1 Introduction

There is an extensive literature on the location of critical points of polynomials in terms of their zeros ([19, Part I] and [21]), whose main pillars are Rolle’s theorem, Gauss–Lucas Theorem, and their refinements. However, actual converses of these theorems have not been found yet. It is obvious that given one of the zeros of a polynomial and its critical points, the remaining zeros are uniquely determined. Nonetheless, there are few results about zero location of polynomials in terms of their critical points and a given zero, most of them contained in [19, §4.5]. In general, these follow from the Schur–Szegő composition theorem [19, Th. 3.4.1d]. Perhaps, the most relevant results in this sense are the theorems of Walsh [19, Th. 4.5.1] and Biernacki [19, Th. 4.5.2].

Let \(P_n^{(\alpha ,\beta )}\) be the n-th monic Jacobi polynomial with parameters \(\alpha ,\beta \in \mathbb {R}\)

$$\begin{aligned} P^{(\alpha ,\beta )}_{n}(z) = \sum _{k=0}^{n} {n+\alpha \atopwithdelims ()n-k} {n+\beta \atopwithdelims ()k} {2n+\alpha +\beta \atopwithdelims ()n}^{-1} (z-1)^k(z +1)^{n-k}, \end{aligned}$$
(1)

where \(2n+\alpha +\beta \ne 0,1,\ldots ,n-1\), \({a\atopwithdelims ()b} = {\Gamma (a+1)}/\left( {\Gamma (a-b+1)\Gamma (b+1)}\right) \) and \(\Gamma (\cdot )\) is the usual Gamma function (see [22, (4.21.6) and (4.3.2)] for more details). These classical polynomials have been extensively used in mathematical analysis and practical applications (cf. [20, 22, 23]). Nowadays, there has been renewed interest in using the Jacobi polynomials in the numerical solution of differential equations. Some of these methods require explicit expressions of the integral of such polynomials and the localization of their zeros (e.g., see [4, 5]). Another area that demands this knowledge is the study of families of polynomials orthogonal in a non-standard sense, particularly the Sobolev-type orthogonality and the orthogonality with respect to a differential operator (e.g., [3, 6, 17] ).

It is well known that \(P^{(\alpha ,\beta )}_{n}\) satisfies the following differentiation relation

$$\begin{aligned} \frac{\mathrm{d}^k \,P^{(\alpha ,\beta )}_{n}}{\mathrm{d}z^k}(z) = \frac{n!}{(n-k)!}\,P^{(\alpha +k,\beta +k)}_{n-k}(z), \quad 0\le k \le n, \end{aligned}$$
(2)

(see [22, (4.21.6)–(4.21.7)] for details). Additionally, if \(\alpha , \, \beta >-1\), the family of polynomials \(\, \{ P^{(\alpha ,\beta )}_{n} \} \) is orthogonal in \([-1,1]\) with respect to the weight \(w(x) = (1-x)^{\alpha } (1+x)^{\beta }\).

For a fixed \(m \in \mathbb {Z}_{+}\), let \(\mathscr {P}^{(\alpha ,\beta )}_{n,m}\) be the monic polynomial of degree \(n+m\) given by

$$\begin{aligned} \mathscr {P}^{(\alpha ,\beta )}_{n,m}= {P}^{(\alpha -m,\beta -m)}_{n+m}. \end{aligned}$$

Then,

$$\begin{aligned} \frac{\mathrm{d}^m\, \mathscr {P}^{(\alpha ,\beta )}_{n,m} }{\mathrm{d}z^m}(z) = \frac{(n+m)!}{n!}\,P^{(\alpha ,\beta )}_{n}(z), \end{aligned}$$

and thus \(\mathscr {P}^{(\alpha ,\beta )}_{n,m}\) is the m-th iterated integral (or a primitive of order m) of \(\frac{(n+m)!\, P^{(\alpha ,\beta )}_{n}}{n!}.\) In what follows, we shall refer to \(\mathscr {P}^{(\alpha ,\beta )}_{n,m}\) as the m-th fundamental iterated integral of \(\frac{(n+m)!}{n!}\, P^{(\alpha ,\beta )}_{n}\).

Given m complex numbers \(\omega _1, \ldots , \omega _m\), let \(\varOmega _k=( \omega _1, \ldots , \omega _k)\) for \(1 \le k \le m\), and \(\mathscr {P}^{(\alpha ,\beta )}_{n,m,\varOmega _m}\) be the m-th iterated integral of \(\frac{(n+m)!}{n!}\;P^{(\alpha ,\beta )}_{n}\) normalized by the conditions

$$\begin{aligned} \frac{\mathrm{d}^k\,\mathscr {P}^{(\alpha ,\beta )}_{n,m,\varOmega _m}}{\mathrm{d}z^k}(\omega _{m-k})=0, \quad k=0,1,\ldots , m-1. \end{aligned}$$
(3)

It is well known that there exists a unique polynomial of degree at most \(m-1\), \(\mathscr {A}_{n,m}(z)=\mathscr {A}_{n,m}(z;\omega _1, \ldots ,\omega _m )\), satisfying the conditions

$$\begin{aligned} \frac{\mathrm{d}^k\,\mathscr {A}_{n,m}}{\mathrm{d}z^k}(\omega _{m-k})= \frac{\mathrm{d}^k\,\mathscr {P}^{(\alpha ,\beta )}_{n,m} }{\mathrm{d}z^k}(\omega _{m-k}), \quad k=0,1,\ldots ,m-1. \end{aligned}$$
(4)

The polynomial \(\mathscr {A}_{n,m}\) is named the Abel–Goncharov interpolation polynomial, associated with conditions (4). The existence and uniqueness of \(\mathscr {A}_{n,m}\) are obvious if we observe that (4) is a triangular system of m equations and m unknowns (the coefficients of \(\mathscr {A}_{n,m}\)) whose determinant is equal to \(\prod _{k=0}^{m-1} k!\). The Abel–Goncharov interpolation polynomials are a generalization of Taylor’s polynomials, which correspond to the case \(\omega _m=\omega _{m-1}=\cdots =\omega _1\). In Sect. 3, you can see explicit expressions of Abel–Goncharov polynomials and some of their properties; for more details, see [1, 10, 23].

