lajos/Extrapolation methods recursion

Coefficients of the \(Q\)-polynomials in extrapolation methods

We fix \(2\le p\in\mathbb{N}\) and assume that \(j\), \(k\), \(\ell\) and \(m\) are always integers. In order to generate the coefficients of the \(Q\)-polynomials, we are interested in the \(\alpha_m\) coefficients in the sum

\[ T_{p,p}=\sum_{m=1}^p \alpha_m T_{m,1}, \]

where the \(T_{j,k}\) quantities obey the following recursion for \(j\ge 2\) and \(2\le k\le j\):

\[ T_{j,k}=\frac{j}{k-1}T_{j,k-1}-\frac{j-k+1}{k-1}T_{j-1,k-1}. \]

For any \(j\ge 1\) and \(1\le k\le j\), let us define \(S_{j,k}\) as

\[ S_{j,k}:=\sum _{m=j-k+1}^j \frac{(-1)^{j+m} m^{k-1} }{(j-m)! (k-j+m-1)!}T_{m,1}. \]

Claim. For any \(j\ge 1\) and \(1\le k\le j\), we have \(T_{j,k}=S_{j,k}\).

Proof. For any \(j\ge 1\) we have \(S_{j,1}=T_{j,1}\). On the other hand, for any \(j\ge 2\) and \(2\le k\le j\), a routine computation shows that \[ S_{j,k}=\frac{j}{k-1}S_{j,k-1}-\frac{j-k+1}{k-1}S_{j-1,k-1}, \] finishing the proof.

For any \(p\ge 2\) and \(z\in\mathbb{C}\), the array of \(Q\)-polynomials, consisting of \(\binom{p}{2}\) elements, are given by \[ \alpha_m\left(1+\frac{z}{m}\right)^{\ell-1}, \] where \(m\) runs from \(2\) to \(p\), and for any fixed \(m\), \(\ell\) runs from \(2\) to \(m\). Taking into account that we have to estimate the absolute value of the \(Q\)-polynomials, and by expressing the \(\alpha_m\) coefficients from the explicit formula for \(S_{p,p}\), we see that the absolute value of the members of the \(Q\)-array is \[ \frac{m^{p-\ell} }{(p-m)! (m-1)!}|z+m|^{\ell-1} \] for \(m=2,\ldots,p\), and \(\ell=2,\ldots,m\).

We note that for fixed \(m\), the \(\frac{m^{p-\ell} }{(p-m)! (m-1)!}\) coefficients are monotone decreasing in \(\ell\).