On Analytical Continuation of the Horn’s Hypergeometric Functions H₃ and Their Ratios

Abstract

This paper considers the Horn’s hypergeometric function $H_3$, which is closely related to other hypergeometric functions and has various mathematical or physical applications. The problem of analytical extension of this function is solved using a special family of functions—branched continued fractions. A new domain of analytical extension of the Horn’s hypergeometric functions $H_3$ and their ratios under certain conditions to real parameters are established. This paper also contains an example of the presentation and continuation of some special function and an analysis of numerical results.

1. Introduction

Special functions arise naturally in problems of physics, engineering, chemistry, computer sciences, statistics, and many other fields of science and industry [1,2,3,4,5,6].

In 1812, Johann Carl Friedrich Gauss considered the hypergeometric function,

$$\begin{array}{cc}\hfill F(\alpha ,\beta ;\gamma ;z)& =1+{\displaystyle \frac{\alpha \beta }{1·\gamma }}z+{\displaystyle \frac{\alpha (\alpha +1)\beta (\beta +1)}{1·2·\gamma (\gamma +1)}}{z}^2\hfill \\ & +{\displaystyle \frac{\alpha (\alpha +1)(\alpha +2)\beta (\beta +1)(\beta +2)}{1·2·3·\gamma (\gamma +1)(\gamma +3)}}{z}^3+\dots ,\hfill \end{array}$$

where $\alpha ,$ $\beta ,$ and $\gamma$ are complex constants herewith $\gamma \notin \{0,-1,-2,\dots \},$ and obtained the continued fraction (see [7]),

$$\begin{array}{c}\hfill {\displaystyle \frac{F(\alpha ,\beta +1;\gamma +1;z)}{F(\alpha ,\beta ;\gamma ;z)}}={\displaystyle \frac{1}{1-{\displaystyle \frac{{\displaystyle \frac{\alpha (\gamma -\beta )}{\gamma (\gamma +1)}}z}{1-{\displaystyle \frac{{\displaystyle \frac{(\beta +1)(\gamma -\alpha +1)}{(\gamma +1)(\gamma +2)}}z}{1-{\displaystyle \frac{{\displaystyle \frac{(\alpha +1)(\gamma -\beta +1)}{(\gamma +2)(\gamma +3)}}z}{1-{\displaystyle \frac{{\displaystyle \frac{(\beta +2)(\gamma -\alpha +2)}{(\gamma +3)(\gamma +4)}}z}{1{-}_{ \ddots }}}}}}}}}}},\end{array}$$

which was called the Gauss continued fraction.

If all the coefficients of the Gauss continued fraction are different from zero, then it converges in the domain

$$\mathfrak{H}=\{z\in \mathbb{C}:z\notin [1,+\infty \left)\right\},$$

except for possibly certain isolated points, is equal to the function

$${\displaystyle \frac{F(\alpha ,\beta +1;\gamma +1;z)}{F(\alpha ,\beta ;\gamma ;z)}}$$

in the neighborhood of the origin and provides an analytic continuation of this function in $\mathfrak{H}.$ The continued fraction converges uniformly on every compact subset the domain $\mathfrak{H},$ which contains none of the above-mentioned isolated points. These isolated points, if they exist, are the poles of the function represented by the continued fraction of Gauss [8,9,10] (see also Theorem 98.1 [11]).

Examples of representing special functions as continued fractions can be found in [11,12,13,14].

In 1966, Nadiya Dronyuk constructed a branched continued fraction for the Appell’s hypergeometric function $F_1$ [15]. Since then, the study of the representation of functions of the Appell, Horn, and Lauricella families by branched continued fractions has begun and continues (see, for example, [16,17,18,19,20,21] and also [22,23,24,25,26]).

This paper considers the Horn’s hypergeometric function $H_3,$ defined as follows [27].

$$\begin{array}{c}\hfill H_3(\alpha ,\beta ;\gamma ;\mathbf{z})=\sum _{r,s=0}^{\infty }{\displaystyle \frac{{\left(\alpha \right)}_{2r+s}{\left(\beta \right)}_s}{{\left(\gamma \right)}_{r+s}}}{\displaystyle \frac{z_1^r}{r!}}{\displaystyle \frac{z_2^s}{s!}},|z_1|<p,|z_2|<q,\end{array}$$

(1)

where $\alpha ,$ $\beta ,$ and $\gamma$ are complex constants herewith $\gamma \notin \{0,-1,-2,\dots \},$ p and q are positive numbers such that $4p+{(2q-1)}^2=1,$ ${(·)}_k$ is the Pochhammer symbol, and $\mathbf{z}=(z_1,z_2)\in {\mathbb{C}}^2$.

Let

$$\mathfrak{I}=\{i\left(k\right)=(i_0,i_1,i_2,\dots ,i_k):i_r=1,2,0\le r\le k,k\ge 0\}.$$

In [28], it is shown that, for $i_0\in \mathfrak{I},$

$$\begin{array}{cc}\hfill & (1+4z_1{\delta }_{i_0}^2){\displaystyle \frac{H_3(\alpha ,\beta ;\gamma ;-\mathbf{z})}{H_3(\alpha +{\delta }_{i_0}^1,\beta +{\delta }_{i_0}^2;\gamma +1;-\mathbf{z})}}\hfill \\ \hfill & =1+d_{i_0}^1{z}_1+d_{i_0}^2{z}_2+\sum _{i_1=1}^2{\displaystyle \frac{c_{i\left(1\right)}^1{z}_{i_1}+c_{i\left(1\right)}^2{z}_1{z}_2}{{\displaystyle 1+d_{i\left(1\right)}^1{z}_1+d_{i\left(1\right)}^2{z}_2+\sum _{i_2=1}^2{\displaystyle \frac{c_{i\left(2\right)}^1{z}_{i_2}+c_{i\left(2\right)}^2{z}_1{z}_2}{1+d_{i\left(2\right)}^1{z}_1+d_{i\left(2\right)}^2{z}_2{+}_{ \ddots }}}}}},\hfill \end{array}$$

(2)

where ${\delta }_i^j$ is the Kronecker delta, i.e.,

$${\delta }_i^j=\left\{\begin{array}{c}1,\mathrm{if}i=j,\hfill \\ 0,\mathrm{if}i\ne j,\hfill \end{array}\right.$$

and

$$\begin{array}{c}{d}_{i_0}^1=2{\delta }_{i_0}^2{\displaystyle \frac{2\gamma -\alpha }{\gamma }},d_{i_0}^2={\delta }_{i_0}^2{\displaystyle \frac{\alpha -\beta -1}{\gamma }},\end{array}$$

(3)

$$\begin{array}{c}{c}_{i_0,1}^1={\displaystyle \frac{(2\gamma -\alpha +{\delta }_{i_0}^2)(\alpha +{\delta }_{i_0}^1)}{\gamma (\gamma +1)}},c_{i_0,1}^2={\displaystyle \frac{2{\delta }_{i_0}^2(2\gamma -\alpha -\beta )(\alpha +{\delta }_{i_0}^1)}{\gamma (\gamma +1)}},\end{array}$$

(4)

$$\begin{array}{c}{c}_{i_0,2}^1={\displaystyle \frac{(\beta +{\delta }_{i_0}^2)(\gamma -\alpha +{\delta }_{i_0}^2)}{\gamma (\gamma +1)}},c_{i_0,2}^2=4c_{i_0,2}^1,\end{array}$$

(5)

and, for $i\left(k\right)\in \mathfrak{I}$ and $k\ge 1,$

$$\begin{array}{c}{d}_{i\left(k\right)}^1=2{\delta }_{i_k}^2{\displaystyle \frac{2\gamma -\alpha +k+{\sum }_{r=0}^{k-1}{\delta }_{i_r}^2}{\gamma +k}},d_{i\left(k\right)}^2={\delta }_{i_k}^2{\displaystyle \frac{\alpha -\beta -1+{\sum }_{r=0}^{k-1}({\delta }_{i_r}^1-{\delta }_{i_r}^2)}{\gamma +k}},\end{array}$$

