Full indefinite Stieltjes moment problem and Pad\'{e} approximants

Full indefinite Stieltjes moment problem is studied via the step-by-step Schur algorithm. Naturally associated with indefinite Stieltjes moment problem are generalized Stieltjes continued fraction and a system of difference equations, which, in turn, lead to factorization of resolvent matrices of indefinite Stieltjes moment problem. A criterion for such a problem to be indeterminate in terms of continued fraction is found and a complete description of its solutions is given in the indeterminate case. Explicit formulae for diagonal and sub-diagonal Pad\'{e} approximants for formal power series corresponding to indefinite Stieltjes moment problem and convergence results for Pad\'{e} approximants are presented.


Introduction
The classical Stieltjes moment problem consists in the following: given a sequence of real numbers s j (j ∈ Z + := N ∪ {0}) find a positive measure σ with a support on R + , such that The notation z →∞ means that z → ∞ nontangentially, that is inside the sector ε < arg z < π − ε for some ε > 0.
In the pioneering paper [40] by T. Stieltjes  where τ ranges over the Stieltjes class S. A subset of its canonical solutions was described already in [40]. Now let us remind the indefinite versions of the classes N and S, see [30].
Definition 1.1. ( [30]) A function f meromorphic on C\R with the set of holomorphy h f is said to be in the generalized Nevanlinna class N κ (κ ∈ N), if for every set z j ∈ C + ∩ h f (z i = z j , i, j = 1, . . . , n) the form n i,j=1 has at most κ and for some choice of z j (j = 1, . . . , n) exactly κ negative squares. A functions f ∈ N κ is said to belong to the generalized Stieltjes class N + κ , if zf ∈ N. Similarly, in [9,10] the class N k κ (κ, k ∈ N) was introduced as the set of functions f ∈ N κ , such that zf (z) belongs to the class N k , see also [18,20], where the class N k 0 was studied.
In the present paper we consider the following problems. Full indefinite moment problem M P κ (s). Given κ ∈ Z + , and an infinite sequence s = {s j } ∞ j=0 of real numbers, describe the set M κ (s) of functions f ∈ N κ , which satisfy (1.10) for all ℓ ∈ N. Full indefinite moment problem M P k κ (s). Given κ, k ∈ Z + , and an infinite sequence s = {s j } ∞ j=0 of real numbers, describe the set M k κ (s) := M κ (s) ∩ N k κ .
Indefinite moment problems M P κ (s) and M P + κ (s) were studied in [31], [32] by the methods of extension theory of Pontryagin space symmetric operators developed in [30], [31]. In particular, it was shown in [31] that the moment problem M P + κ (s) is solvable if and only if the number ν − (S n ) of negative eigenvalues of S n does not exceed κ and S + n > 0 for all n ∈ N. Further applications of the operator approach to the moment problem M P k κ (s) were given in [10]. A reproducing kernel approach to the moment problems M P κ (s) was presented in [21]. A step-by-step algorithm of solving the moment problems M P κ (s) was elaborated in [6], [7] and [2]. Applications of the Schur algorithm to degenerate moment problem in the class N κ were given in [14].
Truncated indefinite moment problem M P k κ (s, ℓ). Given ℓ, κ, k ∈ Z + , and a sequence s = {s j } ℓ j=0 of real numbers, describe the set M k κ (s, ℓ) of functions f ∈ N k κ , which satisfy (1.10). A truncated moment problem is called even or odd regarding to the oddness of the number ℓ + 1 of given moments.
Let H be the set of all infinite real sequences s = {s j } ∞ j=0 and let H κ be the set of sequences s ∈ H, such that (1.11) ν − (S n ) = κ for all n big enough.
Denote by H k κ the set of real sequences s ∈ H κ , such that {s j+1 } ∞ j=0 ∈ H k , i.e. (1.12) ν − (S + n ) = k for all n big enough. A number n j ∈ N is said to be a normal index of the sequence s ∈ H, if det S nj = 0. The ordered set of normal indices n 1 < n 2 < · · · < n N of the sequence s is denoted by N (s).