Therefore, the polynomial \(\mathscr {P}^{(\alpha ,\beta )}_{n,m,\varOmega _m}\) can be written as

$$\begin{aligned} \mathscr {P}^{(\alpha ,\beta )}_{n,m,\varOmega _m}(z)=\mathscr {P}^{(\alpha ,\beta )}_{n,m}(z)-\mathscr {A}_{n,m}(z), \end{aligned}$$
(5)

and we can interpret the polynomial \(\mathscr {P}^{(\alpha ,\beta )}_{n,m,\varOmega _m}\) as the polynomial solution of the Abel–Goncharov boundary value problem (see [1, §3.5])

$$\begin{aligned} \left\{ \begin{array}{ll} \frac{\mathrm{d}^m Y}{\mathrm{d}z^m}(z) = \frac{(n+m)!}{n!}\,P^{(\alpha ,\beta )}_{n}(z), &{} n > m,\\ \frac{\mathrm{d}^k Y}{\mathrm{d}z^k}(\omega _{m-k}) = 0, &{} k=0,1,\ldots ,m-1. \end{array}\right. \end{aligned}$$

Moreover, if \(\alpha ,\beta >-1\), then \(\mathscr {P}^{(\alpha ,\beta )}_{n,m,\varOmega _m}\) is the \((n+m)\)-th monic orthogonal polynomial with respect to the discrete–continuous Sobolev bilinear form (see [2, 3]) given by

$$\begin{aligned} \langle f,g \rangle _{S}= \sum _{k=0}^{m-1} \frac{\mathrm{d}^k f}{\mathrm{d}z^k}(\omega _{m-k}) \frac{\mathrm{d}^k g}{\mathrm{d}z^k}(\omega _{m-k}) + \int _{-1}^{1} \frac{\mathrm{d}^m f}{\mathrm{d}z^m}(x) \frac{\mathrm{d}^m g}{\mathrm{d}z^m}(x) (1-x)^{\alpha } (1+x)^{\beta } \mathrm{d}x. \end{aligned}$$

The main goal of this paper is to study the algebraic and asymptotic properties of the family of monic polynomials \(\{ \mathscr {P}_{n,m,\varOmega _m}^{(\alpha , \beta )}\}_n\), for \(m \in \mathbb {Z}_+,\)\(\{\omega _1, \ldots , \omega _m\} \subset \mathbb {C} \setminus [-1,1]\) and \(\alpha , \beta >-1\). The case \(\alpha =\beta =\omega _1= \cdots =\omega _m=0\) was early studied in [7], where the authors wrote “It would be interesting to obtain results, analogous to Theorem [7, Th. 2], for these polynomials” referring to the Gegenbauer (or ultraspherical) polynomials (\(\alpha =\beta >-1\)). Our Theorem 4 is an extension of [7, Th. 2] for Jacobi polynomials when all the constants of integration \(\omega _i\) are out of the interval \([-1,1]\).

In the next section we review some of the standard facts on Jacobi polynomials and we give the proof of some auxiliary results. The third section is devoted to study the Abel–Goncharov interpolation polynomial \(\mathscr {A}_{n,m}(z)\) of the m-th fundamental iterated integral of Jacobi polynomials. In the last section, our main results on asymptotic behavior of the sequence of polynomials \(\left\{ \mathscr {P}^{(\alpha ,\beta )}_{n,m,\varOmega _m} \right\} _n\) and its zeros are stated and proved.

2 Fundamental Iterated Integrals of Jacobi Polynomials

Recall that, for a fixed \(m,n \in \mathbb {Z}_{+}\), we denote by \(\mathscr {P}^{(\alpha ,\beta )}_{n,m}\) the Jacobi monic polynomial of degree \(n+m\) given by \({P}^{(\alpha -m,\beta -m)}_{n+m}\). From [20, §135 (12) and §138 (14)–(15)], we have the next lemma.

Lemma 1

For a fixed \(m\in \mathbb {Z}_{+}\), let \(\mathscr {P}^{(\alpha ,\beta )}_{n,m}\) be the \((n+m)\)-th fundamental primitive of n-th monic Jacobi polynomial with parameters \(\alpha ,\beta \in \mathbb {R}\). Then

$$\begin{aligned} \mathscr {P}^{(\alpha ,\beta )}_{n,m}(z)= & {} \mathscr {P}^{(\alpha ,\beta )}_{n+1,m-1}(z)+ a_{n,m}^{(\alpha ,\beta )} \, \mathscr {P}^{(\alpha ,\beta )}_{n,m-1}(z) + b_{n,m}^{(\alpha ,\beta )} \, \mathscr {P}^{(\alpha ,\beta )}_{n-1,m-1}(z);\nonumber \\ \hbox {where}\quad a_{n,m}^{(\alpha ,\beta )}= & {} \frac{2 (n+m) (\alpha -\beta )}{(2n+\alpha +\beta +2)(2n+\alpha +\beta )}, \nonumber \\ b_{n,m}^{(\alpha , \beta )}= & {} \frac{-4(n+m)(n+m-1)(n+\alpha )(n+\beta )}{(2n+\alpha +\beta )^2((2n+\alpha +\beta )^2-1)} \quad \hbox {and}\nonumber \\ \mathscr {P}^{(\alpha ,\beta )}_{n,0}(z)= & {} {P}^{(\alpha ,\beta )}_{n}(z). \end{aligned}$$
(6)

The asymptotic behavior of the sequence of polynomials \(\{\mathscr {P}^{(\alpha ,\beta )}_{n,m}\}_n\), stated in the following lemma, is a direct consequence of [22, Th. 8.21.7 & Eqn. (4.21.6)].

