Representation of a one class function of two variables by bicontinued fractions

Let function $u (z, w) = f (z) h (w)$ be defined on the compact set  $\mathbf{K} \subset \mathbb{C}^2$. We study the problem of representation of functions of this class by the product of two continued fractions, which is called a bicontinued fraction. Some properties of Thiele reciprocal derivatives,  Thiele continued fractions and  regular C-fractions are proved. The possibility of representation of functions of this class by bicontinued fractions is shown. Examples are considered, domains of convergence and uniform convergence of obtained bicontinued fractions to the function are indicated.

The Thiele formula is an analogue of a Taylor formula in the theory of continued fractions. The Thiele formula has advantages over other methods of expansion of a function in continued fractions since coefficients of expansion are determined by reciprocal derivatives of a function [11,12,13].
Bicontinued fraction is a new concept that is introduced in this article. Let u be a function of two variables of the form u(z, w) = f (z)h(w). The problem of a function approximation by bicontinued fractions investigated in the paper. The new properties of Thiele continued fractions and Thiele reciprocal derivatives are proved. These properties are used when representing functions by bicontinued fractions.

Approximation of functions by continued fractions
Note that in addition to the set of natural numbers N, we also use the following sets of integer numbers N 0 = N ∪ {0} and N 2 = N\{1}.
The necessary concepts of the theory of continued fractions [9] will be given below.

Definition 1.
A continued fraction is an ordered pair {a n }, {b n } , {f n } , where {a n : a n ∈ C, a n = 0, n ∈ N}, {b n : b n ∈ C, n ∈ N 0 } are sequences of complex numbers, and {f n : f n ∈ C = C ∪ {∞}, n ∈ N 0 } is a sequence in the extended complex plane defined as follows: f n = S n (0), n ∈ N 0 , S 0 (w) = s 0 (w), S n (w) = S n−1 (s n (w)), n ∈ N, s 0 (w) = b 0 + w, s n (w) = a n /(b n + w), n ∈ N.
Similarly, the finite continued fraction, nth approximant of infinite continued fraction Theorem 1. Continued fractions b 0 + K(a i /b i ) and d 0 + K(c i /d i ) are equivalent iff there exists a sequence of non-zero constants {r i : r 0 = 1, r i = 0, i ∈ N} such that (2.1) Let the function of complex variable g is defined on the compact set Z ⊂ C. Function g can be interpolated on set of nodes Z = {z i : z i ∈ Z, z i = z j , i, j = 0, n} by a Thiele interpolation continued fraction (T-ICF) of the form Canonical numerator P n (z; g) and canonical denominator Q n (z; g) of a T-ICF (2.2) are polynomials whose degrees satisfy inequalities deg P n (z; g) ≤ n+1 2 , deg Q n (z; g) ≤ n 2 . By ρ k [z 0 , z 1 , . . . , z k ; g], k = 0, n, denote the reciprocal difference of the kth order.
The reciprocal differences are calculated as follows , k = 2, n.
Coefficients of a T-ICF (2.2) are determined by through reciprocal differences according to the formulae It is known [12,19] that the reciprocal difference of the kth order ρ k [z 0 , z 1 , . . . , z k ; g] is determined by the relation of two determinants formed from interpolation nodes and function values in nodes. The reciprocal difference is the symmetric function of arguments z 0 , z 1 , . . . , z k .
Theorem 2. Let D n (z; g) be a T-ICF of a function g and C = const. Then D n (z; Cg) = C D n (z; g).

REPRESENTATION OF A ONE CLASS FUNCTIONS
Consider the a T-ICF of a function Cg. We have We shall use the relation (2.7). Then we shall have that . . , z k ; g))).
Since the reciprocal difference of the kth order ρ k [z 0 , . . . , z k ; g], k = 0, n, is a symmetric function of the arguments z 0 , z 1 , . . . , z k , then we can go to the limiting case. Definition 3. If exist limit (finite number or infinity) of a reciprocal difference kth order ρ k [z 0 , . . . , z k ; g] when the nodes z 0 , . . . , z k → z * , where z * ∈ Z then the limit value is called the Thiele reciprocal derivative of a kth order of a function g at the point z * and is denoted (k) g(z * ).
It follows from the definition that (k) g(z * ) = lim z 0 ,...,z k →z * ρ k [z 0 , . . . , z k ; g]. The Thiele reciprocal derivatives are calculated by the recurrent formula [11] (1) g(z * ) = 1 ). If the function g is analytic at Z ⊂ C then Thiele reciprocal derivatives of a function g at z * ∈ Z are defined as follows where Hankel determinants H k (z * ), where k = 0, n, be non-zero then a function g at the point z * has finite Thiele reciprocal derivatives up to (2n)th order inclusive which may be calculated by formula (2.8) or by formula (2.9).
Let the function g have Thiele reciprocal derivatives at the point z * ∈ Z. Then we get the expansion of a function g into a Thiele continued fraction (T-CF) in some neighbourhood of the point z * , i.e.
The Thiele reciprocal derivatives has properties that follows directly from properties of reciprocal differences. In particular From Theorem 2, definition of a Thiele reciprocal derivatives, formulae (2.11) and properties (2.12) it follows that D(z; Cg) = C D(z; g), C = const. (2.13) Theorem 5 ( [13]). Let the function w = f (z) has a Thiele reciprocal derivative at the point z 0 ∈ G and let the function u = g(w) has a Thiele reciprocal derivative at the point w 0 ∈ E, where w 0 = f (z 0 ), then the composed function F (z) = g (f (z)) has a Thiele reciprocal derivative at point z 0 and (1) F (z 0 ) = (1) We shall prove the following property of Thiele reciprocal derivatives. Theorem 6. Let the function g has Thiele reciprocal derivatives and C = const then (2.14) Proof. From theorem 5 follows that (1) It follows from here, from (2.8) and (2.12) that Suppose that the formulas (2.14) are true when k = 1, s − 1. Then when k = s from (2.12) we have that The relation A T-CF (2.10) can be given by an equivalent continued fraction with partial denominators equal to one. We have The coefficients of a continued fraction (2.17) can be determined by the coefficients of a T-CF as follows Let the function g is expanded in a formal power series at the neighbourhood of a point z * . It is proved in [13] that (2.17) is equal to a regular continued C-fraction (C-CF). Since the a C-CF is corresponding to the formal power series, then a T-CF will also correspond to the formal power series. It's easy to make sure that F (z; Cg) = CF (z; g), C = const.
Theorem 7 ( [13]). Let the function g be expanded into a C-CF (2.17) in the some neighbourhood of a point z * ∈ Z and lim n→∞ a n (z * ; g) = a, a ∈ C, a = 0.
(A) The continued fraction (2.17) converge to the function g which is meromorphic in the domain R a = {z ∈ C : | arg (a(z − z * ) + 1/4)| < π}. (B) The convergence will be uniform on an arbitrary compact set C ⊂ R a which not contains poles of a function g. (C) The function g is holomorphic at the point z * . Theorem 8 ( [13]). Let the function g be expanded into a C-CF (2.17) in the some neighbourhood of the point z * ∈ Z, a n (z * ; g) = 0 and lim n→∞ a n (z * ; g) = 0.
(A) The continued fraction (2.17) converge to the function g. (B) The convergence will be uniform on an arbitrary compact set Z ⊂ C which not contains poles of a function g. (C) The function g is holomorphic at the point z * and g(z * ) = a 0 (z * ; g).