As was shown in [7] for every s ∈ H κ there exists a sequence of real numbers b i ∈ R\{0}, i ∈ N and real monic polynomials such that the convergents of the continued fraction for sufficiently large n have the asymptotic expansion (1.10) for every ℓ ∈ N. This fact was known already to L. Kronecker [33] and then it was reinvented in [6]. The pairs (a i , b i ) are called atoms, see [24] and the continued fraction (1.14) is called the P -fraction, [36].
Consider the three-term recurrence relation associated with the sequence of atoms {a i , b i }, i ∈ N, and define polynomials P j (z) and Q j (z) as solutions of the system (1.15) subject to the initial conditions The polynomials P j and Q j are called Lanzcos polynomials of the first and second kind. Moreover, the n-th convergent of the continued fraction (1.14) takes the form (see [24,Section 8.3.7]).
As was shown in [7] the set M κ (s, 2n j − 2) can be described in terms of the Lanzcos polynomials of the first and second kind. A sequence s ∈ H k κ is called regular (see [15]), and is designated as s ∈ H k,reg κ , if (1.17) D + nj = det S + nj = 0 for all j ∈ N. In [16] it was shown that an even indefinite Stieltjes moment problems M P k κ (s, 2n N − 1) for regular sequence s is solvable if and only if (1.18) κ N := n − (S nN ) ≤ κ and k + N := n − (S + nN ) ≤ k. For this problem one step of the Schur algorithm was split in [16] into two substeps and this leads to the expansion of f ∈ M k κ (s, 2n N − 2) into a generalized Stieltjes continued fraction where m j are polynomials, l j ∈ R\{0} and f N is a function from the generalized Stieltjes i.e. f N is a solution of an induced moment problem M P k−kN κ−κN (s (N ) ) generated by the sequence s (N ) = (s ). Generalized Stieltjes continued fractions were studied in [15]. Associated to the continued fraction (1.19) there is a system of difference equations (see [42, Define the generalized Stieltjes polynomials P + j and Q + j of the first and second kind as solutions of the system (1.21) subject to the initial conditions where f N (z) satisfies the conditions In what follows for every 2×2 matrix W = (w ij ) 2 i,j=1 we associate the linear-fractional transformation (1.25) T W [τ ] := w 11 τ + w 12 w 21 τ + w 22 .
Denote by W 2N (z) the coefficient matrix of the linear fractional transform (1.23): .
Then the formula (1.23) can be rewritten as The structure of the continued fraction (1.19) leads to the following factorization of the matrix valued function W 2N (z): where the matrices M j (z) and L j are defined by Continued fractions of the form (1.19) with positive and negative masses m j were studied by Beals, Sattinger and Szmigielski [4] in connection with the theory of multipeakon solutions of the Camassa-Holm equation. In [22] Eckhardt and Kostenko showed that inverse spectral problem for multi-peakon solutions of the Camassa-Holm equation is solvable in the class of continued fractions of the form (1.19) with polynomials m j (z) = d j z + m j of formal degree 1. In [23] the spectral theory of continued fractions (1.19) was treated within a classical Hamburger moment problem associated with this sequence.
The full indefinite Stieltjes moment problem M P k κ (s) for regular sequences s ∈ H k,reg κ was considered in [17] via the operator approach. The moment problem M P k κ (s) was treated in [17] as a problem of extension theory for a symmetric operator generated by this generalized Jacobi matrix. In that paper we used quite advanced tools: the theory of boundary triples developed in [25,18,19], and the M. G. Kreȋn theory of resolvent matrices extended in [9,12] to the case of indefinite inner spaces.