(6)

$$\begin{array}{c}{c}_{i\left(k\right),1}^1={\displaystyle \frac{(2\gamma -\alpha +k+{\sum }_{r=0}^k{\delta }_{i_r}^2)(\alpha +{\sum }_{r=0}^k{\delta }_{i_r}^1)}{(\gamma +k)(\gamma +k+1)}},c_{i\left(k\right),1}^2={\displaystyle \frac{2{\delta }_{i_k}^2(2\gamma -\alpha -\beta +k)(\alpha +{\sum }_{r=0}^k{\delta }_{i_r}^1)}{(\gamma +k)(\gamma +k+1)}},\end{array}$$

(7)

$$\begin{array}{c}{c}_{i\left(k\right),2}^1={\displaystyle \frac{(\beta +{\sum }_{r=0}^k{\delta }_{i_r}^2)(\gamma -\alpha +{\sum }_{r=0}^k{\delta }_{i_r}^2)}{(\gamma +k)(\gamma +k+1)}},c_{i\left(k\right),2}^2=4c_{i\left(k\right),2}^1.\end{array}$$

(8)

In addition, in [28,29], several criteria of the convergence of (2) under certain conditions to the real parameters of (1) are established. It is proved that obtained domains of convergence, in particular (see Theorem 2 [28]),

$$\begin{array}{c}\hfill \mathfrak{G}=\{\mathbf{z}\in {\mathbb{C}}^2:2\mathrm{Re}\left(z_1\right)-\mathrm{Re}\left(z_2\right)<{\lambda }_1,\mathrm{Re}\left(z_2\right)<{\lambda }_2,|z_1{z}_2|-\mathrm{Re}\left(z_1{z}_2\right)<{\nu }_3,|z_k|+\mathrm{Re}\left(z_k\right)<{\nu }_k,k=1,2\},\end{array}$$

where ${\lambda }_1,$ ${\lambda }_2,$ ${\mu }_1,$ ${\mu }_2,$ ${\nu }_1,$ ${\nu }_2,$ and ${\nu }_3$ are positive numbers such that

$$\begin{array}{c}\hfill {\displaystyle \frac{2{\nu }_1}{{\mu }_1}}+{\displaystyle \frac{{\nu }_2+4{\nu }_3}{{\mu }_2}}\le 2(1-{\mu }_1),{\displaystyle \frac{2{\nu }_1+4{\nu }_3}{{\mu }_1}}+{\displaystyle \frac{{\nu }_2+4{\nu }_3}{{\mu }_2}}\le 2(1-2{\lambda }_1-2{\lambda }_2-{\mu }_2),\end{array}$$

of branched continued fraction expansions are domains of analytical continuation of the corresponding the Horn’s hypergeometric functions $H_3$ and their ratios. It should be noted that the authors used and developed various techniques of expanding the domain of convergence from the already known to a wider one. Here, the domain is an open connected set.

In this paper, in Section 2, we establish a new domain of analytical extension of the Horn’s hypergeometric functions $H_3$ and their ratios under certain conditions to real parameters. This is obtained by representing them as branched continued fractions. In Section 3, we provide an example of the presentation and extension of some special function and an analysis of numerical results.

2. Domains of Analytical Extension

The method will be used here, according to which the domains of analytical continuation of the special functions are the domains of convergence of their branched continued fraction expansions (see [30,31,32]).

We recall that the branched continued fraction is said to converge if its sequence of approximants converges to a finite limit, which is called its value.

The branched continued fraction, whose elements are functions in the certain domain $\mathfrak{D}$ ($\mathfrak{D}\subset {\mathbb{C}}^2$) is called uniformly convergent on set $\mathfrak{E}$ ($\mathfrak{E}\subset \mathfrak{D}$) if its sequence of approximants converges uniformly on $\mathfrak{E}.$ If this occurs for an arbitrary set $\mathfrak{E}$ such that $\overline{\mathfrak{E}}\subset \mathfrak{D}$, then we say that the branched continued fraction converges uniformly on every compact subset of $\mathfrak{D}.$

Note that an analysis of the study on the convergence problem of branched continued fractions can be found in [33].

The following is true:

Theorem 1.

Let $i_0=1.$ Assume that $\alpha ,$$\beta ,$ and γ are real constants that satisfy the inequalities

$$\begin{array}{c}\hfill \alpha \ge 0,\beta \ge 0,\gamma \ne 0,\gamma \ge \alpha ,and\gamma \ge \beta .\end{array}$$

(9)

Then,

(A) 

The branched continued fraction (2), where $d_{i_0}\left(\mathbf{z}\right),$ $c_{i\left(k\right)}\left(\mathbf{z}\right),$ and $d_{i\left(k\right)}\left(\mathbf{z}\right),$ $i\left(k\right)\in \mathfrak{I},$ $k\ge 1,$ defined by (3)–(8), converges uniformly on every compact subset of the domain

$$\begin{array}{c}\hfill {\mathfrak{D}}_{{\kappa }_1,{\kappa }_2;{\tau }_1,{\tau }_2}={\mathfrak{D}}_{{\kappa }_1,{\kappa }_2}\bigcup {\mathfrak{D}}_{{\tau }_1,{\tau }_2},\end{array}$$

(10)

where

$$\begin{array}{c}{\mathfrak{D}}_{{\kappa }_1,{\kappa }_2}=\{\mathbf{z}\in {\mathbb{C}}^2:4{\kappa }_r(1-\cos \left(\arg z_r\right))\left|z_r\right|<{\cos }^2\left(\arg z_r\right),\hfill \\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}|\arg z_r|<\pi /2,r=1,2,\arg z_1=\arg z_2,2|z_1|\ge |z_2\left|\right\}\end{array}$$

(11)

where

$$\begin{array}{c}\hfill {\kappa }_1=\underset{i\left(k\right)\in \mathfrak{I},k\ge 0}{\sup }{c}_{i\left(k\right),1}^1,{\kappa }_2=\underset{i\left(k\right)\in \mathfrak{I},k\ge 0}{\sup }{c}_{i\left(k\right),2}^1,\end{array}$$

(12)

and

$$\begin{array}{c}\hfill {\mathfrak{D}}_{{\tau }_1,{\tau }_2}=\left\{\mathbf{z}\in {\mathbb{C}}^2:\left|z_r\right|<{\tau }_r,r=1,2\right\},\end{array}$$

(13)

where ${\tau }_1$ and ${\tau }_2$ are positive numbers such that

$$\begin{array}{c}\hfill 16\zeta {\tau }_1(1+2{\tau }_2)\le 1-\eta (4{\tau }_1+{\tau }_2),8\zeta (1+4{\tau }_1){\tau }_2\le {(1-\eta (4{\tau }_1+{\tau }_2))}^2,\end{array}$$

(14)

where

$$\begin{array}{c}\zeta =\underset{i\left(k\right)\in \mathfrak{I},k\ge 0}{\sup }\max \left\{{\displaystyle \frac{c_{i\left(k\right),1}^1}{2}},{\displaystyle \frac{c_{i\left(k\right),1}^2}{4}},c_{i\left(k\right),2}^1,{\displaystyle \frac{c_{i\left(k\right),2}^2}{4}}\right\},\end{array}$$

(15)

$$\begin{array}{c}\eta =\underset{i\left(k\right)\in \mathfrak{I},k\ge 1}{\sup }\max \left\{{\displaystyle \frac{|d_{i\left(k\right)}^1|}{4}},\left|d_{i\left(k\right)}^2\right|\right\},\end{array}$$

(16)

to the function $f^{i_0}\left(\mathbf{z}\right)$ holomorphic in the domain ${\mathfrak{D}}_{{\kappa }_1,{\kappa }_2;{\tau }_1,{\tau }_2}.$

(B) 

The function $f^{i_0}\left(\mathbf{z}\right)$ is an analytic continuation of the function on the left side of (2) in the domain (10).