Lemma 2

If \(\alpha , \beta \in \mathbb {R}\) and \(m\in \mathbb {Z}_{+}\), then

  1. (1)

    (Outer strong asymptotic). Uniformly on compact subsets of \(\overline{\mathbb {C}} \setminus [-1,1]\)

    $$\begin{aligned} \lim _{n \rightarrow \infty }\frac{\mathscr {P}^{(\alpha ,\beta )}_{n,m}(z)}{\varphi ^{n}(z)}= & {} \psi _{\alpha ,\beta ,m}(z) \; \sqrt{\varphi (z)},\quad \hbox { where} \end{aligned}$$
    (7)
    $$\begin{aligned} \varphi (z)= & {} \frac{1}{2} \left( z+\sqrt{z^2-1}\right) \, \hbox { with } \,\sqrt{z^2-1}>0 \, \hbox { when } \, z>1 \nonumber \\ \hbox { and} \, \psi _{\alpha ,\beta ,m} (z)= & {} \frac{2^{2m-\alpha -\beta } \,\left( {\sqrt{z-1}+\sqrt{z+1}}\right) ^{\alpha +\beta -2m}}{ \root 4 \of {(z-1)^{2(\alpha -m)+1}}\; \root 4 \of {(z+1)^{2(\beta -m)+1}}}. \end{aligned}$$
    (8)
  2. (2)

    (n-th root asymptotic behavior). Uniformly on compact subsets of \(\overline{\mathbb {C}} \setminus [-1,1]\)

    $$\begin{aligned} \lim _{n \rightarrow \infty } \left| \mathscr {P}^{(\alpha ,\beta )}_{n,m}(z)\right| ^{\frac{1}{n}}\,= \left| \varphi (z)\right| . \end{aligned}$$
    (9)
  3. (3)

    (Comparative asymptotic behavior). Uniformly on compact subsets of \(\overline{\mathbb {C}} \setminus [-1,1]\)

    $$\begin{aligned} \lim _{n \rightarrow \infty }\frac{\mathscr {P}^{(\alpha ,\beta )}_{n,m}(z)}{{P}^{(\alpha ,\beta )}_{n}(z)}= \left( \frac{1}{\varphi ^{\prime }(z)}\right) ^m. \end{aligned}$$
    (10)

Note that \(2\varphi (z)= z + \sqrt{z^2-1}\) is the conformal mapping of \(\mathbb {C}\setminus [-1,1]\) onto the exterior of the unit circle, where the ellipses \(|z + \sqrt{z^{2} - 1}| = \rho , \,\, \rho >0\) are its level curves.

Formula (9) of n-th root asymptotic behavior of the fundamental iterated integral \(\mathscr {P}^{(\alpha ,\beta )}_{n,m}(z)\) is the same for classical Jacobi polynomials since m is fixed.

The two lemmas listed below are deduced from the well-known Rouché’s theorem (cf. [18, Th. 1.1.1]) and the Biernacki’s theorem (cf. [19, Th. 4.5.2]), respectively.

Lemma 3

Let f and g be polynomials, and \(\gamma \) a closed curve in the complex plane without self-intersections. If \(|f(z)|<|g(z)|\) for all \(z \in \gamma \), then \(f+g\) and g have the same number of zeros in the interior of \(\gamma \).

Lemma 4

Let f be a polynomial whose critical points lie in a compact subset \(K\subset \mathbb {C}\). If there exists \(\zeta \in \mathbb {C}\) such that \(f(\zeta )=0\), then the zeros of f lie in the compact set \([K]_{{\zeta }}=\{z \in \mathbb {C}: \inf _{w \in K}|z-w|\le \mathbf {d}_{K_{\zeta }}\}\), where \(\mathbf {d}_{K_{\zeta }}\) is the diameter of the compact set \(K_{\zeta }=K \cup \{\zeta \}\) (i.e., \(\mathbf {d}_{K_{\zeta }}=\sup _{u,v \in K_{\zeta }}|u-v|\)).

Of course, for all \(\zeta \in \mathbb {C}\) we get \(K \subset K_{\zeta } \subset [K]_{\zeta }\).

We denote by \(\mathbf {Z}^{(\alpha ,\beta )}_{n,m}(A)\) the set of zeros of \( \mathscr {P}^{(\alpha ,\beta )}_{n,m}\) on the set \(A \subset \mathbb {C}\). In the next theorem, we state some aspect of interest about the asymptotic behavior of the zeros of the fundamental iterated integrals of Jacobi polynomials.

Theorem 1

Let \(\alpha ,\beta >-1\), \(m\in \mathbb {N}\) fixed and \(I=(-1,1)\) , then

  1. (1)

    For each \(n>2m\), at least \((n-2m)\) distinct zeros of \(\mathscr {P}^{(\alpha ,\beta )}_{n,m}\) lie in I.