In the present paper we are going to use elementary tools in order to make the presentation available for a wider audience. The main idea is to use the factorization formula for the coefficients matrix W 2j (z) , j > N, j, N ∈ N, which allows to reduce the indefinite Stieltjes moment problem M P k κ (s, 2j) to some classical Stieltjes moment problem M P 0 0 (s (N ) , 2(j−N )) with the resolvent matrix W If (1.30) is in force, then the matrix valued functions W 2j (z) converge to an entire matrix valued function W + ∞ (z) of order 1/2 and the linear fractional transformation (1.27) generated by the matrix valued function W ∞ (z) provides a description of the set M k κ (s). In Section 5 Padé approximants for formal power series corresponding to an indefinite Stieltjes moment problem are calculated. As was shown in [8] the diagonal Padé approximants for formal power series corresponding to an indefinite Hamburger moment problem are represented as a ratio of the Lanzcos polynomials of the 2-nd and the 1-st kind, [8]. In Theorem 5.4 we show that the sub-diagonal Padé approximants of the corresponding formal power series is a ratio of the generalized Stieltjes polynomials of the 2-nd and the 1-st kind. In Theorem 5.6 convergence of Padé approximants is derived from the classical results using the formula (1.29).
In Section 6 the results are illustrated by an example of indefinite moment problem associated with Laguerre polynomials L n (z, α) in the non-classical case α < −1.

Preliminaries
2.1. Generalized Nevanlinna functions. A function f ∈ N κ is said to belong to the class N κ,−ℓ (κ, ℓ ∈ Z + := N ∪ {0}) if f admits the asymptotic expansion (1.10) for some real numbers s 0 , . . . , s ℓ . Let us also set Every real polynomial P (z) = p ν z ν + · · · + p 1 z + p 0 of degree ν belongs to a class N κ−(P ) , where the index κ − (P ) can be evaluated by (see [30,Lemma 3.5]) Recall, that a function f ∈ N κ is said to be from the generalized Stieltjes class N ±k κ , if z ±1 f (z) belongs to N k (κ, k ∈ Z + ). Let us collect some properties of generalized Nevanlinna functions, see [30], [9]. Proposition 2.1. ( [30]) Let κ, κ 1 , k ∈ Z + . Then the following statements hold: ( if a function f ∈ N κ has an asymptotic expansion (1.5) for every n ∈ N, then there exists κ ′ ≤ κ, such that {s j } ∞ j=0 ∈ H κ ′ . The notions of generalized poles of non-positive type of a function f ∈ N κ were introduced in [32]. The following definitions are based on [34]. A point α ∈ R is called a generalized pole of non-positive type (GPNT) of the function f ∈ N κ with multiplicity Similarly, the point ∞ is called a generalized pole of f of nonpositive type (GPNT) with The following fundamental result was proved in [32, Theorem 3.5].
Then the total multiplicity of the poles of f in C + and the generalized poles of negative type of f in R ∪ {∞} is equal to κ.

2.2.
Regular sequences and step-by-step algorithm. In the present paper we will consider so-called regular sequences s from H k κ introduced in [15]. Definition 2.3. The sequence s is called regular (and is denoted as s ∈ H k,reg κ ), if one of the following equivalent conditions holds: (i) P j (0) = 0 for every j ∈ N; (ii) D + nj := det S + nj = 0 for every j ∈ N; The following step-by-step algorithm of solving the indefinite Stieltjes moment problem for regular sequences s ∈ H k,reg κ was developed in [17]. For an indefinite Hamburger moment problem such an algorithm was elaborated in [6,7].
Let f ∈ M k κ (s) and let n 1 be the first normal index, i.e.
The formula (2.6) yields the following representation of f ∈ M k κ (s): with some real numbers s (1) i , i ∈ Z + . Explicit formulas for calculation of the sequence i=0 are presented in [17, Remarks 3.4, 3.6], the polynomial m 1 (z) and the number l 1 can be found by (see [17, (2.16), (3.27)]): This completes the first step of the Schur algorithm.
