Abstract
In this paper, we consider the representation and extension of the analytic functions of three variables by special families of functions, namely branched continued fractions. In particular, we establish new symmetric domains of the analytical continuation of Lauricella–Saran’s hypergeometric function with certain conditions on real and complex parameters using their branched continued fraction representations. We use a technique that extends the convergence, which is already known for a small domain, to a larger domain to obtain domains of convergence of branched continued fractions and the PC method to prove that they are also domains of analytical continuation. In addition, we discuss some applicable special cases and vital remarks.
1. Introduction
Special functions, including Lauricella–Saran’s hypergeometric functions, occur naturally in various problems in mathematics, statistics, physics, chemistry, and engineering. This paper discusses the representation and analytical extension of these functions. Domains of analytical continuation will be symmetric domains of convergence of special families of functions, namely branched continued fractions. Note that here, the domain is an open connected subset of
Lauricella–Saran’s family of 14 functions ( or ) owes its appearance mainly to two papers [,]. These functions are defined by triple power series, particularly
where and where
Various applications and studies of different properties of Lauricella–Saran’s functions are discussed in many scientific works. In particular, Lauricella–Saran’s hypergeometric function was used to compute the canonical partition function of the model of heteropolymers in the form of a freely jointed chain [,] and the generalized Nordsieck integral [] to investigate the compound gamma bivariate distribution [,] and the propagator seagull diagram []. The problem of analytical continuation and asymptotics for the function using the integral representation was considered in [,]. Asymptotic expansions of the function were studied in [].
The authors of [] gave the formal expansion
where
as well as the following expansion, which is symmetrical to it:
where
It is also established here that
is the domain of the analytical continuation of the function on the left side of Equation (1) (or Equation (3)), provided that (or ), where The problem of representing and extending Lauricella–Saran’s hypergeometric functions and through branched continued fractions was considered in [,,], respectively.
In Section 2 of this paper, we give the necessary definitions and preliminary results. New symmetric domains of the analytical continuation of Lauricella–Saran’s hypergeometric function with certain conditions for real and complex parameters, using their branched continued fraction representations, are established in Section 3.
General information on branched continued fractions can be found in [,,].
2. Preliminary Definitions and Results
The concept of a branched continued fraction can be approached in different ways, particularly through the sequence of its approximants. A brief description follows.
Let and let
denote the sets of multiindices.
The ordered pair of sequences
of complex numbers satisfies the following conditions:
- (1)
- for ;
- (2)
- If for there exists a multiindex such that then for and .
We then generate the sequence as follows:
and so on, in addition to
and so on.
The ordered pair
is the branched continued fraction denoted by
Furthermore, considering the branched continued fraction in Equation (6), we admit a confluent case where there are no constraints (1).
We shall need the following:
Definition 1.
A branched continued fraction (Equation (6)) converges if, at most, a finite number of its approximants do not make sense and if the limit of its sequence of approximants
exists and is finite.
Note that the approximant makes sense if the 0/0 uncertainty does not arise when computing its value. We assume that
Definition 2.
A branched continued fraction (Equation (6)) converges absolutely if
The convergence criteria for branched continued fractions are often given in terms of convergence sets. The definition is as follows:
Definition 3.
A convergence set Ω is a set where and such that if for all where then Equation (6) converges.
Definition 4.
A uniform convergence set Ω is a convergence set to which there corresponds a sequence of positive numbers depending only on Ω and converging to zero such that
for every branched continued fraction (Equation (6)) with all
Theorem 1 below was proven in []. Here, for convenience, we give its formulation using multiindex notation in the same way as in the triple power series. Note that this is possible because for any , there is a mapping where
and
where and is the Kronecker delta, such that for all Also, it can be shown that the mapping is bijective.
Theorem 1.
Let where , be real constants satisfying
Then, the following are true:
- (1)
- The branched continued fraction
- (2)
- The values of the branched continued fraction and of its approximants are in the closed domain
Indeed, Theorem 1 (1) follows directly from Theorem 1 [], with
Next, by setting
it is clear that
where and
are valid for and Therefore, the nth approximants of Equation (7) can be written as follows:
In the proof of Theorem 1 [], it is shown that
and hence, according to Equations (8) and (10), any yields
which proves Theorem 1 (2).
Note that this theorem is analogous to Theorem 11.1 in []. Also, the fact that the majorant method [] (p. 51) and the formula for the difference of two approximants of the branched continued fraction [] (p. 28) were used in the proof of Theorem 1 [].
An important application of branched continued fractions is the representation of holomorphic functions by branched continued fractions, the elements of which are functions, particularly polynomials. And here, we need the following definition:
Definition 5.
