1. Introduction
In some references [1] [2] [3] , the boundary value problem (Riemann-Hilbert problem) of analytic functions on finite curves is discussed, but the research on infinite curves is not deep enough. In [4] , the author discusses the Riemann boundary value problem on the positive real axis and generalizes the concept of the generalized principal part.
The Riemann-Hilbert method is a brand-new method for studying orthogonal polynomials formed in recent 20 years. In 1992, FoKas A S, Its A R and Kitaev A V constructed a matrix-valued Riemann-Hilbert boundary value problem in [5] , the only solution of which is the orthogonal polynomial on the real axis. In 1993, Deift P and Zhou X introduced the Riemann-Hilbert boundary value problem of oscillatory type in [6] , and applied it to the study of orthogonal polynomials. Therefore, the Riemann-Hilbert method was formed [6] .
2. Preliminary
In this paper, the right branch of the Hyperbola
is denoted by default to L, which is regarded as the image of the function
, and L is oriented from top to bottom.
Denote by
the point
and
respectively its upper and lower infinite ends. Then
consists of two connected components, the right part
and the left part
.
We use bilinear form to replace inner product on hyperbola, which is a common way. For example, [Lu J K, 1993] gives the solvable condition of singular integral equation by this way; for example, Delft P. defined a polynomial group similar to orthogonal polynomials in bilinear form in [7] , and we studied similar polynomial groups on hyperbola:
Let
be a nonzero weight function. We introduce bilinear form in polynomial space
with degree no more than n:
(1)
Take a group of bases
in
and make Schmidt orthogonalization on this group of bases, then we have
where
, If
is always not zero, then this process can always be carried out. Finally, we get a pseudo-orthogonal polynomial group with a weight function of
on L:
(2)
where
is the first coefficient of
, then
is a pseudo-orthogonal polynomial of degree k with the first coefficient of 1. Obviously, the pseudo-orthogonal polynomial group
is unique.
Definition 1. Let f is defined on L, if there is some positive real number a, such that
(3)
where M and
are definite constants, then denoted by
, and if
, then denoted by
. If
and
, then denoted by
, or
. Moreover, if
, then denoted by
.
Definition 2 Let f is a function defined on L. There exists
such that
,
where v is a real number and
is a bounded function, then denoted by
.
Definition 3 If F is holomorphic in the complex plane cut by the Hyperbola, then denoted by
.
Definition 4 Let f be a locally integrable function on L. If
(4)
is integrable, it is called the Cauchy-type integral with kernel density f on L, and the Cauchy principal value integral with kernel density f is defined by
(5)
where
, if the integral exists.
Ref [Wang Ying, 2017], below we introduce the concept of a generalized main part.
Definition 5 Let
. If there exists an entire function
such that
. (6)
and then
is called the generalized principal part of
at
, denoted by
.
Reference [8] proves the generalized principal part of Cauchy integral at infinity and Plemelj formula.
Theorem 1 [8] If
is locally integrable on L. Then
. (7)
Theorem 2 [8] If
, then the boundary values of the Cauchy-type integral
exist and have the following Plemelj formula:
. (8)
3. Matrix Value Riemann Boundary Value Problem
In this paper, we consider the Riemann boundary value problem of lower trigonometric matrix on hyperbola.
Let
(9)
be a matrix-valued function defined on subset
of the complex plane
, and each element
be a function defined on
. If every element
of
satisfies the same property, then
is said to have its corresponding property, such as
.
Problem (boundary value problem of lower trigonometric matrix value function) Find the matrix-valued partitioned holomorphic function
with L as the jump curve, such that
(10)
where
(11)
I is the identity matrix of 2 × 2,
.
We can convert (10) into four related Riemann boundary value problems:
(12)
(13)
(14)
(15)
Obviously, (12) is a Liouville problem. It is known from Painlevé theorem that
is analytic over the entire complex plane. Because
, it is known from the generalized Liouville theorem that
(16)
where
is a polynomial with a leading coefficient of 1 and a degree of n.
By (16), we have
(17)
Obviously (17) is a jump problem with L as the jump curve. Let
, (18)
by
,
. (19)
Therefore, by Plemelj formula (8) and Theorem 1, we can know that
is a partitioned holomorphic function with L as the jump curve, and satisfies:
(20)
let
, then F is a partitioned holomorphic function with L as the jump curve and satisfies:
(21)
Obviously problem (21) is a zero-order Liouville problem, its solution is
, so
(22)
if and only if condition
is satisfied. By
(23)
and Theorem 1 and (19), it can be seen that
is equivalent to
. (24)
Obviously (13) is the Liouville problem, similar to (12) we have
(25)
where
is a polynomial of order not exceeding
.
By (16), we have
(26)
Obviously, (26) is a fixed-order jump problem, similar to (15). It can be seen that its solution is
(27)
if and only if condition
(28)
is satisfied.
Let
, then
(29)
that is,
(30)
then
is a pseudo-orthogonal polynomial of degree
on L with respect to the weight function w.
Definition 6
(31)
we call it the companion function of f with respect to the weight function w.
Theorem 3 If
, then the lower triangular matrix-valued Riemann boundary value problem (10) has a solution, and its solution has the following form:
(32)
where
is a polynomial with a leading coefficient of 1 and a degree of n, and
is the companion function of
with respect to the middle weight function w.
Proof: If (10) has a solution, it can be seen from the previous discussion that its solution is of the form (32).
Conversely, the polynomial with pseudo-orthogonal and leading coefficient 1 is unique, and by reversing each previous step, we get that
is the solution of (10), that is, (10) has and only one set of solutions (32).
The matrix-valued boundary value problem (10) is characterized by the pseudo-orthogonal polynomial
on L with respect to the weight function w and the leading coefficient is 1. Therefore, we call this problem the Riemann-Hilbert characteristic characterization of the orthogonal polynomial of the weight function w on hyperbola, or
is the characteristic orthogonal polynomial of the matrix-valued boundary value problem (10), please refer to [Deift P, 2011] for details.