Applying this algorithm repeatedly N times one obtains a function f N ∈ N k−k + N κ−κN , with κ N and k N given by (1.18), connected with f by the formula (1.19). Moreover, the function f N satisfies the asymptotic expansion Then there exist sequences of polynomials m j (z) and numbers l j such that the 2j−th convergent u2j v2j of the continued fraction coincides with the j−th convergent of the P −fraction (1.14) corresponding to the sequence s. Let functions f and f N (N ∈ N) be connected by (1.19) and let s (N ) is the N −th induced sequence. Then , where κ N and k + N are given by (1.18). In the case of a regular sequence s ∈ H k κ the parameters l j and m j (z) in (2.12) can be calculated recursively by the above Schur algorithm in terms of the sequence s: The N −th induced sequence s (N ) can be found as the sequence of coefficients of the

Generalized Stieltjes continued fractions.
In [15] the expansion (2.12) of f ∈ M k κ (s) into a generalized Stieltjes fraction for s ∈ H k,reg κ was derived from the expansion of its unwrapping transform zf (z 2 ) into the P −fraction.
and let the P −fraction (1.14) and the generalized S−fraction (2.12) correspond to the sequence s. Then the parameters l j and m j (z) (j ∈ Z + ) of the generalized S−fraction (2.12) are connected with the parameters b j and a j (z) (j ∈ N) of the P −fraction (1.14) by the equalities In the case when m j (z) ≡ m j are constants the generalized S−fraction reduces to the classical S-fraction (1.7) and the formulas (2.18) and (2.19) coincide with the well known classical formulas from [40], see also [1,Appendix,(3), (4)].
Conversely, m j and l j can be represented in terms of the P −fractions, see [15, Corollary 4.1].
Corollary 2.6. Let s ∈ H k,reg κ , let s be associated with the S−fraction (2.12), let d i be the leading coefficient of the polynomial m i (z) and let Then

A system of difference equations and generalized Stieltjes polynomials.
Let us consider a system of difference equations associated with the continued fraction (2.12) If the j-th convergent of this continued fraction is denoted by uj vj , then u j , v j can be found as solutions of the system (3.1) (see [42,Section 1]) subject to the following initial conditions The first two convergents of the continued fraction (2.12) take the form The polynomials P + i (z), Q + i (z) are called the generalized Stieltjes polynomials corresponding to the sequence s.
As was noticed in [16] the generalized Stieltjes polynomials coincide with the solutions u i and v i of the system (3.1).
Proposition 3.2. Let s ∈ H k,reg κ and let P + j (z) and Q + j (z) be the generalized Stieltjes polynomials defined by (3.3). Then the solutions {u j } N j=0 and {v j } N j=0 of the system (3.1), (3.2) take the form 3. The Stieltjes polynomials satisfy the following properties Here we used the generalized Liouville-Ostrogradsky formula (see [17, (2.9 Lemma 3.4. Let P + i and Q + i be the Stieltjes polynomials defined by (3.3). Then This completes the proof.
Lemma 3.5. Let s ∈ H k,reg κ and let P i (z) and Q i (z) (i ∈ Z + ) be Lanczos polynomials of the first and second kind and let l j and m j (z) (j ∈ N) be parameters of the generalized S−fraction (2.12). Then (i) The constants l i can be calculated by (ii) For every N ∈ N the following formulas hold Proof. 1) Let Q + i (z) be Stieltjes polynomials defined by (3.3). Substituting in (3.1) y j = Q + j and z = 0, we obtain By Definition 3.1 and by the generalized Liouville-Ostrogradsky formula (3.7) This implies (3.9).
