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Finite q-identities related to well-known theorems of Euler and Gauss Johann Cigler Fakultät für Mathematik Universität Wien A-1090 Wien, Nordbergstraße 15 email: johann.cigler@univie.ac.at Abstract We give generalizations of a finite version of Euler’s pentagonal number theorem and of a q −identity of Gauss by introducing a new parameter. 0. Introduction n−k +1 n n ⎡ ⎤ q q (1− ) (1− ) Let ⎢ ⎥ k be a q −binomial coefficient. k (1− ) (1− ) ⎣ ⎦ q q A. Berkovich and F. G. Garvan [1] and with another method S.O. Warnaar [4] have proved that 2L j (3j +1) ⎡2L −j ⎤ j 2 ∑(−1) q ⎢ ⎥ 1 j =−L ⎣L +j ⎦ for all L ∈. This is a finite version of Euler’s pentagonal number theorem, because for L →∞ it reduces to the pentagonal number theorem ∞ ∞ j (3j +1) n j 2 ∏(1−q ) ∑(=−1) q . n 1 j =−∞ n ⎡ ⎤ k n−k Let ( , ) . A famous theorem of Gauss states that r x a ∑ x a n ⎢ ⎥ k k ⎣ ⎦ r (1, −1) 0 and 2n+1 (q; q)2n 3 2n−1 r (1, 1) (1 q)(1 q ) (1 q ). −