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Bases of the space of solutions of some fourth-order linear difference equations: applications in rational approximation. (English) Zbl 1278.42035

It is well known that orthogonal polynomials with respect to the discrete Sobolev inner product \[ <f,g>_S=\int_I f g d\mu +\lambda f^{(r)} (c) f^{(r)} (c), \] where \(\lambda\in \mathbb R^{+},\) \(c\in \mathbb R\) and \(\mu\) is a finite Borel measure on an interval (finite or infinite) \(I\subset \mathbb R,\) satisfies a \(2r+3\) term recurrence relation. The authors consider the five-term recurrence relation satisfied by the sequence of polynomials \(\{Q_n\}_{n=0}^{\infty}\) orthogonal with respect to the above inner product, where \(r=1\) and \(c=0.\) They analyse the set of solutions of the fourth-order linear difference equation defined by the corresponding five-term recurrence relation. Indeed, they prove that the set of four sequences \(\{Q^{[i]}_{n-i}\}_{n-i\geq 0}\) \((i=0,~1,~2,~3),\) where \(Q^{[0]}_{n}=Q_{n}\) and \[ Q^{[i]}_{n-i}(z)=\frac{1}{\int_I R_{i-1}^2(t)d\mu_2(t)}\int_{I} \frac{Q^{[i]}_{n}(z)-Q^{[i]}_{n}(t)}{z-t} R_{i-1}(t)d\mu_2(t), ~ i=1,~2,~3, \] \(\{R_n\}_n\) is the sequence of monic orthogonal polynomials with respect to the measure \(d\mu_2(t)=t^2 d\mu(t),\) is a basis of the linear space of polynomial solutions of that fourth-order linear difference equation. Also, some convergence results for a sequence of rational functions \(\frac{Q^{[1]}_{n-1}}{Q_{n}}\) are obtained.

MSC:

42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
33C47 Other special orthogonal polynomials and functions
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References:

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