3.2. The class U k κ (J) and linear fractional transformations. Let J and Z be the 2 × 2 matrices Definition 3.6. Let W (z) be a 2 × 2 matrix valued function meromorphic in C + and let h + W be the domain of holomorphy of W in C + , κ ∈ Z + . Then W (z) is called a generalized J-inner matrix valued function from the class U κ (J), if (i) the kernel has κ negative squares in h + W × h + W ; (ii) J − W (µ)JW (µ) * = 0 for a.e. µ ∈ R.
A matrix valued function W ∈ U κ (J) is said to belong to the class U k κ (J), κ, k ∈ Z + , if (3.14) ZW Z −1 ∈ U k (J).
Consider the linear fractional transformation associated with the matrix valued function W (z) = (w i,j (z)) 2 i,j=1 . The linear fractional transformation associated with the product W 1 W 2 of two matrix valued functions W 1 (z) and W 2 (z), coincides with the composition T W1 • T W2 .
The following statement is an easy corollary of Definition 3.6.
Let us denote f (z) = T W (z)[τ (z)]. Then Since ZW Z −1 ∈ U k1 and zτ ∈ N k2 then f ∈ N k ′ for some In the present paper two special types of matrix valued functions from U k κ (J) play an important role (see [17, Lemma 2.11, Lemma 2.12]). Lemma 3.8. Let m(z) be a real polynomial such that κ − (zm(z)) = κ 1 , κ − (m(z)) = k 1 , let M (z) be a 2 × 2 matrix valued function and let τ be a meromorphic function, such that Then M ∈ U k1 κ1 (J) and the following equivalences hold: and let φ be a meromorphic function, such that Then L ∈ U k1 κ1 (J) and the following equivalences hold:    Proof. It follows from (3.1) that

The statement (ii) follows from (3.26) and (3.27).
Let us formulate an analog of Theorems 1.2 and 3.10 for the odd truncated indefinite moment problem (see [17]).
be generalized Stieltjes polynomials, let M i (z) and L i be given by (1.28) and let the matrix valued function W 2N −1 be defined by .
(iii) The matrix valued function W 2N −1 (z) belongs to the class U kN κN (J) and admits the factorization Substituting in (3.30) τ (z) = ∞ one obtains from Theorem 3.11 the following:  are not satisfied. In these cases the number of moments interpolating by the linear fractional transformations T W [τ ] can be reduced and also their indices can decrease. We will start with the linear fractional transformations T M in the simplest cases when m is a positive constant and hence M ∈ U 0 0 (J). Lemma 3.14. Let M (z) be a 2 × 2 matrix valued function Verification of (i) for τ ∈ N k κ , such that: By Lemma 2.1 −τ −1 ∈ N −k κ . If (3.35) holds then mz − τ (z) −1 has GPNT (generalized pole of negative type) at ∞ of the same multiplicity as −τ (z) −1 , i.e.
In the case (a3) we apply Lemma 3.14 to the function φ 1 (z) = T Mn [τ 1 (z)] and then again there are three possibilities: In the case (b3) one should continue this process based on Lemma 3.14, Lemma 3.15, Theorem 1.2 and Theorem 3.11.
Similarly, one can treat the case κ < k by using Lemma 3.14 and Theorem 1.2.

4.
