**The Block Theorem and Normal Approximants**

Because of the way the (*m*, *n*)th approximant is constructed, the difference

*Q*(_{n}*z*)*f*(*z*) −*P*(_{m}*z*)

is a power series whose first term is of degree no less than

*m*+*n*+ 1.

If the first term of that difference is of degree

*m*+*n*+*r*+ 1,*r*> 0,

then the rational function *R _{m, n}* occupies

- (
*r*+ 1)2

cells in the Padé table, from position (*m*, *n*) through position (*m*+*r*, *n*+*r*), inclusive. In other words, if the same rational function appears more than once in the table, that rational function occupies a square block of cells within the table. This result is known as the **block theorem**.

If a particular rational function occurs exactly once in the Padé table, it is called a **normal** approximant to *f*(*z*). If every entry in the complete Padé table is normal, the table itself is said to be normal. Normal Padé approximants can be characterized using determinants of the coefficients *c _{n}* in the Taylor series expansion of

*f*(

*z*), as follows. Define the (

*m*,

*n*)th determinant by

with *D*_{m,0} = 1, *D*_{m,1} = *c _{m}*, and

*c*= 0 for

_{k}*k*< 0. Then

- the (
*m*,*n*)th approximant to*f*(*z*) is normal if and only if none of the four determinants*D*_{m,n−1},*D*,_{m,n}*D*_{m+1,n}, and*D*_{m+1,n+1}vanish; and - the Padé table is normal if and only if none of the determinants
*D*are equal to zero (note in particular that this means none of the coefficients_{m,n}*c*in the series representation of_{k}*f*(*z*) can be zero).

Read more about this topic: Padé Table

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