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Arithmetic properties of an infinite product related to theta functions. (Propriétés arithmétiques d’un produit infini lié aux fonctions thêta.) (French) Zbl 0853.11056

It is proved that the infinite product \(\prod^{+ \infty}_{n= 1} (1- q^{-n})\) is not algebraic of degree less than 2 for \(q\) belonging to \(\mathbb{Z} \setminus \{-1, 0, 1\}\). The proof is elementary and uses only two classical identities which can be deduced from Jacobi’s triple product identity. A more general theorem is also proved; this theorem leads to a second proof of the above result. It also allows for the proof that the infinite product \(\prod^{+ \infty}_{n= 0} (1- q^{-5n-1}) (1- q^{-5 n-4}) (1- q^{-5n-5})\) is not algebraic of degree less than 2 for \(q\) belonging to \(\mathbb{Z} \setminus \{-1, 0, 1\}\).
Reviewer: D.Duverney (Lille)

MSC:

11J72 Irrationality; linear independence over a field
11E25 Sums of squares and representations by other particular quadratic forms
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