Note that the values ${\kappa }_1,$ ${\kappa }_2,$ $\zeta ,$ and $\eta$ exist and are finite, since there exist finite limits of the coefficients of (2). The conditions (14) are correct, for instance, if

$${\tau }_1={\tau }_2={\displaystyle \frac{2}{21\tau +\sqrt{128\tau +441{\tau }^2}}},\tau =\max \{\zeta ,\eta \}.$$

To prove Theorem 1, we include the following result with a short proof (see [34], Theorem 3.15).

Theorem 2.

Let $i_0=1$ (or $i_0=2$) and let

$$\begin{array}{c}\hfill b_0+\sum _{i_1=1}^2{\displaystyle \frac{a_{i\left(1\right)}}{{\displaystyle 1+\sum _{i_2=1}^2{\displaystyle \frac{a_{i\left(2\right)}}{1{+}_{ \ddots }}}}}}\end{array}$$

(17)

be a branched continued fraction whose elements such that

$$\begin{array}{c}\hfill |a_{i\left(k\right)}|\le {\displaystyle \frac{q_{i\left(k\right)}(1-q_{i(k-1)})}{2}},i\left(k\right)\in \mathfrak{I},k\ge 1,\end{array}$$

where

$$\begin{array}{c}\hfill q_{i\left(0\right)}=0,0<q_{i\left(k\right)}<1,i\left(k\right)\in \mathfrak{I},k\ge 1.\end{array}$$

Then the branched continued fraction (17) converges, and, in addition, the values of (17) and of its approximants are in the closed disk

$$\begin{array}{c}\hfill \mathfrak{O}=\{z\in \mathbb{C}:|z-b_0|\le 1\}.\end{array}$$

Proof. 

We show that the majorant of (17) is a branched continued fraction,

$$\begin{array}{c}\hfill |b_0|+\sum _{i_1=1}^2{\displaystyle \frac{-q_{i\left(1\right)}/2}{{\displaystyle 1+\sum _{i_2=1}^2{\displaystyle \frac{-q_{i\left(2\right)}(1-q_{i\left(1\right)})/2}{{\displaystyle 1+\sum _{i_3=1}^2{\displaystyle \frac{-q_{i\left(3\right)}(1-q_{i\left(2\right)})/2}{1{+}_{ \ddots }}}}}}}}}.\end{array}$$

(18)

Let $F_{i\left(n\right)}^{\left(n\right)}={\widehat{F}}_{i\left(n\right)}^{\left(n\right)}=1,$ $i\left(n\right)\in \mathfrak{I},$$n\ge 1,$ and

$$\begin{array}{c}{F}_{i\left(k\right)}^{\left(n\right)}=1+\sum _{i_{k+1}=1}^2{\displaystyle \frac{a_{i(k+1)}}{{\displaystyle 1+\sum _{i_{k+2}=1}^2\frac{a_{i(k+2)}}{{\displaystyle {1+}_{{\displaystyle \ddots }{\displaystyle +\sum _{i_n=1}^2\frac{a_{i\left(n\right)}}{1}}}}}}}},\\ {\widehat{F}}_{i\left(k\right)}^{\left(n\right)}=1+\sum _{i_{k+1}=1}^2{\displaystyle \frac{-q_{i(k+1)}(1-q_{i\left(k\right)})/2}{{\displaystyle 1+\sum _{i_{k+2}=1}^2\frac{-q_{i(k+2)}(1-q_{i(k+1)})/2}{{\displaystyle {1+}_{{\displaystyle \ddots }{\displaystyle +\sum _{i_n=1}^2\frac{-q_{i\left(n\right)}(1-q_{i(n-1)})/2}{1}}}}}}}},\end{array}$$

where $i\left(k\right)\in \mathfrak{I},$ $1\le k\le n-1,$ $n\ge 2,$ be the ‘tails’ of approximants of (17) and (18), respectively. Then,

$$\begin{array}{c}{F}_{i\left(k\right)}^{\left(n\right)}=1+\sum _{i_{k+1}=1}^2{\displaystyle \frac{a_{i(k+1)}}{F_{i(k+1)}^{\left(n\right)}}},\end{array}$$

(19)

$$\begin{array}{c}{\widehat{F}}_{i\left(k\right)}^{\left(n\right)}=1+\sum _{i_{k+1}=1}^2{\displaystyle \frac{-q_{i(k+1)}(1-q_{i\left(k\right)})/2}{{\widehat{F}}_{i(k+1)}^{\left(n\right)}}},\end{array}$$

(20)

where $i\left(k\right)\in \mathfrak{I},$ $1\le k\le n-1,$ $n\ge 2,$ and, thus, for each $n\ge 1$ we write the nth approximants of (17) and (18) as

$$\begin{array}{c}\hfill f_n=b_0+\sum _{i_1=1}^2{\displaystyle \frac{a_{i\left(1\right)}}{F_{i\left(1\right)}^{\left(n\right)}}}and{\widehat{f}}_n=\left|b_0\right|+\sum _{i_1=1}^2{\displaystyle \frac{-q_{i\left(1\right)}}{2{\widehat{F}}_{i\left(1\right)}^{\left(n\right)}}},\end{array}$$

respectively.

Using (19) and (20) for an arbitrary $n\ge 1$ by induction on k for each multiindex $i\left(k\right)\in \mathfrak{I},$ $1\le k\le n,$ it is easy to show that

$$\begin{array}{c}\hfill |F_{i\left(k\right)}^{\left(n\right)}|\ge {\widehat{F}}_{i\left(k\right)}^{\left(n\right)}>q_{i\left(k\right)}.\end{array}$$

(21)

Thus, $F_{i\left(k\right)}^{\left(n\right)}\ne 0$ and ${\widehat{F}}_{i\left(k\right)}^{\left(n\right)}\ne 0$ for all $i\left(k\right)\in \mathfrak{I},$ $1\le k\le n,$ $n\ge 1.$

Applying the formula for the difference of two approximants of the branched continued fraction (see [28]), for (17) we obtain

$$\begin{array}{cc}\hfill |f_{n+k}-f_n|& \le \sum _{i_1=1}^2\sum _{i_2=1}^2\dots \sum _{i_{n+1}=1}^2{\displaystyle \frac{{\prod }_{r=1}^{n+1}\left|a_{i\left(r\right)}\right|}{{\prod }_{r=1}^{n+1}|F_{i\left(r\right)}^{(n+k)}|{\prod }_{r=1}^n\left|F_{i\left(r\right)}^{\left(n\right)}\right|}}\hfill \\ \hfill & \le {(-1)}^{n+1}\sum _{i_1=1}^2\sum _{i_2=1}^2\dots \sum _{i_{n+1}=1}^2{\displaystyle \frac{{(-1)}^{n+1}{q}_{i\left(1\right)}{\prod }_{r=2}^{n+1}{q}_{i\left(r\right)}(1-q_{i(r-1)})}{2^{n+1}{\prod }_{r=1}^{n+1}{\widehat{F}}_{i\left(r\right)}^{(n+k)}{\prod }_{r=1}^n{\widehat{F}}_{i\left(r\right)}^{\left(n\right)}}}\hfill \\ \hfill & =-({\widehat{f}}_{n+k}-{\widehat{f}}_n),\hfill \end{array}$$

(22)

where $n\ge 1$ and $k\ge 1.$ It follows that the sequence $\left\{{\widehat{f}}_n\right\}$ monotonically decreases. In addition, from (21) we have for $n\ge 1$

$${\widehat{f}}_n=|b_0|+\sum _{i_1=1}^2{\displaystyle \frac{-q_{i\left(1\right)}}{2{\widehat{F}}_{i\left(1\right)}^{\left(n\right)}}}\ge \left|b_0\right|-1,$$

i.e., the sequence $\left\{{\widehat{f}}_n\right\}$ is bounded below. Therefore, there exists a limit,

$$\widehat{f}=\underset{n\to \infty }{lim}{\widehat{f}}_n.$$

Now, using (22), we have for $k\ge 1$

$$\sum _{n=1}^k|f_{n+1}-f_n|\le \sum _{n=1}^k({\widehat{f}}_n-{\widehat{f}}_{n+1})=\left|b_0\right|-\sum _{i_1=1}^2{\displaystyle \frac{q_{i\left(1\right)}}{2}}-{\widehat{f}}_{k+1}.$$

If $k\to \infty ,$ than (17) converges.