  2. (2)

    There exists a compact subset of the complex plane K, such that \((-1,1)\subset K\) and \(\displaystyle \bigcup _{n\ge 1}\mathbf {Z}^{(\alpha ,\beta )}_{k,m}(\mathbb {C}) \subset K.\)

  3. (3)

    All the roots of \(\mathscr {P}^{(\alpha ,\beta )}_{n,m}\) accumulate at \([-1,1]\).

Proof

  1. (1)

    From (6), for consecutive values of m, we get that there exist \((2m+1)\) constants \(a_0,\,a_1, \ldots ,\, a_{2m}\) such that \(\mathscr {P}^{(\alpha ,\beta )}_{n,m}(z)=\sum _{k=0}^{2m}\,a_k\,{P}^{(\alpha ,\beta )}_{n-m+k}(z)\). Hence, \(\mathscr {P}^{(\alpha ,\beta )}_{n,m}\) is a quasi-orthogonal polynomial of order 2m with respect to the measure \((1-x)^{\alpha } (1+x)^{\beta } dx\) on I. Hence, from [9, Th. 2] we have the first assertion of the theorem.

  2. (2)

    If \(m=1\), all the critical points of \(\mathscr {P}^{(\alpha ,\beta )}_{n,1}\) lie in \((-1,1)\) and by the first sentence of the theorem at least \(n-2\) of its zeros are on \(I=[-1,1]\). Let \(x_0 \in I\) such that \(\mathscr {P}^{(\alpha ,\beta )}_{n,1}(x_0)=0\). Then, according to the notations in Lemma 4, we get that \(I_{x_0}=I\) and \(\mathbf {d}_{I_{x_0}}=2\). Hence, from Lemma 4 we get \(\left( \bigcup _{n\ge 1}\mathbf {Z}^{(\alpha ,\beta )}_{k,1}(\mathbb {C})\right) \subset [I]_\mathbf {x_0}\).

    Suppose that for a fixed \(m \in \mathbb {N}\), there exists a compact set \(K^{(\alpha ,\beta )}_m\) such that \(\left( \bigcup _{n\ge 1}\mathbf {Z}^{(\alpha ,\beta )}_{k,m}(\mathbb {C})\right) \subset K^{(\alpha ,\beta )}_m\). As the zeros of \(\mathscr {P}^{(\alpha ,\beta )}_{n,m}\) are the critical points of \(\mathscr {P}^{(\alpha ,\beta )}_{n,m+1}\), from Theorem 1-1) and Lemma 4 we get the desired statement.

  3. (3)

    For a fixed \(m \in \mathbb {N}\), from Theorem 1-2) we know that the set of all zeros of \(\{\mathscr {P}^{(\alpha ,\beta )}_{n,m}\}\) are uniformly bounded.

Note that for all \(n \in \mathbb {Z}_+\) the functions \(\frac{\mathscr {P}^{(\alpha ,\beta )}_{n,m}(z)}{{P}^{(\alpha ,\beta )}_{n}(z)}\) and \(\left( \frac{1}{\varphi ^{\prime }(z)}\right) ^m=\left( \frac{\sqrt{z^2-1}}{\varphi (z)}\right) ^m\) are analytic on \(\overline{\mathbb {C}} \setminus [-1,1]\), where \(\varphi \) is given by (8). Furthermore, \(\left( \frac{\sqrt{z^2-1}}{\varphi (z)}\right) ^m \ne 0\) if \(z\in \overline{\mathbb {C}} \setminus [-1,1]\), and hence, the sentence 3 is a consequence of (10). \(\square \)

In the classical Szegő’s book [22, §6.72], the reader can find a full description of the distribution of the zeros of \(P^{(\alpha -m,\beta -m)}_{n+m}\), i.e., \(\mathscr {P}^{(\alpha ,\beta )}_{n,m}\), when \(\alpha , \beta \in \mathbb {R}\) and \(n,m\in \mathbb {Z}_{+}\) are fixed. In this sense, it is convenient to cite [12], where this analysis is embedded in a more general framework, using non-standard versions of orthogonality like the so-called quasi-orthogonality.

Additionally, there is a broad literature about zero location and asymptotic behavior of classical orthogonal polynomials with varying parameters. In particular, in the case of Jacobi polynomials (\(\alpha =\alpha _n\) and \(\beta =\beta _n\)), the reader can see [8, 11, 14,15,16] and references therein. The case considered in the current paper is different, because the parameters in the fundamental iterated integrals of Jacobi polynomials are constant.

3 Abel–Goncharov Interpolation Polynomials

Given m complex numbers \(\omega _1, \ldots , \omega _m \), and \({\varOmega _k}\) for \(1 \le k \le m\), as in (3). As we show in the first section, there exists a unique polynomial \(\mathscr {A}_{n,m}\) of degree at most \(m-1\), such that equations (4) are satisfied. This polynomial is given by

$$\begin{aligned} \mathscr {A}_{n,m}(z)= \mathscr {P}^{(\alpha ,\beta )}_{n,m}(\omega _m) + \sum _{k=1}^{m-1} \frac{1}{k!}\frac{ d^k \mathscr {P}^{(\alpha ,\beta )}_{n,m}}{dz^k}(\omega _{m-k}) \; \mathscr {G}_{k,m}(z) \end{aligned}$$
(11)

where \(\mathscr {G}_{k,m}(z)=\mathscr {G}_{k,m}(z;\omega _m,\omega _{m-1}, \ldots , \omega _{m-k}) \) is the monic polynomial of degree k , generate by the k-th iterated integral

$$\begin{aligned} \mathscr {G}_{k,m}(z) = k!\,\int _{\omega _m}^{z} \int _{\omega _{m-1}}^{s_{m-1}} \ldots \int _{\omega _{m-(k-1)}}^{s_{m-(k-1)}} \mathrm{d}s_{m-1}\,\mathrm{d}s_{m-2} \ldots \mathrm{d}s_{m-k}, \end{aligned}$$
(12)

see [23, §4.1.4 (15)–(16)] for more details. The polynomial \(\mathscr {G}_{k,m}\) is called the k-th Goncharov’s polynomial associated with \(\{\omega _1, \ldots , \omega _m \}\).

