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Vectors of type II Hermite-Padé approximations and a new linear independence criterion. (English) Zbl 1476.11100

Let \(m\in\mathbb Z^+\), \(\gamma_1,\dots , \gamma_m\in\mathbb R\) and \(l\in\{ 1,\dots ,m\}\). Let \(q_n^{(\nu)},p_n^{(1,\nu)},\dots ,p_n^{(m,\nu)}\) with \(\nu =1,\dots ,l\) be \(l(m+1)\) sequences of integers. For \(\mu=1,\dots ,m\) and \(\nu=1,\dots ,l\) set \(\varepsilon_n^{(\mu,\nu)}=q_n^{(\nu)}\gamma_n-p_n^{(\mu,\nu)}\). Suppose that for all choices of \(l\) distinct indeces \((\mu_1,\dots ,\mu_l)\) from \(1\) to \(m\) we have \(\lim_{n\to\infty} \det [\varepsilon_n^{(\mu,\nu)}]_{\mu=\mu_1,\dots ,\mu_l, \nu=1,\dots ,l}=0 \). Assume that for all \(\lambda^{(i,j)}\in\mathbb Z\), \(i=1,\dots ,l\), \(j=1,\dots ,m\) such that the matrix \([\lambda^{(i,j)}]_{i=1,\dots ,l, j=1,\dots ,m}\in\mathcal{M}(l,m;\mathbb Z)\) has rank \(l\), the square matrix \([\lambda^{(i,j)}]_{i=1,\dots ,l, j=1,\dots ,m} [\varepsilon_n^{(\mu,\nu)}]_{\mu=1,\dots ,m, \nu=1,\dots ,l}\in\mathcal{M}(l,l;\mathbb Z)\) is non-singular for infinitely many \(n\). Then the author proves that \(\dim_{\mathbb Q}(\mathbb Q+\mathbb Q \gamma_1+\dots +\mathbb Q \gamma_m)\geq 2+m-l\). Some examples, variations of this result and consequences are included also.

MSC:

11J72 Irrationality; linear independence over a field
11B83 Special sequences and polynomials
11C20 Matrices, determinants in number theory
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
39A60 Applications of difference equations
41A28 Simultaneous approximation
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis

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