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# A few (conjectural) modular properties of determinants associated to Pascal's triangle

Roland Bacher

started in september 04, last update: september 04

Under construction.

The aim of these notes is to present a few problems and conjectures related to the my joint paper with R. Chapman Symmetric Pascal matrices modulo (European J. of Combinatorics 25 (2004) 459-473).

Denote by the infinite symmetric matrix with coefficients

given by the binomial coefficients arising as entries in Pascal's triangle

We denote by the symmetric submatrix of defined by the first rows and columns. The matrix has thus coefficients with . Denote by the characteristic polynomial of (where denotes of course the identity matrix of order ).

The main result of [1] (Theorem 1.3 in [1]) reads now:

Theorem 0.1   When is a power of a prime and then

where satisfies .

This result determines the reduction modulo of the characteristic polynomial completely.

Theorem 0.1 seems to be the tip of an iceberg. Many similar formulae seem to exist. The easiest one (Conjecture 1.6 of [1]) can be stated as follows:

Conjecture 0.2   For each integer there exists a monic polynomial of degree such that with the following property: if is a power of a prime and then

where satisfies .

The equality states of course that the polynomials are palindromic: and their roots are symmectric with respect to the unit circle: is a root of if and only if is a root of .

The first few of these conjectural polynomials are seemingly

If this conjecture holds then

by Theorem 0.1 (and some easy corollaries of it).

Computations with for suggest (cf. Conjecture 1.7 in [1]):

Conjecture 0.3   We have

Theorem 0.1 and Conjectures 0.2 and 0.3 define now completely modulo from the trivial initial values and .

Proofs of these conjectures would thus be very interesting.

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Roland Bacher 2004-09-13