Finally, by (21) for any $n\ge 1$ we have

$$\begin{array}{cc}\hfill |f_n-b_0|& \le \sum _{i_1=1}^2{\displaystyle \frac{|a_{i\left(1\right)}|}{|F_{i\left(1\right)}^{\left(n\right)}|}}\hfill \\ \hfill & \le \sum _{i_1=1}^2{\displaystyle \frac{q_{i\left(1\right)}}{2{\widehat{F}}_{i\left(1\right)}^{\left(n\right)}}}\hfill \\ \hfill & \le 1,\hfill \end{array}$$

which proves the theorem. □

Proof of Theorem 1.

Let $i_0\in \mathfrak{I}.$ We set

$$\begin{array}{c}\hfill F_{i\left(n\right)}^{\left(n\right)}\left(\mathbf{z}\right)=1+d_{i\left(n\right)}^1{z}_1+d_{i\left(n\right)}^2{z}_2,i\left(n\right)\in \mathfrak{I},n\ge 1,\end{array}$$

(23)

and

$$\begin{array}{c}\hfill F_{i\left(k\right)}^{\left(n\right)}\left(\mathbf{z}\right)=1+d_{i\left(k\right)}^1{z}_1+d_{i\left(k\right)}^2{z}_2+\sum _{i_{k+1}=1}^2{\displaystyle \frac{c_{i(k+1)}^1{z}_{i_{k+1}}+c_{i(k+1)}^2{z}_1{z}_2}{{\displaystyle 1+d_{i(k+1)}^1{z}_1+d_{i(k+1)}^2{z}_2{+}_{{\displaystyle \ddots }{\displaystyle +\sum _{i_n=1}^2{\displaystyle \frac{c_{i\left(n\right)}^1{z}_{i_n}+c_{i\left(n\right)}^2{z}_1{z}_2}{1+d_{i\left(n\right)}^1{z}_1+d_{i\left(n\right)}^2{z}_2}}}}}}},\end{array}$$

where $i\left(k\right)\in \mathfrak{I},$ $1\le k\le n-1,$ $n\ge 2.$ Then, the following relation holds.

$$\begin{array}{c}\hfill F_{i\left(k\right)}^{\left(n\right)}\left(\mathbf{z}\right)=1+d_{i\left(k\right)}^1{z}_1+d_{i\left(k\right)}^2{z}_2+\sum _{i_{k+1}=1}^2{\displaystyle \frac{c_{i(k+1)}^1{z}_{i_{k+1}}+c_{i(k+1)}^2{z}_1{z}_2}{F_{i(k+1)}^{\left(n\right)}\left(\mathbf{z}\right)}},\end{array}$$

(24)

where $i\left(k\right)\in \mathfrak{I},$ $1\le k\le n-1,$ $n\ge 2.$ This allows us to write the nth approximant of (2) as follows:

$$\begin{array}{c}\hfill f_n^{i_0}\left(\mathbf{z}\right)=1+d_{i_0}^1{z}_1+d_{i_0}^2{z}_2+\sum _{i_1=1}^2{\displaystyle \frac{c_{i\left(1\right)}^1{z}_{i_1}+c_{i\left(1\right)}^2{z}_1{z}_2}{F_{i\left(1\right)}^{\left(n\right)}\left(\mathbf{z}\right)}},n\ge 1.\end{array}$$

We set

$$z_k={\rho }_k{e}^{i\phi },{\rho }_k=\left|z_k\right|,k=1,2,\arg z_1=\arg z_2=\phi .$$

Let n be an arbitrary natural number and $\mathbf{z}$ be an arbitrary fixed point in (11). By induction on $k,$ $1\le k\le n,$ for each $i\left(k\right)\in \mathfrak{I},$ we prove that

$$\begin{array}{c}\hfill \mathrm{Re}\left(F_{i\left(k\right)}^{\left(n\right)}\left(\mathbf{z}\right)e^{-i\phi }\right)>{\displaystyle \frac{\cos \phi }{2}}\ge c>0.\end{array}$$

(25)

From an arbitrary fixed point $\mathbf{z},$ $\mathbf{z}\in {\mathfrak{D}}_{{\kappa }_1,{\kappa }_2},$ it follows that for any neighborhood there exists positive real $\psi$ such that $0<\psi \le \pi /2,$ $\left|\phi \right|\le \pi /2-\psi ,$ and thus,

$$\begin{array}{cc}\hfill {\displaystyle \frac{\cos \phi }{2}}& \ge {\displaystyle \frac{\cos (\pi /2-\psi )}{2}}\hfill \\ & ={\displaystyle \frac{sin\psi }{2}}\hfill \\ & =c\hfill \\ & >0.\hfill \end{array}$$

Using (9) and (11) from (23) for $k=n$ and any $i\left(n\right)\in \mathfrak{I},$ we have

$$\begin{array}{cc}\hfill \mathrm{Re}\left(F_{i\left(n\right)}^{\left(n\right)}\left(\mathbf{z}\right)e^{-i\phi }\right)& =\cos \phi +{\rho }_1{d}_{i\left(n\right)}^1+{\rho }_2{d}_{i\left(n\right)}^2\hfill \\ & =\cos \phi +(2{\rho }_1-{\rho }_2){\delta }_{i_n}^2{\displaystyle \frac{2\gamma -\alpha +n+{\sum }_{r=0}^{n-1}{\delta }_{i_r}^2}{\gamma +n}}+{\rho }_2{\delta }_{i_n}^2{\displaystyle \frac{2\gamma -\beta -1+n+{\sum }_{r=0}^{n-1}{\delta }_{i_r}^1}{\gamma +n}}\hfill \\ & >{\displaystyle \frac{\cos \phi }{2}}.\hfill \end{array}$$

Assuming that inequalities (25) hold for $k=s+1$ and for all $i(s+1)\in \mathfrak{I}$ such that $s+1\le n,$ from (24) for $k=s$ and for any $i\left(s\right)\in \mathfrak{I}$, we obtain

$$\begin{array}{cc}\hfill F_{i\left(s\right)}^{\left(n\right)}\left(\mathbf{z}\right)e^{-i\phi }& =e^{-i\phi }+d_{i\left(s\right)}^1{\rho }_1+d_{i\left(s\right)}^2{\rho }_2+\sum _{i_{k+1}=1}^2{\displaystyle \frac{c_{i(k+1)}^1{\rho }_{i_{k+1}}}{F_{i(k+1)}^{\left(n\right)}\left(\mathbf{z}\right)}}+\sum _{i_{k+1}=1}^2{\displaystyle \frac{c_{i(k+1)}^2{\rho }_1{\rho }_2}{F_{i(k+1)}^{\left(n\right)}\left(\mathbf{z}\right)e^{-i\phi }}}\hfill \\ \hfill & =e^{-i\phi }+(2{\rho }_1-{\rho }_2){\delta }_{i_s}^2{\displaystyle \frac{2\gamma -\alpha +s+{\sum }_{r=0}^{s-1}{\delta }_{i_r}^2}{\gamma +s}}+{\rho }_2{\delta }_{i_s}^2{\displaystyle \frac{2\gamma -\beta -1+s+{\sum }_{r=0}^{s-1}{\delta }_{i_r}^1}{\gamma +s}}\hfill \\ & +\sum _{i_{s+1}=1}^2{\displaystyle \frac{c_{i(s+1)}^1{\rho }_{i_{s+1}}{e}^{-i\phi }}{F_{i(s+1)}^{\left(n\right)}\left(\mathbf{z}\right)e^{-i\phi }}}+\sum _{i_{s+1}=1}^2{\displaystyle \frac{c_{i(s+1)}^2{\rho }_1{\rho }_2}{F_{i(s+1)}^{\left(n\right)}\left(\mathbf{z}\right)e^{-i\phi }}}.\hfill \end{array}$$