Example 1

(Abel’s polynomials) If \(\omega _1, \ldots , \omega _m\) form an arithmetic progression, i.e., \(\omega _{m-k}=\omega + k \vartheta \), where \(\omega , \vartheta \in \mathbb {C}\) are fixed and \(k=0,1,\ldots ,m-1\), it is well known that in this case the k-th Goncharov polynomials

$$\begin{aligned} \mathscr {G}_{k,m}(z)=(z-\omega ) (z-\omega -(m-k)\vartheta )^{k-1}, \end{aligned}$$
(13)

are the so-called k-th Abel’s polynomial.

If \(\vartheta =0\), we have the special case \(\mathscr {G}_{k,m}(z)=(z-\omega )^{k}\) (Taylor’s case), and then, the m-th Abel–Goncharov interpolation polynomial (11) becomes the Taylor’s expansion of \(\mathscr {P}^{(\alpha ,\beta )}_{n,m}\) in \(\omega \) , as we mentioned in Introduction.

According to (2), it follows that \(\frac{1}{k!}\frac{ \mathrm{d}^k \mathscr {P}^{(\alpha ,\beta )}_{n,m}}{\mathrm{d}z^k}(\omega _{m-k})= {n+m\atopwithdelims ()k} \mathscr {P}^{(\alpha ,\beta )}_{n,m-k}(\omega _{m-k})\), and replacing this formula in (11), we get

$$\begin{aligned} \mathscr {A}_{n,m}(z)= \mathscr {P}^{(\alpha ,\beta )}_{n,m}(\omega _m) + \sum _{k=1}^{m-1} {n+m\atopwithdelims ()k} \mathscr {P}^{(\alpha ,\beta )}_{n,m-k}(\omega _{m-k}) \; \mathscr {G}_{k,m}(z). \end{aligned}$$
(14)

Theorem 2

Given \(m >0\) and \(\omega _1, \ldots , \omega _m \in \mathbb {C}\setminus [-1,1]\) fixed, let \( \mathscr {A}_{n,m}(z)\) be the Abel–Goncharov polynomial of interpolation associated with conditions (4),

$$\begin{aligned} \sigma _{m}=\max _{0\le k \le m-1}|\varphi (\omega _{m-k})|,\; U=\{k: |\varphi (\omega _{m-k})|=\sigma _{m} \}\; \text {and} \; \hat{k}=\max _{k \in U}|k|. \end{aligned}$$
(15)

Then, uniformly on compact subsets of \(\overline{\mathbb {C}}\)

$$\begin{aligned} \lim _{n \rightarrow \infty } \frac{\mathscr {A}_{n,m}(z)}{n^{\hat{k}} \;\mathscr {P}^{(\alpha ,\beta )}_{n,m-\hat{k}}(\omega _{m-\hat{k}})}= & {} \frac{\mathscr {G}_{\hat{k},m}(z)}{\hat{k}!}, \end{aligned}$$
(16)
$$\begin{aligned} \lim _{n \rightarrow \infty } |\mathscr {A}_{n,m}(z)|^{\frac{1}{n}}= & {} \sigma _{m}. \end{aligned}$$
(17)

The branch of the square root in (8) is chosen so that \(\left| \varphi (\omega _{m-k})\right| >1\), for each \(0 \le k \le m-1\).

Proof

Let \(V=\{k: |\varphi (\omega _{m-k})|< \sigma _{m} \}\). Obviously \(U\cap V= \emptyset \) and \(U \cup V=\{1,2,\ldots ,m\}\). From (14), we get

$$\begin{aligned} \left( \frac{(n+m-\hat{k})!}{(n+m)!}\right) \frac{\mathscr {A}_{n,m}(z)}{\mathscr {P}^{(\alpha ,\beta )}_{n,m-\hat{k}}(\omega _{m-\hat{k}})}= & {} \frac{\mathscr {G}_{\hat{k},m}(z)}{\hat{k}!} + \sum _{k \in U \setminus \{ \hat{k}\}} \; A_{n,m,k} \; \frac{\mathscr {G}_{k,m}(z)}{k!} \nonumber \\&+ \sum _{k \in V} \; A_{n,m,k} \; \frac{\mathscr {G}_{k,m}(z)}{k!}, \end{aligned}$$
(18)

where    \(\displaystyle A_{n,m,k}= \frac{(n+m-\hat{k})!}{(n+m-k)!} \frac{\mathscr {P}^{(\alpha ,\beta )}_{n,m-k}(\omega _{m-k})}{\mathscr {P}^{(\alpha ,\beta )}_{n,m-\hat{k}}(\omega _{m-\hat{k}})}.\)

Firstly, we will prove that for all \(k \in \left( U \cup V\right) \setminus \{\hat{k}\}\)

$$\begin{aligned} \lim _{n \rightarrow \infty } A_{n,m,k}= 0. \end{aligned}$$
(19)

If \(k \in V\), then \(|\varphi (\omega _{m-k})|<|\varphi (\omega _{m-\hat{k}})|\),

$$\begin{aligned} A_{n,m,k} = \frac{(n+m-\hat{k})!}{(n+m-k)!} \left( \frac{\varphi (\omega _{m-k})}{\varphi (\omega _{m-\hat{k}})}\right) ^n \frac{\mathscr {P}^{(\alpha ,\beta )}_{n,m-k}(\omega _{m-k})}{\varphi ^n(\omega _{m-k})} \, \frac{\varphi ^n(\omega _{m-\hat{k}})}{\mathscr {P}^{(\alpha ,\beta )}_{n,m-\hat{k}}(\omega _{m-\hat{k}})} \end{aligned}$$

and from (7), we can assert that for \(k \in V\) we get (19).

