I calculate the claim of the theorem below for a couple of times, always I get $\sum_{i=1}^{\infty} c_i z^{L+i}$ and not $\sum_{i=1}^{\infty} c_i z^{L+M+i}$ as the coefficient of RHS of last determinant (i.e. the first determinant of 1.11 seems incorrect?) :
Is the theorem incorrect or am I mistaken?
Note : rows in P and Q are the same except for the last one.
$\endgroup$ 61 Answer
$\begingroup$Let us go step by step.
\begin{align*} Q^{[L/M]}&\sum_{i=0}^\infty c_i z - P^{[L/M]}(z)= \\ & =\begin{vmatrix} c_{L-M+1} & c_{L-M+2} & \cdots & c_{L+1}\\ c_{L-M+2} & c_{L-M+3} & \cdots & c_{L+2}\\ \vdots & \vdots & \vdots & \vdots \\ c_{L-1} & c_{L} & \cdots & c_{L+M -1}\\ c_{L} & c_{L+1} & \cdots & c_{L+M}\\ \sum_{i=L-M+1}^\infty c_iz^{M+i} & \sum_{i=L-M+2}^\infty c_iz^{M+i-1} & \cdots & \sum_{i=L+1}^\infty c_iz^{i} \end{vmatrix} = \\ & \\ & =\begin{vmatrix} c_{L-M+1} & c_{L-M+2} & \cdots & c_{L+1}\\ c_{L-M+2} & c_{L-M+3} & \cdots & c_{L+2}\\ \vdots & \vdots & \vdots & \vdots \\ c_{L-1} & c_{L} & \cdots & c_{L+M -1}\\ c_{L} & c_{L+1} & \cdots & c_{L+M}\\ \sum_{i=L-M+1}^{L} c_iz^{M+i} & \sum_{i=L-M+2}^{L+1} c_iz^{M+i-1} & \cdots & \sum_{i=L+1}^{L+M} c_iz^{i} \end{vmatrix} + \\ &\phantom{mmm}+ \begin{vmatrix} c_{L-M+1} & c_{L-M+2} & \cdots & c_{L+1}\\ c_{L-M+2} & c_{L-M+3} & \cdots & c_{L+2}\\ \vdots & \vdots & \vdots & \vdots \\ c_{L-1} & c_{L} & \cdots & c_{L+M -1}\\ c_{L} & c_{L+1} & \cdots & c_{L+M}\\ \sum_{i=L+1}^\infty c_iz^{M+i} & \sum_{i=L+2}^\infty c_iz^{M+i-1} & \cdots & \sum_{i=L+M+1}^\infty c_iz^{i} \end{vmatrix} = \\ & \\ & =0+\begin{vmatrix} c_{L-M+1} & c_{L-M+2} & \cdots & c_{L+1}\\ c_{L-M+2} & c_{L-M+3} & \cdots & c_{L+2}\\ \vdots & \vdots & \vdots & \vdots \\ c_{L-1} & c_{L} & \cdots & c_{L+M -1}\\ c_{L} & c_{L+1} & \cdots & c_{L+M}\\ \sum_{i=L+1}^\infty c_iz^{M+i} & \sum_{i=L+2}^\infty c_iz^{M+i-1} & \cdots & \sum_{i=L+M+1}^\infty c_iz^{i} \end{vmatrix} = \\ &\\ & =\begin{vmatrix} c_{L-M+1} & c_{L-M+2} & \cdots & c_{L+1}\\ c_{L-M+2} & c_{L-M+3} & \cdots & c_{L+2}\\ \vdots & \vdots & \vdots & \vdots \\ c_{L-1} & c_{L} & \cdots & c_{L+M -1}\\ c_{L} & c_{L+1} & \cdots & c_{L+M}\\ \sum_{i=1}^\infty c_{L+i}z^{M+L+i} & \sum_{i=1}^\infty c_{L+1+i}z^{M+L+i} & \cdots & \sum_{i=1}^\infty c_{L+M+i}z^{M+L+i} \end{vmatrix} = \\ &\\ & =\sum_{i=1}^\infty z^{M+L+i}\begin{vmatrix} c_{L-M+1} & c_{L-M+2} & \cdots & c_{L+1}\\ c_{L-M+2} & c_{L-M+3} & \cdots & c_{L+2}\\ \vdots & \vdots & \vdots & \vdots \\ c_{L-1} & c_{L} & \cdots & c_{L+M -1}\\ c_{L} & c_{L+1} & \cdots & c_{L+M}\\ c_{L+i} & c_{L+1+i} & \cdots & c_{L+M+i} \end{vmatrix} = \\ &\\ & =\sum_{i=1}^\infty z^{L+M+i}\begin{vmatrix} c_{L-M+1} & c_{L-M+2} & \cdots & c_{L+1}\\ c_{L-M+2} & c_{L-M+3} & \cdots & c_{L+2}\\ \vdots & \vdots & \vdots & \vdots \\ c_{L-1} & c_{L} & \cdots & c_{L+M -1}\\ c_{L} & c_{L+1} & \cdots & c_{L+M}\\ c_{L+i} & c_{L+1+i} & \cdots & c_{L+M+i} \end{vmatrix} \end{align*}
$\endgroup$ 1