Then, using ([16], Corollary 2), (9), (11), and (12), we have

$$\begin{array}{cc}\hfill \mathrm{Re}\left(F_{i\left(s\right)}^{\left(n\right)}\left(\mathbf{z}\right)e^{-i\phi }\right)& \ge \mathrm{Re}\left(e^{-i\phi }\right)+\sum _{i_{s+1}=1}^2{c}_{i(s+1)}^1{\rho }_{i_{s+1}}\mathrm{Re}\left({\displaystyle \frac{e^{-i\phi }}{F_{i(s+1)}^{\left(n\right)}\left(\mathbf{z}\right)e^{-i\phi }}}\right)+\sum _{i_{s+1}=1}^2{c}_{i(s+1)}^2{\rho }_1{\rho }_2\mathrm{Re}\left({\displaystyle \frac{1}{F_{i(s+1)}^{\left(n\right)}\left(\mathbf{z}\right)e^{-i\phi }}}\right)\hfill \\ \hfill & \ge \cos \phi -\sum _{i_{s+1}=1}^2{\displaystyle \frac{c_{i(s+1)}^1{\rho }_{i_{s+1}}(1-\cos \phi )}{2\mathrm{Re}\left(F_{i(s+1)}^{\left(n\right)}\left(\mathbf{z}\right)e^{-i\phi }\right)}}+\sum _{i_{s+1}=1}^2{\displaystyle \frac{c_{i(s+1)}^2{\rho }_1{\rho }_2\mathrm{Re}\left(F_{i(s+1)}^{\left(n\right)}\left(\mathbf{z}\right)e^{-i\phi }\right)}{|F_{i(s+1)}^{\left(n\right)}\left(\mathbf{z}\right)e^{-i\phi }{|}^2}}\hfill \\ & >\cos \phi -{\displaystyle \frac{(1-\cos \phi )}{\cos \phi }}\sum _{i_{s+1}=1}^2{\kappa }_{i_{s+1}}{\rho }_{i_{s+1}}\hfill \\ & >{\displaystyle \frac{\cos \phi }{2}}.\hfill \end{array}$$

Thus, inequalities (25) hold and, therefore, $F_{i\left(k\right)}^{\left(n\right)}\left(\mathbf{z}\right)\ne 0$ for all $i\left(k\right)\in \mathfrak{I},$ $1\le k\le n,$$n\ge 1,$ and $\mathbf{z}\in {\mathfrak{D}}_{{\kappa }_1,{\kappa }_2}.$ This means that $\left\{f_n^{i_0}\left(\mathbf{z}\right)\right\}$ is a sequence of functions holomorphic in the domain (11).

Let $\mathfrak{Q}$ be an arbitrary compact subset of the domain (11). Then there exists an open ball with center at the origin and radius R containing $\mathfrak{Q}.$ Using (9), (11), and (25), for arbitrary $\mathbf{z}\in {\mathfrak{D}}_{{\kappa }_1,{\kappa }_2}$ and $n\in \mathbb{N},$ we obtain

$$\begin{array}{cc}\hfill |f_n^{i_0}\left(\mathbf{z}\right)|& \le 1+|d_{i_0}^1\left|\right|z_1|+|d_{i_0}^2\left|\right|z_2|+\sum _{i_1=1}^2{\displaystyle \frac{c_{i\left(1\right)}^1|z_{i_1}|+c_{i\left(1\right)}^2|z_1||z_2|}{\mathrm{Re}\left(F_{i\left(1\right)}^{\left(n\right)}\left(\mathbf{z}\right)e^{-i\phi }\right)}}\hfill \\ \hfill & <1+R\left(\right|d_{i_0}^1|+|d_{i_0}^2\left|\right)+{\displaystyle \frac{2R}{\cos \phi }}\sum _{i_1=1}^2(c_{i\left(1\right)}^1+c_{i\left(1\right)}^{2}R)\hfill \\ & =C\left(\mathfrak{Q}\right),\hfill \end{array}$$

i.e., the sequence $\left\{f_n^{i_0}\left(\mathbf{z}\right)\right\}$ is uniformly bounded on the domain ${\mathfrak{D}}_{{\kappa }_1,{\kappa }_2},$ and, at the same time, is uniformly bounded on every compact subset of the domain (11).

By the equivalent transformation ([34], pp. 29–33),

$$\begin{array}{c}\hfill r_{i\left(k\right)}={\displaystyle \frac{1}{1+d_{i\left(k\right)}^1{z}_1+d_{i\left(k\right)}^2{z}_2}},i\left(k\right)\in \mathfrak{I},k\ge 1,\end{array}$$

we write (2) in the form

$$\begin{array}{c}\hfill 1+d_{i_0}^1{z}_1+d_{i_0}^2{z}_2+\sum _{i_1=1}^2{\displaystyle \frac{{\displaystyle \frac{c_{i\left(1\right)}^1{z}_{i_1}+c_{i\left(1\right)}^2{z}_1{z}_2}{1+d_{i\left(1\right)}^1{z}_1+d_{i\left(1\right)}^2{z}_2}}}{{\displaystyle 1+\sum _{i_2=1}^2{\displaystyle \frac{{\displaystyle \frac{c_{i\left(2\right)}^1{z}_{i_2}+c_{i\left(2\right)}^2{z}_1{z}_2}{(1+d_{i\left(1\right)}^1{z}_1+d_{i\left(1\right)}^2{z}_2)(1+d_{i\left(2\right)}^1{z}_1+d_{i\left(2\right)}^2{z}_2)}}}{1{+}_{ \ddots }}}}}}.\end{array}$$

(26)

We set

$$L={\displaystyle \frac{-2\zeta +\sqrt{2\zeta +4{\zeta }^2}}{8\zeta }}.$$

Then, the domain

$$\begin{array}{c}\hfill {\mathfrak{L}}_R=\left\{\mathbf{z}\in {\mathbb{R}}^2:0<z_2\le 2z_1<R<L\right\}\end{array}$$

is contained in ${\mathfrak{D}}_{{\kappa }_1,{\kappa }_2}$ for each $0<R<L,$ in particular ${\mathfrak{L}}_{L/2}\subset {\mathfrak{D}}_{{\kappa }_1,{\kappa }_2}.$ Using (4)–(9), (11), and (15) for arbitraries $\mathbf{z}\in {\mathfrak{L}}_R,$ ${\mathfrak{L}}_R\subset {\mathfrak{D}}_{{\kappa }_1,{\kappa }_2},$ and $i\left(1\right)\in \mathfrak{I},$ we have

$$\begin{array}{cc}\hfill \left|{\displaystyle \frac{c_{i\left(1\right)}^1{z}_{i_1}+c_{i\left(1\right)}^2{z}_1{z}_2}{1+d_{i\left(1\right)}^1{z}_1+d_{i\left(1\right)}^2{z}_2}}\right|& ={\displaystyle \frac{|c_{i\left(1\right)}^1{z}_{i_1}+c_{i\left(1\right)}^2{z}_1{z}_2|}{\left|1+\left({\delta }_{i_1}^2{\displaystyle \frac{2\gamma -\alpha +1+{\delta }_{i_0}^2}{\gamma +1}}\right)(2z_1-z_2)+\left({\delta }_{i_1}^2{\displaystyle \frac{2\gamma -\beta +{\delta }_{i_0}^1}{\gamma +1}}\right)z_2\right|}}\hfill \\ & <2\zeta (L+2L^2)\hfill \\ & \le {\displaystyle \frac{1}{8}},\hfill \end{array}$$

