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5-dissections and sign patterns of Ramanujan's parameter and its companion

Shane Chern, Dazhao Tang (2021)

Czechoslovak Mathematical Journal

In 1998, Michael Hirschhorn discovered the 5-dissection formulas of the Rogers-Ramanujan continued fraction R ( q ) and its reciprocal. We obtain the 5-dissections for functions R ( q ) R ( q 2 ) 2 and R ( q ) 2 / R ( q 2 ) , which are essentially Ramanujan’s parameter and its companion. Additionally, 5-dissections of the reciprocals of these two functions are derived. These 5-dissection formulas imply that the coefficients in their series expansions have periodic sign patterns with few exceptions.

A remark on Construire un noyau de la fonctorialité by Lafforgue

Hervé Jacquet (2012)

Annales de l’institut Fourier

Lafforgue has proposed a new approach to the principle of functoriality in a test case, namely, the case of automorphic induction from an idele class character of a quadratic extension. For technical reasons, he considers only the case of function fields and assumes the data is unramified. In this paper, we show that his method applies without these restrictions. The ground field is a number field or a function field and the data may be ramified.

An inequality for local unitary Theta correspondence

Z. Gong, L. Grenié (2011)

Annales de la faculté des sciences de Toulouse Mathématiques

Given a representation π of a local unitary group G and another local unitary group H , either the Theta correspondence provides a representation θ H ( π ) of H or we set θ H ( π ) = 0 . If G is fixed and H varies in a Witt tower, a natural question is: for which H is θ H ( π ) ≠ 0 ? For given dimension m there are exactly two isometry classes of unitary spaces that we denote H m ± . For ε ∈ { 0 , 1 } let us denote m ε ± ( π ) the minimal m of the same parity of ε such that θ H m ± ( π ) ≠ 0 , then we prove that m ε + ( π ) + m ε - ( π ) ≥ 2 n + 2 where n is the dimension of π .

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