Representation of functions of two variables by bicontinued fractions
The results obtained in the previous section, allow us to specify a way of representation functions of two complex variables of the form by the product of two continued fractions for each variable. We shall call this product a bicontinued fraction. Let the function u be defined on the compact set K = Z × W ⊂ C 2 . Suppose that functions f and h are analytic on the compact sets Z and W respectively. Then, according to Theorem 3 and Theorem 4, each of these functions has Thiele reciprocal derivatives.
Letw ∈ W be a fixed point and H = h(w). The auxiliary function F(z) = Hf (z) can be expanded into a T-CF about variable z on the compact set Z. If considers properties of the Thiele continued fraction (2.13) than we obtain the expansion of a function F in a neighbourhood of the point z * ∈ Z into a T-CF about variable z Similarly, letz ∈ Z be a fixed point and F = f (z), then the auxiliary function H(w) = F h(w) can be expanded into a T-CF in the neighbourhood of the point Since the point (z,w) is an arbitrary point of the compact set K then we get a representation of a function u as the product of two T-CF in the neighbourhood of a point (z * , w * ) ∈ K We call this product a Thiele bicontinued fraction (T-BCF). Similarly, the function u can be represented by a bicontinued C-fraction (C-BCF) We shall consider examples of function representations by bicontinued fractions and we shall show the domains of convergence and domains of uniform convergence of such representations too. Example 1. Consider the function u 1 (z, w) = (δ+βz) α tg(ε+γw), where α ∈ C\{Z}, β, γ, δ, ε ∈ C\{0}.
Proof. It is proved in [13] that the coefficients of expansion function (δ + z) α into a T-CF in the neighbourhood of a point z * ∈ C\{−δ} are equals (3.6) From (2.16) and (3.6) it follows that the coefficients of expansion of a function (δ +βz) α into a T-CF in the neighbourhood of a point z * ∈ C\{−δ/β} will be equals n ∈ N.
We make the notation ξ = δ + βz * . Let the sequence {r i } be defined as follows .

REPRESENTATION OF A ONE CLASS FUNCTIONS
If the sequence {r i } is defined as follows r 2n−1 = αe αz * , r 2n = e −αz * , n ∈ N, then after equivalent transformations we obtain the expansion of a function e αz into a T-CF The expansion of a function into a C-CF will be written as follow it follows from Theorem 8 that continued fractions (3.15) and (3.16) converge to the function e αz on complex plane C and on an arbitrary compact set Z ⊂ C continued fractions converge uniformly. In the monograph [13] was proved that the coefficients of the expansion of a function ln(β + w) into a T-CF in the neighbourhood of a point w * ∈ C\{−β} takes values b 2n−1 (w * ; ln(β + w)) = (2n − 1)(β + w * ), b 2n (w * ; ln(β + w)) = 2 n , n ∈ N. (3.17) Similar to the previous case, it follows from (2.16) and (3.17) that the coefficients of the expansion of a function ln(β + γw) into a T-CF in the neighbourhood of a point w * ∈ W = C\{−β/δ} are defined by formulae b 2n−1 (w * ; ln(β + γw)) = 2n + 1 γ (β + γw * ), b 2n (w * ; ln(β + γw)) = 2 n , n ∈ N.

Conclusions
The new properties of Thiele reciprocal derivatives which proved in this work allow obtaining a representation of functions of two variables in the form of the product of two continued fractions. This technique can be naturally extended to the case of functions of three, four or more variables. Also, the properties proved in this work together with other properties of Thiele reciprocal derivatives allow us to find the extension of a function f (α + βz) from the extension of a function f (z).