and for arbitrary $i\left(k\right)\in \mathfrak{I},$ $k\ge 2,$

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{c_{i\left(k\right)}^1{z}_{i_k}+c_{i\left(k\right)}^2{z}_1{z}_2}{(1+d_{i(k-1)}^1{z}_1+d_{i(k-1)}^2{z}_2)(1+d_{i\left(k\right)}^1{z}_1+d_{i\left(k\right)}^2{z}_2)}}\right|\hfill \\ \hfill & ={\displaystyle \frac{|c_{i\left(k\right)}^1{z}_{i_k}+c_{i\left(k\right)}^2{z}_1{z}_2|}{\left|1+\left({\delta }_{i_{k-1}}^2{\displaystyle \frac{2\gamma -\alpha +k-1+{\sum }_{r=0}^{k-2}{\delta }_{i_r}^2}{\gamma +k-1}}\right)(2z_1-z_2)+\left({\delta }_{i_{k-1}}^2{\displaystyle \frac{2\gamma -\beta -2+k+{\sum }_{r=0}^{k-2}{\delta }_{i_r}^1}{\gamma +k-1}}\right)z_2\right|}}\hfill \\ \hfill & \times {\displaystyle \frac{1}{\left|1+\left({\delta }_{i_k}^2{\displaystyle \frac{2\gamma -\alpha +k+{\sum }_{r=0}^{k-1}{\delta }_{i_r}^2}{\gamma +k}}\right)(2z_1-z_2)+\left({\delta }_{i_k}^2{\displaystyle \frac{2\gamma -\beta -1+k+{\sum }_{r=0}^{k-1}{\delta }_{i_r}^1}{\gamma +k}}\right)z_2\right|}}\hfill \\ \hfill & <2\zeta (L+2L^2)\hfill \\ \hfill & \le {\displaystyle \frac{1}{8}},\hfill \end{array}$$

i.e., the elements of the branched continued fraction (26) satisfy the conditions of Theorem 2, with

$$\begin{array}{c}\hfill q_{i\left(k\right)}={\displaystyle \frac{1}{2}}foralli\left(k\right)\in \mathfrak{I},k\ge 1.\end{array}$$

(27)

It follows from Theorem 2 that the branched continued fraction (26) converges in the domain ${\mathfrak{L}}_R,$ ${\mathfrak{L}}_R\subset {\mathfrak{D}}_{{\kappa }_1,{\kappa }_2}.$ In addition, it follows from the concept of an equivalent transformation ([34], pp. 29–33) that the branched continued fraction (2) converges also in the domain ${\mathfrak{L}}_R,$ ${\mathfrak{L}}_R\subset {\mathfrak{D}}_{{\kappa }_1,{\kappa }_2}.$ Thus, by ([29], Theorem 3), the branched continued fraction (2) converges uniformly on compact subsets of the domain (11) to a holomorphic function in this domain.

Using (4)–(9) and (13)–(16) for arbitrary $\mathbf{z}\in {\mathfrak{D}}_{{\tau }_1,{\tau }_2}$, we obtain

$$\begin{array}{cc}\hfill \left|{\displaystyle \frac{c_1^1{z}_1+c_1^2{z}_1{z}_2}{1+d_1^1{z}_1+d_1^2{z}_2}}\right|& =|c_1^1{z}_1|\hfill \\ & <2\zeta {\tau }_1\hfill \\ & <{\displaystyle \frac{1}{8}},\hfill \\ \hfill \left|{\displaystyle \frac{c_2^1{z}_2+c_2^2{z}_1{z}_2}{1+d_2^1{z}_1+d_2^2{z}_2}}\right|& <{\displaystyle \frac{\zeta {\tau }_1(1+4{\tau }_2)}{1-\eta (4{\tau }_1+{\tau }_2)}}\hfill \\ & <{\displaystyle \frac{1}{8}},\hfill \end{array}$$

and for arbitrary $i\left(k\right)\in \mathfrak{I},$ $k\ge 1,$

$$\begin{array}{cc}\hfill \left|{\displaystyle \frac{c_{i\left(k\right),1}^1{z}_1+c_{i\left(k\right),1}^2{z}_1{z}_2}{(1+d_{i\left(k\right)}^1{z}_1+d_{i\left(k\right)}^2{z}_2)(1+d_{i\left(k\right),1}^1{z}_1+d_{i\left(k\right),1}^2{z}_2)}}\right|& =\left|{\displaystyle \frac{c_{i\left(k\right),1}^1{z}_1+c_{i\left(k\right),1}^2{z}_1{z}_2}{1+d_{i\left(k\right)}^1{z}_1+d_{i\left(k\right)}^2{z}_2}}\right|\hfill \\ \hfill & <{\displaystyle \frac{2\zeta {\tau }_1(1+2{\tau }_2)}{1-\eta (4{\tau }_1+{\tau }_2)}}\hfill \\ & \le {\displaystyle \frac{1}{8}},\hfill \\ \hfill \left|{\displaystyle \frac{c_{i\left(k\right),2}^1{z}_2+c_{i\left(k\right),2}^2{z}_1{z}_2}{(1+d_{i\left(k\right)}^1{z}_1+d_{i\left(k\right)}^2{z}_2)(1+d_{i\left(k\right),2}^1{z}_1+d_{i\left(k\right),2}^2{z}_2)}}\right|& <{\displaystyle \frac{\zeta (1+4{\tau }_1){\tau }_2}{{(1-\eta (4{\tau }_1+{\tau }_2))}^2}}\hfill \\ & \le {\displaystyle \frac{1}{8}}.\hfill \end{array}$$

In other words, the elements of the branched continued fraction (26) satisfy the conditions of Theorem 2, with $q_{i\left(k\right)},$ $i\left(k\right)\in \mathfrak{I},$ $k\ge 1,$ that satisfy the condition (27). It follows that the branched continued fraction (26) converges for all $\mathbf{z}\in {\mathfrak{D}}_{{\tau }_1,{\tau }_2}$, where ${\mathfrak{D}}_{{\tau }_1,{\tau }_2}$ is defined by (13), and that all approximants of (26) lie in the closed disk

$$\mathfrak{W}=\{z\in \mathbb{C}:|z-1-d_{i_0}^1{z}_1-d_{i_0}^2{z}_2|\le 1\}$$

if $\mathbf{z}\in {\mathfrak{D}}_{{\tau }_1,{\tau }_2}$. Thus, by Theorem 2, the branched continued fraction (26) and at the same time, by the concept of an equivalent transformation ([34], pp. 29–33), the branched continued fraction (2) converge uniformly on compact subsets of the domain (13). This is a complete proof of (A).

The proof of (B) is similar to the proof of ([28], Theorem 2); hence it is omitted. □

From Theorem 1 we have the following results.

Corollary 1.