If \(k \in U \setminus \{ \hat{k}\}\), then \(k < \hat{k}\) and \(|\varphi (\omega _{m-k})|= |\varphi (\omega _{m-\hat{k}})|\). Writing \(\varphi (\omega _{m-k})= |\varphi (\omega _{m-\hat{k}})| e^{i \theta } \) and \(\varphi (\omega _{m-\hat{k}})= |\varphi (\omega _{m-\hat{k}})| e^{i \hat{\theta }}\), with \(\theta , \hat{\theta } \in [0.2\pi )\), we get

$$\begin{aligned} A_{n,m,k} = \left( \frac{(n+m-\hat{k})!}{(n+m-k)!}\right) e^{n(\theta -\hat{\theta })\,i} \left( \frac{\mathscr {P}^{(\alpha ,\beta )}_{n,m-k}(\omega _{m-k})}{\varphi ^n(\omega _{m-k})}\right) \left( \frac{\varphi ^n(\omega _{m-\hat{k}})}{\mathscr {P}^{(\alpha ,\beta )}_{n,m-\hat{k}}(\omega _{m-\hat{k}})}\right) \end{aligned}$$

and as in the previous reasoning, from (7), we can assert that if \(k \in U \setminus \{ \hat{k}\}\) we have (19).

Now, according to (18)–(19), we get (16). Finally, (17) is a consequence of (16) and (9). \(\square \)

4 General Primitive of Jacobi Polynomials and Its Zeros

For \(\rho \in \mathbb {R}_{+}\), let \(\mathbf {E}_{\rho }\) be the ellipse \(|z-1|+|z+1|= \rho + \rho ^{-1}\). Obviously, \(\mathbf {E}_{\rho }\) divides the complex plane into the following two disjoint regions

$$\begin{aligned} \overline{\mathbf {E}}_{\rho }= & {} \left\{ z\in \mathbb {C}: |z-1|+|z+1|> \rho + \rho ^{-1} \right\} , \\ \underline{\mathbf {E}}_{\rho }= & {} \left\{ z\in \mathbb {C}: |z-1|+|z+1|\le \rho + \rho ^{-1} \right\} . \end{aligned}$$

Analogously to the notations introduced in Theorem 1, we denote

$$\begin{aligned} \mathbf {Z}^{(\alpha ,\beta )}_{n,m,\varOmega _m} = \left\{ z \in \mathbb {C}: \mathscr {P}^{(\alpha ,\beta )}_{n,m,\varOmega _m}(z)=0 \right\} \end{aligned}$$

( i.e., the set of \((n+m)\) zeros of \(\mathscr {P}^{(\alpha ,\beta )}_{n,m,\varOmega _m}\)) and by \(\mathbf {Z}^{(\alpha ,\beta )}_{m,\varOmega _m}\) the set of accumulation points of zeros of \(\{\mathscr {P}_{n,m,\varOmega }^{(\alpha ,\beta )}\}\).

Lemma 5

Let \(\alpha ,\beta >-1\), \(m\in \mathbb {N}\) and \(\varOmega _m=(\omega _1, \ldots , \omega _m) \in \mathbb {C}^m\) fixed. Then, there exists a compact subset \(K\subset \mathbb {C}\), such that \((-1,1)\subset K\) and \(\mathbf {Z}^{(\alpha ,\beta )}_{n,m,\varOmega _m} \subset K\) for all n.

Proof

We proceed analogously to the proof of Theorem 1. If \(m=1\), for all \(n \ge 1\) the critical points of \(\mathscr {P}^{(\alpha ,\beta )}_{n,1,\omega _1}\) are on \(I=[-1,1]\). Then, from Lemma 4, we get \(\mathbf {Z}^{(\alpha ,\beta )}_{n,1,\varOmega _1}\) is a subset of the compact set \([I]_{\omega _1}\), which was defined in Lemma 4.

Suppose that for a fixed \(m \in \mathbb {N}\) there exists a compact subset \(K_{m-1}\) such that \(\mathbf {Z}^{(\alpha ,\beta )}_{n,m-1,\varOmega _{m-1}} \subset K_{m-1}\). As the zeros of \(\mathscr {P}^{(\alpha ,\beta )}_{n,m-1,\varOmega _{m-1}}\) are the critical points of \(\mathscr {P}^{(\alpha ,\beta )}_{n,m,\varOmega _{m}}\), from Lemma 4 we get \(\mathbf {Z}^{(\alpha ,\beta )}_{n,m,\varOmega _{m}} \subset [K_{m-1}]_{{\omega }_m}\). \(\square \)

Theorem 3

Given \(m >0\) and \(\omega _1, \ldots , \omega _m \in \mathbb {C}\setminus [-1,1]\) fixed, let \(\rho _{m }=2\sigma _{m}\), where \(\sigma _{m}\) is given by (15). Then, uniformly on compact subsets of \(\overline{\mathbf {E}}_{\rho _{m}} \)

$$\begin{aligned} \lim _{n \rightarrow \infty }\frac{\mathscr {P}^{(\alpha ,\beta )}_{n,m,\varOmega _m}(z)}{{P}^{(\alpha ,\beta )}_{n}(z)}= \left( \frac{1}{\varphi ^{\prime }(z)}\right) ^m. \end{aligned}$$
(20)

Furthermore, \(\mathbf {Z}^{(\alpha ,\beta )}_{m,\varOmega _m} \subset \underline{\mathbf {E}}_{\rho _{m}}\).