Full indefinite moment problem M P k κ (s) 4.1. Indefinite moment problem M P k κ (s) for s ∈ H 0 0 . Recall, that the moment problem M P k κ (s) is called indeterminate if it has more then one solution. A sequence s is called nondegenerate, if (4.1) there is N ∈ N, such that D n = 0, D + n = 0 for all n ≥ N. In this subsection we consider the indefinite moment problem M P k κ (s) for a nondegenerate sequence s ∈ H 0 0 . As is known, see [1, Theorem 0.5], for a nondegenerate sequence s ∈ H 0 0 the corresponding moment problem M P 0 0 (s) is indeterminate, if and only if Theorem 4.1. Let s be a nondegenerate sequence from H 0 0 and let (4.2) holds. Then (i) The moment problem M P k κ (s) is solvable and indeterminate for any pair of κ, k ∈ Z + . (ii) The sequence of resolvent matrices W 2n (z) converges to an entire matrix valued κ (s, 2n − 1) for every n ∈ N. By Theorem 1.2 there exists a sequence of functions φ N (z) ∈ N k κ such that (3.33) holds and It follows from (4.4) that which contradicts the generalized Liouville-Ostrogradsky identity (3.7). As was mentioned above the matrix valued functions W 2N (z) converges locally uni- By Corollary 3.12 is not identically equal to 0. Let Ω be the open set of points in C + such that w + 11 (z) − f (z)w + 21 (z) = 0. Then for every point z ∈ Ω the sequence of functions is correctly defined in a neighborhood of z and converges locally uniformly in Ω to a function φ(z). Since φ N ∈ N k κ then φ ∈ N k ′ κ ′ with κ ′ ≤ κ and k ′ ≤ k. It follows from (4.4) that Since f ∈ N k κ this implies κ ′′ = κ ′ = κ and k ′′ = k ′ = k and hence τ ∈ N k κ .

3.
Proof of the fact that T W + ∞ [τ ] satisfies (3.33) for every τ ∈ N k κ , κ, k ∈ Z + . Application of the Schur algorithm to the Stieltjes moment problem M k κ (s) gives on the N −th step an induced sequence s (N ) , which can be found as the sequence of coefficients , j ∈ N.
Then for every j ∈ N the resolvent matrix W 2j (z) admits the factorization (see [17,Proposition 4.7]) 2(j−N ) (z). Taking the limit as j → ∞ one obtains the following factorization of the entire matrix valued function W + ∞ (z) (4.9) is the resolvent matrix of the induced Stieltjes moment problem M 0 0 (s (N ) ).
∈ M k κ (s): Due to the above item 3 the function φ N from (4.10) satisfies the condition (3.33).
Next By Theorem 1.2 f ∈ M k ′ κ ′ (s, 2N − 1). Since N can be chosen arbitrarily large f ∈ M k ′ κ ′ (s). Now it follows from the item 2 that τ ∈ N k ′ κ ′ and hence κ ′ = κ and k ′ = k. 5. Verification of (i): Solvability of the indefinite Stieltjes moment problem M P k κ (s) for any pair of κ, k ∈ Z + follows from the item 4. The formula (4.3) gives two different solutions of the problem M P k κ (s) for different parameter functions τ 1 , τ 2 ∈ N k κ and thus the problem M P k κ (s) is indeterminate.

4.2.
Indefinite moment problem M P k κ (s), general case. Theorem 4.2. Let s be a nondegenerate sequence from H k0,reg κ0 , κ 0 , k 0 ∈ N and let l j and m j (z) (j ∈ N) be parameters of the generalized S−fraction (2.12). Then the moment problem M P k0 κ0 (s) is indeterminate, if and only if If (4.11) holds, then (i) The sequence of resolvent matrices W 2n (z) converges to an entire matrix valued function W + ∞ (z) = (w + ij (z)) 2 i,j=1 of order ≤ 1/2. (ii) The moment problem M P k κ (s) is solvable, if and only if (4.12) κ 0 ≤ κ, and k 0 ≤ k.
Proof. 1. Redaction of the indefinite moment problem M P k κ (s) to a classical one: Let us choose N big enough, so that (4.14) ν Then the induced sequence s (N ) which arises on the N −th step of the Schur algorithm (see Section 2.2) belongs to the class H 0 0 . The corresponding Stieltjes moment problem M 0 0 (s (N ) ) is classical. By Lemma 2.4 (4.15) f N ) ). In particular, Hence the problem M k0 κ0 (s (N ) ) is indeterminate if and only if the Stieltjes moment problem M 0 0 (s (N ) ) is indeterminate. The latter is equivalent to (4.11).
