3.5 Common Identities and Recursion Formulas

The identities we discuss in this section apply either to the Pade ap­proximants themselves, or to the numerators and denominators; conse­quently there are two quite different types of relationships to be dis­tinguished.

One of the most remarkable relationships which occurs in the theory is that the numerators and denominators of neighboring Pade approximants obey the same recurrence relations. This fact is the key to the connection with continued-fraction theory. The other relations we will prove have diverse applications elsewhere in this book.

The generalization to (5.10c) is possible if G(z) and H(z) are functions of z only, and are independent of L and M\ it is easily justified because the Frobenius identities are linear in the sense that (5.10c) is linear. Our conclusion is that with the definitions (5.10), we have identities among the elements S{,/m] of Figure 2 as follows: Frobenius identities.

From these important results, we may obtain further identities. Using the symbol • to denote entries to be eliminated, we examine the configuration (*' * *) using (J*) twice and (.*). This leads to a single identity for the (* * *) configuration which turns out to be which is useful for Kronecker's algorithm (see Section 2.4). Similarly, there is a (* * *) identity, which is

This identity concludes our survey of identities for Q[L/M\z) and P[L/M](z). Next we turn to identities for the Pade approximants themselves. First, there are the fundamental two-term identities between neighboring approximants. Consider If the Pade approximants are degenerate, then Equations (5.13), (5.14) are to be understood as being multiplied up, in which case they become correct. The results, being algebraic, are essentially unchanged. Subtracting (5.13) and (5.14),

 

 

Therefore

 

 

 

Since the left-hand side of (5.15) is a polynomial of order L+M+ 1, (5.15) becomes

 

 

By inspection of the determinantal forms (5.1), (5.8), we find the leading coefficients to be as follows:

 

The coefficient of

 

 

The coefficient of

 

Substituting in (5.16) for the leading coefficients, we obtain

 

 

 

 

and so (5.16) after division by

 

finally becomes

 

 

which is a (* *) identity. By working in a precisely similar way, and using Sylvester's identity for (5.18), we find

 

 

This identity concludes the derivation of the two-term identities (5.17)- (5.20). From these may be derived some invariants, called cross ratios, which are independent of z. We find

 

 

which is a (|l|) identity. It is not the only (JJ) identity; there are others given by interconnections and by (|x|) and by more complicated patterns. Again we refer to EPA for details.