A branched continued fraction whose elements are functions in a certain domain converges uniformly on the set if its sequence of approximants converges uniformly on If this occurs for an arbitrary set E such that , (Here, is the closure of the set E) then the branched continued fraction converges uniformly on every compact subset of
3. Branched Continued Fractions and Analytic Continuation
In this section, we prove that the branched continued fraction in Equation (1) (as well as Equation (3)) converges in new symmetric domains and provides the analytic continuation in these domains. One of the key results here is Theorem 2. Its proof reveals a technique for extending the convergence, which is already known for a small domain, to a larger domain, and it uses some of the results for Theorem 6 []. It also shows that one of the reasons why parabolic regions are so important is that they form the basis of the cardioid domains. Note that here, the region is a domain together with all, part, or none of its boundary.
Theorem 2.
Let and be complex constants which satisfy the conditions
where are defined by Equation (2), p is a positive number, and Then, the following are true:
Proof.
We show that (1) is valid in the domain in Equation (13), where for convenience we write the sets
as
Let
Then, it is clear that the recurrence relations
and
are valid for and
Let us show that the approximants of the branched continued fraction in Equation (1) form a sequence of functions holomorphic in the domain in Equation (13). Since the numerator and denominator of each approximant are polynomials, they are the entire functions of three variables. And the quotient of two entire functions, where the denominator is not equal to zero, is a holomorphic function. Therefore, it suffices to show that
Let n be an arbitrary natural number and be an arbitrary fixed point in Equation (13). We set Then the inequalities
and
are valid for
Indeed, since is an arbitrary fixed point in then for its arbitrary neighborhood, there exists such that
and therefore
Next, we prove the first inequality in Equation (20). From Equation (16), it is clear that Equation (20) is valid for Let the first inequality in Equation (20) hold for , where Then, from Equation (17), one finds that
and hence, under Equations (11) and (13) and Corollary 2 [], we have
and
Similarly, we obtain the first inequality in Equation (21). Thus, the inequalities in Equation (19) hold, and therefore, is a sequence of functions holomorphic in the domain in Equation (13).
Next, for an arbitrary compact subset of , there exists an open triple-disk
such that and hence, under Equations (18), (20), and (21), for any and , we have
In other words, is a sequence of functions uniformly bounded on every compact subset of Equation (13).
It is clear that for each L that satisfies the inequalities
the domain
is contained in Equation (13), particularly Moreover, for any using Equation (15), one finds that
In other words, the elements of Equation (1) satisfy Theorem 1 with where Thus, under Theorem 1 (1), the branched continued fraction in Equation (1) converges in An application of Theorem 5 [] then yields the uniform convergence of Equation (1) to a holomorphic function on every compact subset of Equation (13).
Therefore, to prove (1), it suffices to show that this assertion is also valid in the domain in Equation (14). An application of Theorem 1 (1) with where shows that the branched continued fraction in Equation (1) converges for all . Theorem 1 (2) implies that the approximants of Equation (1) all lie in the closed domain in Equation (9) if . Hence, under Theorem 5 [], the convergence of the branched continued fraction in Equation (1) is uniform on every compact subset of the domain in Equation (14).
The proof of (2) is analogous to the proof of Theorem 6 (2) [], and hence it was omitted. Note that from the proof of Theorem 6 (2) [], it follows that the branched continued fraction in Equation (1) corresponds at to the function on the left side of Equation (1). Since, as was proved above, the sequence of its approximants converges uniformly on each compact subset of some neighborhood of the origin to a function holomorphic in this neighborhood, one can apply the PC method (see []) and easily prove that the function to which the branched continued fraction in Equation (1) converges on the domain in Equation (12) is an analytic continuation of the function on the left side of Equation (1) in this domain. □
Corollary 1.
Theorem 3 is symmetric to Theorem 2, and thus it can be proven in much the same way as in Theorem 2.
Theorem 3.
Let and be complex constants which satisfy the conditions
where with are defined by Equation (4), p is a positive number, and Then, thet following are true:
Corollary 2.
An application of Theorem 2 follows:
Theorem 4.
Let and be real constants such that
where u is a positive number and where are defined by Equation (2). Then, the following are true:
Proof.
Note that Equation (32) is the Cartesian product of two planes cut along the real axis from to and one plane cut along the real axis from to where u is a positive number satisfying Equation (31).
Corollary 3.
An application of Theorem 3 is Theorem 5 below, which can be proven in much the same way as Theorem 4.
Theorem 5.
Let and be real constants such that
where v is a positive number and are defined by Equation (4). Then, the following are true:
Corollary 4.
The following result gives the analytical extension domain that is the Cartesian product of two planes cut along the real axis from to and one plane cut along the real axis from to where and u is a positive number that satisfies the following conditions (Equation (35)):
Theorem 6.
Let and be real constants which satisfy the conditions
where are defined by Equation (2) and u is a positive number. Then, the following are true:
Proof.
