and T the unit disc and unit circle in C respectively. We consider the diagonal operator D : O(T m × T ) ∋ f →D(f ) ∈O(T m ) , defined by (1.1) D(f )(z )=  α∈Z m a α,|α| z α , where f (z,w)=  α∈Z m ,n∈Z a α,n z α w n is the Laurent expansion of f (see [2]–[5], [7], [9]). 1991 Mathematics Subject Classification : Primary 32A05, 32A25. Key words and phrases : diagonal series, rational function, Nash function, rationally convex, Hadamard convolution. Supported by KBN Grant 2 1077 91 01.