Let $i_0=1.$ Assume that β and γ are real constants that satisfy the inequalities

$$\begin{array}{c}\hfill \beta \ge 0,\gamma \ne 1,\gamma -1\ge \beta .\end{array}$$

Then the branched continued fraction

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{{\displaystyle 1+\sum _{i_1=1}^2{\displaystyle \frac{c_{i\left(1\right)}^1{z}_{i_1}+c_{i\left(1\right)}^2{z}_1{z}_2}{{\displaystyle 1+d_{i\left(1\right)}^1{z}_1+d_{i\left(1\right)}^2{z}_2+\sum _{i_2=1}^2{\displaystyle \frac{c_{i\left(2\right)}^1{z}_{i_2}+c_{i\left(2\right)}^2{z}_1{z}_2}{1+d_{i\left(2\right)}^1{z}_1+d_{i\left(2\right)}^2{z}_2{+}_{ \ddots }}}}}}}}},\end{array}$$

where

$$\begin{array}{c}{c}_{i_0,1}^1={\displaystyle \frac{(2(\gamma -1)-{\delta }_{i_0}^2){\delta }_{i_0}^1}{(\gamma -1)\gamma }},c_{i_0,1}^2={\displaystyle \frac{2{\delta }_{i_0}^2(2(\gamma -1)-\beta ){\delta }_{i_0}^1}{(\gamma -1)\gamma }},\\ c_{i_0,2}^1={\displaystyle \frac{(\beta +{\delta }_{i_0}^2)(\gamma -1+{\delta }_{i_0}^2)}{(\gamma -1)\gamma }},c_{i_0,2}^2=4c_{i_0,2}^1,\end{array}$$

and, for $i\left(k\right)\in \mathfrak{I}$ and $k\ge 1,$

$$\begin{array}{c}{d}_{i\left(k\right)}^1=2{\delta }_{i_k}^2{\displaystyle \frac{2(\gamma -1)+k+{\sum }_{r=0}^{k-1}{\delta }_{i_r}^2}{\gamma -1+k}},d_{i\left(k\right)}^2={\delta }_{i_k}^2{\displaystyle \frac{-\beta -1+{\sum }_{r=0}^{k-1}({\delta }_{i_r}^1-{\delta }_{i_r}^2)}{\gamma -1+k}},\\ c_{i\left(k\right),1}^1={\displaystyle \frac{(2(\gamma -1)k+{\sum }_{r=0}^k{\delta }_{i_r}^2){\sum }_{r=0}^k{\delta }_{i_r}^1}{(\gamma -1+k)(\gamma +k)}},c_{i\left(k\right),1}^2={\displaystyle \frac{2{\delta }_{i_k}^2(2(\gamma -1)-\beta +k){\sum }_{r=0}^k{\delta }_{i_r}^1}{(\gamma -1+k)(\gamma +k)}},\\ c_{i\left(k\right),2}^1={\displaystyle \frac{(\beta +{\sum }_{r=0}^k{\delta }_{i_r}^2)(\gamma -1+{\sum }_{r=0}^k{\delta }_{i_r}^2)}{(\gamma -1+k)(\gamma +k)}},c_{i\left(k\right),2}^2=4c_{i\left(k\right),2}^1,\end{array}$$

converges uniformly on every compact subset of the domain (10) to the function $f^1\left(\mathbf{z}\right)$ holomorphic in ${\mathfrak{D}}_{{\kappa }_1,{\kappa }_2;{\tau }_1,{\tau }_2},$ and, in addition, the function $f^1\left(\mathbf{z}\right)$ is an analytic continuation of the hypergeometric function $H_3(1,\beta ;\gamma ;-\mathbf{z})$ in the domain (10).

The following theorem can be proved in much the same way as Theorem 1.

Theorem 3.

Let $i_0=2.$ Assume that $\alpha ,$$\beta ,$ and γ are real constants that satisfy the inequalities

$$\begin{array}{c}\hfill \gamma \ge \alpha \ge 0,\gamma -1\ge \beta \ge 0.\end{array}$$

Then, we have the following:

(A) 

The branched continued fraction (2), where $d_{i_0}\left(\mathbf{z}\right),$ $c_{i\left(k\right)}\left(\mathbf{z}\right),$ and $d_{i\left(k\right)}\left(\mathbf{z}\right),$ $i\left(k\right)\in \mathfrak{I},$ $k\ge 1,$ defined by (3)–(8), converges uniformly on every compact subset of (10) to the function $f^{i_0}\left(\mathbf{z}\right)$ holomorphic in ${\mathfrak{D}}_{{\kappa }_1,{\kappa }_2;{\tau }_1,{\tau }_2}.$

(B) 

The function $f^{i_0}\left(\mathbf{z}\right)$ is an analytic continuation of the function on the left side of (2) in (10).

Corollary 2.

Let $i_0=2.$ Assume that α and γ are real constants that satisfy the inequalities

$$\alpha \ge 0,\gamma \ne 1,\gamma -1\ge \alpha .$$

Then the branched continued fraction

$$\begin{array}{c}\hfill {\displaystyle \frac{1+4z_1}{{\displaystyle 1+d_2^1{z}_1+d_2^2{z}_2+\sum _{i_1=1}^2{\displaystyle \frac{c_{i\left(1\right)}^1{z}_{i_1}+c_{i\left(1\right)}^2{z}_1{z}_2}{{\displaystyle 1+d_{i\left(1\right)}^1{z}_1+d_{i\left(1\right)}^2{z}_2+\sum _{i_2=1}^2{\displaystyle \frac{c_{i\left(2\right)}^1{z}_{i_2}+c_{i\left(2\right)}^2{z}_1{z}_2}{1+d_{i\left(2\right)}^1{z}_1+d_{i\left(2\right)}^2{z}_2{+}_{ \ddots }}}}}}}}}\end{array}$$

where

$$\begin{array}{c}{d}_{i_0}^1=2{\delta }_{i_0}^2{\displaystyle \frac{2(\gamma -1)-\alpha }{\gamma -1}},d_{i_0}^2={\delta }_{i_0}^2{\displaystyle \frac{\alpha -1}{\gamma -1}},\\ c_{i_0,1}^1={\displaystyle \frac{(2(\gamma -1)-\alpha +{\delta }_{i_0}^2)(\alpha +{\delta }_{i_0}^1)}{(\gamma -1)\gamma }},c_{i_0,1}^2={\displaystyle \frac{2{\delta }_{i_0}^2(2(\gamma -1)-\alpha )(\alpha +{\delta }_{i_0}^1)}{(\gamma -1)\gamma }},\\ c_{i_0,2}^1={\displaystyle \frac{{\delta }_{i_0}^2(\gamma -1-\alpha +{\delta }_{i_0}^2)}{(\gamma -1)\gamma }},c_{i_0,2}^2=4c_{i_0,2}^1,\end{array}$$

and, for $i\left(k\right)\in \mathfrak{I}$ and $k\ge 1,$

$$\begin{array}{c}{d}_{i\left(k\right)}^1=2{\delta }_{i_k}^2{\displaystyle \frac{2(\gamma -1)-\alpha +k+{\sum }_{r=0}^{k-1}{\delta }_{i_r}^2}{\gamma +k-1}},d_{i\left(k\right)}^2={\delta }_{i_k}^2{\displaystyle \frac{\alpha -1+{\sum }_{r=0}^{k-1}({\delta }_{i_r}^1-{\delta }_{i_r}^2)}{\gamma +k-1}},\\ c_{i\left(k\right),1}^1={\displaystyle \frac{(2(\gamma -1)-\alpha +k+{\sum }_{r=0}^k{\delta }_{i_r}^2)(\alpha +{\sum }_{r=0}^k{\delta }_{i_r}^1)}{(\gamma +k-1)(\gamma +k)}},c_{i\left(k\right),1}^2={\displaystyle \frac{2{\delta }_{i_k}^2(2(\gamma -1)-\alpha +k)(\alpha +{\sum }_{r=0}^k{\delta }_{i_r}^1)}{(\gamma +k-1)(\gamma +k)}},\\ c_{i\left(k\right),2}^1={\displaystyle \frac{{\sum }_{r=0}^k{\delta }_{i_r}^2(\gamma -1-\alpha +{\sum }_{r=0}^k{\delta }_{i_r}^2)}{(\gamma +k-1)(\gamma +k)}},c_{i\left(k\right),2}^2=4c_{i\left(k\right),2}^1,\end{array}$$

converges uniformly on every compact subset of the domain (10) to the function $f^2\left(\mathbf{z}\right)$ holomorphic in ${\mathfrak{D}}_{{\kappa }_1,{\kappa }_2;{\tau }_1,{\tau }_2},$ and, in addition, the function $f^2\left(\mathbf{z}\right)$ is an analytic continuation of the hypergeometric function

$$\begin{array}{c}\hfill {\displaystyle \frac{H_3(\alpha ,1;\gamma ;-\mathbf{z})}{F(\alpha /2,(\alpha +1)/2;\gamma -1;4z_1)}}\end{array}$$

in the domain (10).

