Padé Approximants in Complex Points Revisited

In 1976, Chisholm et al. 1 published a paper concerning the location of poles and zeros of Padé approximants of ln 1 − z developed at the complex point ζ : ln 1 − z ln 1 − ζ − ∑∞ n 1 1/n z − ζ/1 − ζ . They claimed that all poles and zeros of diagonal Padé approximants n/n interlace on the cut z ζ t 1 − ζ , t ∈ 1,∞ . Unfortunately, this result is only partially true, for poles. Klarsfeld remarked in 1981 2 that the zeros do not follow this rule. The study of this problemwas the starting motivation of the present work.We consider the general class of complexsymmetric functions f , that is, functions satisfying the following condition:


Introduction
In 1976, Chisholm et al. 1 published a paper concerning the location of poles and zeros of Padé approximants of ln 1 − z developed at the complex point ζ : ln 1 They claimed that all poles and zeros of diagonal Padé approximants n/n interlace on the cut z ζ t 1 − ζ , t ∈ 1, ∞ .Unfortunately, this result is only partially true, for poles.Klarsfeld remarked in 1981 2 that the zeros do not follow this rule.The study of this problem was the starting motivation of the present work.We consider the general class of complexsymmetric functions f, that is, functions satisfying the following condition: In particular, if ζ and ζ are two complex conjugate points, then that is, the coefficients of these two series are also complex conjugates, This property is the basic element of all our proofs.

International Journal of Mathematics and Mathematical Sciences
Let us introduce some definitions and notations.The 1-point Padé approximant PA , or simply Padé approximant m/n to f at the point ζ is a rational function P m /Q n if and only if Because the existence of PA defined by 1.3 implies the invertibility of Q n , the equivalent definition is This definition leads to the following linear system for the coefficients of polynomials Q n and P m : The PA exists iff the system 1.7 has a solution.The following so-called Padé form 3 defines PA if it exists and other rational functions in the square blocs in the Padé table if 1.7 has no solution ; for simplicity, it is written for the case ζ 0, where

1.10
Let f be a function defined at the points ζ 1 , . . ., ζ N ∈ and having at these points the following power expansions: i 1, 2, . . ., N : 12 where This leads to the following definition like 1.4 : representing m n 1 linear equations.

Zeros and Poles of Pad é Approximants of ln 1 − z
This function studied in 1 is related to the Stieltjes function defined in the cut-plane \ 1, ∞ .The zeros and poles of PA defined by a power series of f expanded at the real points interlace on the cut 1, ∞ .What does happen if PA is defined at the complex point ζ? Klarsfeld remarked 2 that Chisholm result 1 is wrong and showed International Journal of Mathematics and Mathematical Sciences that poles of the PA of ln 1 − z follow the cut z ζ t 1 − ζ , t ≥ 1, as mentioned in the introduction, but not the zeros.We generalize this result to all m ≥ n.For convenience, let us introduce the following notations: 2.4 Proof.We can readily verify this theorem looking at the formula 1.9 and considering the numerator P * m and the denominator Q * n of m/n * and the relation 2.4 .The denominator More exactly, we have That is, in this case, the poles of m/n * follow the way of poles of m/n rotated by some angle around the branch point which gives 2.5 and 2.6 .On the contrary, the definition of P * m always contains c * 0 , and then the zeros of m/n * locate out of 2.6 .
The location of the zeros of m/n * remains an open problem.However, the following remarks can unlock, may be, this question.The particular case of Gilewicz theorem 3, page 217 says that if n/n f is a PA of some function f and α a constant, then This readily leads to the following theorem, where all notations are the same as in Theorem 2.1 except c * 0 which is replaced by an arbitrary constant α.
, where q i are defined by the linear system 1.7 .We identify the solutions q i as coefficients of Q * n due to 3.1 : Then, the zeros of Q * n poles of m/n * are the complex conjugates of those of Q n .The same arguments are used for the system 1.8 and for P m and P * m which completes the proof.

Theorem 2 . 1 .
Let m/n and m/n * , m ≥ n, be the Padé approximants of the function f z ln 1 − z developed at the points z 0 and z ζ, respectively, then if z k k 1, 2, . . ., n denotes a pole of m/n , then z * k ζ z k 1 − ζ 2.5 denotes the pole of m/n * .In other words, the poles of m/n * locate on the cut z ζ t 1 − ζ , t ≥ 1, 2.6 directed by the straight line joining the point of development of f : z ζ with the branch point z 1.The zeros of m/n * locate out of this line.

2 . 10 ln 1 − ζ . 3 .Theorem 3 . 1 .
If α 0, then the poles and also the zeros simulate the cut ζ t 1 − ζ , t ≥ 0. The problem consists to analyze the behavior of the zeros of P * n as a function of α with α ∈ 0, c * 0 Zeros and Poles of Pad é Approximants at Complex Conjugate Points In this section, m/n f z − ζ and m/n * f z − ζ denote the Padé approximants of a complexsymmetric function f at the point ζ and its complex conjugate ζ, respectively.Let m/n and m/n * be Padé approximants of a complexsymmetric function f at ζ and ζ, then the zeros and the poles of m/n are complex conjugates of the corresponding zeros and poles of m/n * .Proof.Equation 1.1 gives