Proof

From (5), we know that

$$\begin{aligned} \frac{\mathscr {P}^{(\alpha ,\beta )}_{n,m,\varOmega _m}(z)}{{P}^{(\alpha ,\beta )}_{n}(z)}= \frac{\mathscr {P}^{(\alpha ,\beta )}_{n,m}(z)}{{P}^{(\alpha ,\beta )}_{n}(z)} -\frac{\mathscr {A}_{n,m}(z)}{{P}^{(\alpha ,\beta )}_{n}(z)}. \end{aligned}$$

The uniform limit of the first quotient in the right side is given by (10). Hence, to proof (20) it is sufficient to proof that

$$\begin{aligned} \lim _{n \rightarrow \infty } \frac{\mathscr {A}_{n,m}(z)}{P_{n}^{(\alpha ,\beta )} (z)}=0, \quad \hbox {uniformly on compact subsets of } \; \overline{\mathbf {E}}_{\rho _{m}}. \end{aligned}$$
(21)

From (14), we have

$$\begin{aligned} \frac{\mathscr {A}_{n,m}(z)}{P_{n}^{(\alpha ,\beta )}(z)}=\sum _{k=0}^{m-1} \frac{\mathscr {P}^{(\alpha ,\beta )}_{n,m-k}(\omega _{m-k})}{P_{n}^{(\alpha ,\beta )} (\omega _{m-k})} \left( \frac{(n+m)!}{(n+m-k)!}\frac{P_{n}^{(\alpha ,\beta )} (\omega _{m-k})}{P_{n}^{(\alpha ,\beta )} (z)} \right) \frac{\mathscr {G}_{k,m}(z)}{k!}, \end{aligned}$$

where \(\mathscr {G}_{m}(z)\equiv 1\). For \(k=0,1,\ldots ,m-1\), we get

$$\begin{aligned} \frac{(n+m)!}{(n+m-k)!}\frac{P_{n}^{(\alpha ,\beta )} (\omega _{m-k})}{P_{n}^{(\alpha ,\beta )} (z)}= & {} \frac{(n+m)!}{(n+m-k)!} \left( \frac{\varphi (\omega _{m-k})}{\varphi (z)}\right) ^{n} \frac{P_{n}^{(\alpha ,\beta )}(\omega _{m-k})}{\varphi ^n(\omega _{m-k})}\\&\cdot \frac{ \varphi ^n(z)}{P_{n}^{(\alpha ,\beta )}(z)}. \end{aligned}$$

As \(|\varphi (\omega _{m-k})|<|\varphi (z)|\) for all \(z \in \overline{\mathbf {E}}_{\rho _*}\), from (10) it follows (21).

Finally, the assertion \(\mathbf {Z}^{(\alpha ,\beta )}_{m,\varOmega _m} \subset \underline{\mathbf {E}}_{\rho _{m}}\) is a consequence of (20) and Lemma 5, using analogous argument as in the proof of 3) in Theorem 1. \(\square \)

Theorem 4

Assume that \(m >0\), \(\omega _1, \ldots , \omega _m \in \mathbb {C}\setminus [-1,1]\) and \( \hat{k}=m-1\) . Then the accumulation points of zeros of \( \{\mathscr {P}_{n,m,\varOmega _{m}}^{(\alpha ,\beta )}\} \) are located on the union of the interval \([-1,1]\) and the ellipse

$$\begin{aligned} \mathbf {E}_{\rho _{m}}= \left\{ z\in \mathbb {C}: |z-1|+|z+1|= \rho _{m} + \rho _{m}^{-1} \right\} , \end{aligned}$$
(22)

where \(\hat{k}\) is defined in (15), \(\displaystyle \rho _{m}\) is as in Theorem 3, and the branch of the square root in (8) is chosen so that \(\left| \varphi (\omega _k)\right| >1\), for each \(1 \le k \le m\) (see Fig. 1).

Proof

From (5), the zeros of the polynomial \(\mathscr {P}_{n,m,\varOmega _m}^{(\alpha ,\beta )}\) satisfy the equation

$$\begin{aligned} \left| \mathscr {P}_{n,m}^{(\alpha ,\beta )}(z)\right| ^{\frac{1}{n}} = \left| \mathscr {A}_{n,m}(z)\right| ^{\frac{1}{n}}. \end{aligned}$$
(23)

Taking the limit as \(n \rightarrow \infty \) on both sides of (23), from (17) and (9), we have that \(\mathbf {Z}^{(\alpha ,\beta )}_{m,\varOmega _m} \subset \mathbf {E}_{\rho _{m}}\) where

$$\begin{aligned} \mathbf {E}_{\rho _{m}}=\left\{ z \in \mathbb {C}: |z+\sqrt{z^2-1}|=\rho _{m}\right\} . \end{aligned}$$

Let \(\tilde{k}\) be an index, \(1 \le \tilde{k} \le m\), such that \(\displaystyle \varphi (\omega _{\tilde{k}})=\rho _{m} e^{i\tilde{\theta }}\), \(0 \le \tilde{\theta } < 2 \pi \). Hence, we have that \(z+\sqrt{z^2-1}= \rho _{m}\, e^{i\tilde{\theta }}\), \(z-\sqrt{z^2-1}=\rho _{m}^{-1}\,e^{-i\tilde{\theta }}\), and taking the difference between both we get \(\sqrt{z^2-1}= (\rho _{m} e^{i\tilde{\theta }}+ \rho _{m}^{-1}e^{-i\tilde{\theta }})/2\). Thus,

$$\begin{aligned} |z-1|+|z+1|= \frac{|\rho _{m} e^{i\tilde{\theta }} -1|^2 + |\rho _{m} e^{i\tilde{\theta }} +1|^2}{2 \rho _{m} }, \end{aligned}$$

which is equivalent to the equation of the ellipse in (22). As the limit that we have taken is uniform on compact subsets of \(\mathbb {C}\setminus [-1,1]\), the theorem is proved. \(\square \)