3. Applications

Consider the special function (see [35])

$$\begin{array}{cc}\hfill H_3\left(1,1;{\displaystyle \frac{3}{2}};-\mathbf{z}\right)& ={\displaystyle \frac{1}{2\sqrt{z_1-z_2-z_2^2}}}\left(\arctan {\displaystyle \frac{2z_1-z_2}{\sqrt{z_1-z_2-z_2^2}}}-\arctan {\displaystyle \frac{z_2}{\sqrt{z_1-z_2-z_2^2}}}\right).\hfill \end{array}$$

(28)

It follows from (1),

$$\begin{array}{c}\hfill H_3\left(1,1;{\displaystyle \frac{3}{2}};-\mathbf{z}\right)=\sum _{r,s=0}^{\infty }{\displaystyle \frac{{\left(1\right)}_{2r+s}{\left(1\right)}_s}{{(3/2)}_{r+s}}}{\displaystyle \frac{{(-z_1)}^r}{r!}}{\displaystyle \frac{{(-z_2)}^s}{s!}}.\end{array}$$

(29)

From Corollary 1, the branched continued fraction

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{{\displaystyle 1+{\displaystyle \frac{{\displaystyle \frac{4}{3}}{z}_1}{{\displaystyle 1+{\displaystyle \frac{{\displaystyle \frac{16}{15}}{z}_1}{1{+}_{ \ddots }}}+{\displaystyle \frac{{\displaystyle \frac{2}{15}}{z}_2+{\displaystyle \frac{8}{15}}{z}_1{z}_2}{1+{\displaystyle \frac{12}{5}}{z}_1{+}_{ \ddots }}}}}}+{\displaystyle \frac{{\displaystyle \frac{2}{3}}{z}_2+{\displaystyle \frac{8}{3}}{z}_1{z}_2}{{\displaystyle 1+{\displaystyle \frac{8}{3}}{z}_1-{\displaystyle \frac{2}{3}}{z}_2+{\displaystyle \frac{{\displaystyle \frac{4}{5}}{z}_1+{\displaystyle \frac{8}{15}}{z}_1{z}_2}{1{+}_{ \ddots }}}+{\displaystyle \frac{{\displaystyle \frac{4}{5}}{z}_2+{\displaystyle \frac{16}{5}}{z}_1{z}_2}{1+{\displaystyle \frac{16}{5}}{z}_1-{\displaystyle \frac{4}{5}}{z}_2{+}_{ \ddots }}}}}}}}}\end{array}$$

(30)

converges and represents a single-valued branch of the analytic function (28) in the domain

$$\begin{array}{c}\hfill {\mathfrak{T}}_{{\kappa }_1,{\kappa }_2;{\tau }_1,{\tau }_2}={\mathfrak{D}}_{{\kappa }_1,{\kappa }_2;{\tau }_1,{\tau }_2}\bigcap \mathfrak{R},\end{array}$$

(31)

where ${\mathfrak{D}}_{{\kappa }_1,{\kappa }_2;{\tau }_1,{\tau }_2}$ is defined by (10) and

$$\begin{array}{c}\hfill \mathfrak{R}=\left\{\mathbf{z}\in {\mathbb{C}}^2:z_1-z_2-z_2^2\notin (-\infty ,0]\right\}.\end{array}$$

Table 1. Relative error of 17th partial sum and 17th approximants for (28).

z	(28)	(30)	(29)
$(0.2165+0.1250i,0.4330+0.2500i)$	$0.6289-0.1159i$	$2.95\times {10}^{-11}$	$1.22\times {10}^{-2}$
$(0.4330+0.2500i,0.2165+0.1250i)$	$0.6075-0.1072i$	$1.50\times {10}^{-10}$	$2.86\times {10}^3$
$(0.1250-0.2165i,0.2500-0.4330i)$	$0.6481+0.2454i$	$1.82\times {10}^{-10}$	$1.32\times {10}^{-2}$
$(0.3750-0.6495i,0.5000-0.8660i)$	$0.4076+0.2359i$	$4.58\times {10}^{-8}$	$2.05\times {10}^{10}$
$(0.1250+0.2165i,0.3750+0.6495i)$	$0.5801-0.2601i$	$1.57\times {10}^{-7}$	$5.45\times {10}^{-1}$
$(8.6603+5.0000i,17.3205+10.0000i)$	$0.0578-0.0256i$	$5.49\times {10}^{-3}$	$4.38\times {10}^{52}$
$(5.0000+8.6603i,10.0000+17.3205i)$	$0.0432-0.0478i$	$8.80\times {10}^{-3}$	$4.39\times {10}^{52}$
$(8.6603-5.0000i,17.3205-10.0000i)$	$0.0578+0.0256i$	$5.49\times {10}^{-3}$	$4.38\times {10}^{52}$
$(-100.0000+173.2051i,-300.0000+519.6152i)$	$-0.0011-0.0036i$	$4.14\times {10}^{-1}$	$1.30\times {10}^{101}$
$(-500.0000-866.0254i,-150.0000-259.8076i)$	$-0.0010+0.0051i$	$2.78\times {10}^{-1}$	$5.52\times {10}^{107}$
$(50.0000-86.6025i,100.0000-173.2051i)$	$0.0057+0.0073i$	$1.47\times {10}^{-1}$	$3.12\times {10}^{87}$
$(500.0000-866.0254i,450.0000-779.4229i)$	$0.0014+0.0018i$	$2.96\times {10}^{-1}$	$1.63\times {10}^{116}$

shows the relative errors of the approximation of special function (28) by approximants of (30) and partial sums (29). The approximation by branched continued fraction (30) is better than double power series approximation (29) at points close to the origin. At points much further from the origin in different directions, the branched continued fraction still converges, albeit worse, while the double power series diverges. Among these directions are those that contain points that do not belong to the domain (31). This indicates that the branched continued fraction (30) converges in a wider domain, and, therefore, there is a wider domain of analytical continuation of special function (30).

Figure 1 shows the curves of the approximant of (30) for fixed variables $z_2$ (Figure 1a) and $z_1$ (Figure 1b). On the selected segments, in both cases we have a consistent and one-sided approximation of the function curve (28). In theses cases, the so-called ’fork property’ [34], characteristic of branched continued fractions with positive elements, is not observed.

Finally, Figure 2a–d and Figure 3a–d depict domains in four different planes, where the ninth and tenth approximations of (30) guarantees certain truncation error bounds for special function (28). In these cases, the domains of even and odd approximants are different clear figures in different planes. Their configurations indicate the type of possible new domains of convergence of the branched continued fraction (30), and at the same time, the domains of analytic expansion of special function (28).

Calculations and plots were performed using the Microsoft Visual Studio 2022 Community Edition IDE using the .Net 8.0 platform.

4. Discussion and Conclusions

This paper concerns the establishment of the new domain of analytical continuation of the Horn’s hypergeometric functions $H_3$ and their ratios through branched continued fractions as special family functions. This domain is obtained as the domain of convergence of the corresponding branched continued fraction expansions (2). Numerical experiments indicate the existence of wider domains of convergence, and, therefore, the domains of analytical expansion of the Horn’s hypergeometric functions $H_3$ and their ratios. However, the problem of establishing them remains open. The problem of establishing truncation error bounds for branched continued fraction expansions (2) also remains open.

Note that the results obtained can be used in the study of special functions represented by the Horn’s hypergeometric functions $H_3$ and their ratios. Furthermore, since the Horn’s hypergeometric function $H_3$ is closely related to the Appell’s hypergeometric functions $F_1$ and $F_4$ (see [35]), these results can also be applied to their study.

Finally, it is worth noting that the methods for studying the convergence of branched continued fractions developed here can be applied to the study of branched continued fraction expansions of other hypergeometric functions.