, let P i (z) and Q i (z) (i ∈ Z + ) be Lanczos polynomials of the first and second kind and let l j and m j (z) (j ∈ N) be parameters of the generalized S−fraction (2.12). Assume additionally that (4.19) l j > 0 for all j ∈ N.
Then the following statements are equivalent: (i) the moment problem M P κ (s) is indeterminate; (ii) the following series converge (iii) the following series converges Proof. By [8,31] (i) and (ii) are equivalent. Let us show, that (ii) and (iii) are equivalent. By the first equality in (3.10) Q i (0) = −(l 1 + l 2 + · · · + l i )P i (0). and hence by (3.11) This proves the implication (ii) ⇒ (iii).
This proves the implication (iii) ⇒ (ii).  as a ratio, where the numerator is a polynomial of formal degree n and the denominator is a polynomial of exact degree n vanishing at 0.
Explicit formula for diagonal Padé approximants for sequences s = {s j } ∞ j=0 ∈ H κ was found in [7], in the classical case s ∈ H 0 see [3,38]. In this section we will formulate the corresponding statements for sequences s = {s j } Proof. We present a proof of this statement from [17] for the convenience of the reader. It follows from (3.3) and Theorem 3.11 that the function belongs to M(s, 2n j − 1). Therefore, the function − Qj (z) Pj (z) has the asymptotic Since Q j (z) is a polynomial of degree n j − n 1 < n j and P j (z) is a polynomial of exact degree n j the function − Qj (z) Pj (z) is the [n j /n j ] Padé approximant for the formal power series (5.1) due to Remark 5.2 and (5.5).
In the following proposition stated in [16] without proof it is shown that the subdiagonal Padé approximants can be calculated in terms of generalized Stieltjes polynomials.
, κ, k ∈ Z + , let W 2N (z) be given by (1.26), N ∈ N and let s (N ) be the induced sequence defined in Lemma 2.4, and let be the corresponding formal power series. Then (i) the diagonal Padé approximants for the formal power series (5.1) are connected with diagonal g [n/n] Padé approximants for the power series (5.8) by the formula (ii) the subdiagonal Padé approximants for the power series (5.1) are connected with subdiagonal g [n/n−1] Padé approximants for the power series (5.8) by Proof. Consider the induced moment problem M P k−k + N κ−κN (s (N ) , 2(n j − n N ) − 1) and let It follows from the factorization formula (5.12) and (5.15) that It follows from the factorization formula (5.14) and (5.16) that , κ, k ∈ Z + . Then (i) If the problem M P k κ (s) is determinate, then diagonal and subdiagonal Padé approximants converge to the unique solution of M P k κ (s) locally uniformly on C \ R + .
(ii) If the problem M P k κ (s) is indeterminate, then the sequence f [n/n−1] of subdiagonal Padé approximants converges locally uniformly on C \ R + , while the sequence f [n/n] of diagonal Padé approximants is not convergent but precompact in the topology of locally uniform convergence.
Proof. Let us choose N big enough, so that Then the induced sequence s (N ) belongs to the class H 0 0 and by Lemma 5.5 the problem of convergence of diagonal and subdiagonal Padé approximants is reduced to the corresponding problem for diagonal and subdiagonal Padé approximants for the series
Here we consider the case when α < −1 and α is not a negative integer. The case when α is a negative integer was treated in [35]. If −k − 1 < α < −k, k ∈ N, then L n (z, α) are orthogonal polynomials with respect to the indefinite inner product (see [11], [37]) Polynomials Q n (x, α) of the second kind are defined as solutions of (6.1) subject to the initial conditions Q −1 (z, α) ≡ −1 and Q 0 (z, α) ≡ 0.
The moments s n for all n ∈ N are defined by (6.4) s n = S(z n ) = Γ(n + α + 1).
Notice that the full moment problem is determinate and in the case α > −1 its unique solution of M P 0 0 (s) is given by where W α,β is the Whittaker function, see [26, 9.222].