Fig. 1
figure 1

Zeros of \(\mathscr {P}_{60,2,\varOmega _{2}}^{(-1/2,1/2)}\) and \(\mathscr {P}_{60,3,\varOmega _{3}}^{(0,0)}\), where \(\varOmega _{2}=(4+i,-2)\) and \(\varOmega _{3}=(4i,-i,2)\). In each case, the ellipse is given by (22)

Full size image

In example 1, if for each \(0\le k \le m-1\) it holds that \((\omega + k \vartheta ) \not \in [-1,1]\), then all the zeros of the Abel’s polynomials (13) are out to the interval \([-1,1]\). What is interesting for the following corollary.

Corollary 1

Under the assumptions of Theorems 2 and 4, if the zeros of the Goncharov polynomial \(\mathscr {G}_{\hat{k},m}\) are outside to the interval \([-1,1]\) then the accumulation points of zeros of \(\{\mathscr {P}_{n,m,\varOmega _m}^{(\alpha ,\beta )}\}\) are located on the ellipse \(\mathbf {E}_{\rho _{m}}\).

Proof

Obviously, from Theorem 4 it is sufficient to prove that there does not exist an accumulation point of zeros of \(\{\mathscr {P}_{n,m,\varOmega _m}^{(\alpha ,\beta )}\}\) located on the interval \([-1,1]\).

Let \(\varepsilon \in \mathbb {R}\) such that \(\omega _1, \ldots , \omega _m\) and the zeros of \(\mathscr {G}_{\hat{k},m}\) are on the exterior of the ellipse \(\mathbf {E}_{1+\varepsilon }\). Thus, if \(w \in \mathbf {E}_{1+\varepsilon }\), from (7) and (16) we get, for sufficiently large values of n,

$$\begin{aligned} \mathscr {A}_{n,m}(w)\approx & {} {n+m\atopwithdelims ()\hat{k}} \; \psi _{\alpha ,\beta ,m - \hat{k}}(\omega _{m-\hat{k}}) \;\varphi ^{n+\frac{1}{2}}(\omega _{m-\hat{k}}) \; \mathscr {G}_{\hat{k},m}(w), \end{aligned}$$
(24)
$$\begin{aligned} \mathscr {P}_{n,m}^{(\alpha ,\beta )}(w)\approx & {} \psi _{\alpha ,\beta ,m}(w) \;\varphi ^{n+\frac{1}{2}}(w). \end{aligned}$$
(25)

As the zeros of the Goncharov polynomial \(\mathscr {G}_{\hat{k},m}\) are on the exterior of the ellipse \(\mathbf {E}_{1+\varepsilon }\), then from (24), there exists \(N_1\in \mathbb {Z}_{+}\) such that for \(n>N_1\) the zeros of the polynomial \( \mathscr {A}_{n,m}\) are on the exterior of the ellipse \(\mathbf {E}_{1+\varepsilon }\) too. From (24)–(25)

$$\begin{aligned} \left| \mathscr {A}_{n,m}(w) \right|\approx & {} {n+m \atopwithdelims ()\hat{k}} \; \left| \frac{\mathscr {G}_{\hat{k},m}(w) \; \psi _{\alpha ,\beta ,m - \hat{k}}(\omega _{m-\hat{k}}) }{ \psi _{\alpha ,\beta ,m}(w) } \right| \; \left| \frac{\varphi (\omega _{m-\hat{k}})}{\varphi (w)} \right| ^{n+\frac{1}{2}} \; \nonumber \\&\cdot \left| \psi _{\alpha ,\beta ,m}(w) \;\varphi ^{n+\frac{1}{2}}(w) \right| \nonumber \\\ge & {} \left| \frac{\mathscr {G}_{\hat{k},m}(w) \; \psi _{\alpha ,\beta ,m}(\omega _{m-\hat{k}})}{\psi _{\alpha ,\beta ,m}(w)} \right| \; \left| \frac{\varphi (\omega _{m-\hat{k}})}{\varphi (w)} \right| ^{n+\frac{1}{2}} \; \left| \mathscr {P}_{n,m}^{(\alpha ,\beta )}(w) \right| \end{aligned}$$
(26)

As it is well know from classical complex analysis (cf. [13, §51]), \(\varphi (z)\) maps the ellipse \(|z-1|+|z+1|=r+ \frac{1}{r}\), with \(r>0\), onto the circumference \(|z|=r\). Hence, as each \(\omega _k\) is on the exterior of the ellipse \(\mathbf {E}_{1+\varepsilon }\) and \(w \in \mathbf {E}_{1+\varepsilon }\), we get that \(|\varphi (\omega _{m-\hat{k}})|>|\varphi (w)|\). Thus, from (26) there exists \(N_2\in \mathbb {Z}_{+}\) such that if \(n>N_2\), then \(\displaystyle \left| \mathscr {A}_{n,m}(w) \right| > \left| \mathscr {P}_{n,m}^{(\alpha ,\beta )}(w) \right| \).

Finally, from Lemma 3 and Theorem 4, the corollary is proven. \(\square \)