First of all, for convenience, we write the domain in Equation (36) as
where
and is defined by Equation (14) with
As in the proof of Theorem 2, we show that (1) is valid in the domain
Let n be an arbitrary natural number, be an arbitrary real value from and be an arbitrary fixed point in Equation (37). Then, we have
and
being valid for
Let us prove Equation (39). From Equation (16), it is clear that the inequalities in Equation (39) are valid for Assuming through the induction that Equation (39) holds for , where then from Equation (17), one obtains
In the same way, we obtain the inequalities in Equation (40). Thus, the inequalities
hold, and therefore, is a sequence of functions holomorphic in the domain in Equation (37).
For an arbitrary compact subset of Equation (38), there exists an open triple-disk (Equation (22)) such that We cover with domains of the form
and choose from this cover a finite subcover
We set
Then, for any and , we have
In other words, is a sequence of functions uniformly bounded on every compact subset of Equation (38).
It is clear that for each L such that
the domain
is contained in Equation (38) (e.g., ). Taking into account Equation (35), for any where is contained in Equation (38), one can find that
In other words, the elements of Equation (1) satisfy Theorem 1, with and Thus, according to Theorem 1 (1), the branched continued fraction in Equation (1) converges in where is contained in Equation (38). It follows from Theorem 5 [] that the convergence is uniform on compact subsets of Equation (38) to a holomorphic function in this domain.
The fact that (1) is also valid in the domain in Equation (14) with can be proven in much the same way as in the proof of Theorem 2 (1). The proof of (2) is analogous to the proof of Theorem 2 (2) and Theorem 6 (2) [], and hence it was omitted. □
Corollary 5.
Finally, we have the following theorem, which is symmetric to Theorem 6:
Theorem 7.
Let and be complex constants which satisfy the conditions
where are defined by Equation (4) and v is a positive number. Then, the following are true:
Corollary 6.
Let and be real constants satisfying Equation (41), where are defined by Equation (28) and v is a positive number. Then, Equation (29) converges uniformly on every compact subset of Equation (42) to a function , which is holomorphic in and is an analytic continuation of Equation (30) in this domain.
4. Discussions and Conclusions
We considered the representation and extension of the analytic functions of three variables by a special family of functions: branched continued fractions. The main results were new symmetric domains of analytical continuation for Lauricella–Saran’s hypergeometric functions with certain conditions for real and complex parameters, which were established using their branched continued fraction representations. In particular, in the case of real parameters, we obtained the Cartesian product of two planes cut along the real axis from to and one plane cut along the real axis from to where u is a positive number. To prove the above, we used a technique that extends the convergence of branched continued fractions, which is already known for a small domain, to a larger domain, as well as the PC method (see []) to prove that they were also domains of analytical continuation. In fact, in pairs, they were proven to be effective tools and could therefore be applied to Lauricella–Saran’s other functions. However, we could not establish such domains of analytical continuation for Lauricella–Saran’s hypergeometric functions with arbitrary admissible real or complex parameters. Unfortunately, the well-developed methods for investigating the convergence of continued fractions are generally not carried over to their multidimensional generalization of branched continued fractions. Therefore, there is a need to develop new methods that would provide effective convergence criteria in both the partial and general cases.
The branched continued fraction, being a generalization of the continued fraction, is interesting in itself because it has good approximate properties, such as wide regions of convergence and numerical stability. Therefore, further investigations can be continued in various directions. First, one can try to extend the domains of convergence of the branched continued fraction expansions with real and complex coefficients in their elements using parabolic regions of convergence and angular domains, which can be found in [,,,], respectively. To render branchedcontinued fractions as more useful in computation, it is necessary to know more about their rate of convergence and numerical stability. Therefore, truncation error analysis and the computational stability of the branched continued fraction expansions are other directions. These are interesting and somewhat new directions, and there are not many results here (see [,,,,,]).
The numerical experiments in [,,,,,] have shown that branched continued fraction expansions provide a useful tool for representing and computing the values of analytic functions. Therefore, a no less interesting and essential direction of research is the application of branched continued fractions to compute special functions, including Lauricella–Saran’s hypergeometric functions, which naturally arise in various problems in various fields of science, especially physics (see, for example, [,,,,] and also [] (Part 5), [] (Chapters 7 and 8), [], and [] (Chapters 5)).
Author Contributions
Conceptualization, R.D.; methodology, R.D.; investigation, R.D. and V.G.; writing—original draft, R.D. and V.G. All authors have read and agreed to the published version of the manuscript.
Funding
This research received no external funding.
Data Availability Statement
No new data were created or analyzed in this study. Data sharing is not applicable to this article.
Acknowledgments
The authors were partially supported by the Ministry of Education and Science of Ukraine, project registration number 0123U101791.
Conflicts of Interest
The authors declare no conflicts of interest.
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