# The Jacobi Operator on $(-1,1)$ and
its Various $m$-Functions

Fritz Gesztesy  Department of Mathematics, Baylor University, Sid Richardson
Bldg., 1410 S. 4th Street, Waco, TX 76706, USA<EMAIL_ADDRESS>http://www.baylor.edu/math/index.php?id=935340 ,  Lance L. Littlejohn
Department of Mathematics, Baylor University, Sid Richardson Bldg., 1410 S.
4th Street, Waco, TX 76706, USA<EMAIL_ADDRESS>http://www.baylor.edu/math/index.php?id=53980 ,  Mateusz Piorkowski
Department of Mathematics, KU Leuven
Celestijnenlaan 200B, 3001 Leuven, Belgium<EMAIL_ADDRESS>https://www.kuleuven.be/wieiswie/en/person/00153535 and  Jonathan Stanfill
Department of Mathematics, The Ohio State University
100 Math Tower, 231 West 18th Avenue, Columbus, OH 43210, USA
<EMAIL_ADDRESS>https://u.osu.edu/stanfill-13/

###### Abstract.

We offer a detailed treatment of spectral and Weyl–Titchmarsh–Kodaira theory
for all self-adjoint Jacobi operator realizations of the differential
expression

$\displaystyle\tau_{\alpha,\beta}=-(1-x)^{-\alpha}(1+x)^{-\beta}(d/dx)\big{(}(1-x)^{\alpha+1}(1+x)^{\beta+1}\big{)}(d/dx),$
$\displaystyle\alpha,\beta\in{\mathbb{R}},\;x\in(-1,1),$

in $L^{2}\big{(}(-1,1);(1-x)^{\alpha}(1+x)^{\beta}dx\big{)}$,
$\alpha,\beta\in{\mathbb{R}}$. In addition to discussing the separated
boundary conditions that lead to Jacobi orthogonal polynomials as
eigenfunctions in detail, we exhaustively treat the case of coupled boundary
conditions and illustrate the latter with the help of the general
$\eta$-periodic and Krein–von Neumann extensions. In particular, we treat all
underlying Weyl–Titchmarsh–Kodaira and Green’s function induced $m$-functions
and revisit their Nevanlinna–Herglotz property. We also consider connections
to other differential operators associated with orthogonal polynomials such as
Laguerre, Gegenbauer, and Chebyshev.

###### Key words and phrases:

Singular Sturm–Liouville operators, Jacobi and hypergeometric differential
equation, boundary values, boundary conditions, Weyl–Titchmarsh–Kodaira
$m$-functions.

###### 2020 Mathematics Subject Classification:

Primary: 33C45, 34B09, 34B24, 34C10, 34L40; Secondary: 34B20, 34B30.

###### Contents

  1. 1 Introduction
  2. 2 The Basics of Weyl–Titchmarsh–Kodaira Theory
  3. 3 The (Singular) Weyl–Titchmarsh–Kodaira $m$-Function
  4. 4 The Jacobi Operator and its Boundary Values
  5. 5 The Regular and Limit Circle Case $\alpha,\beta\in(-1,1)$
    1. 5.1 The Regular and Limit Circle Case for Separated Boundary Conditions
    2. 5.2 The Regular and Limit Circle Case for Coupled Boundary Conditions
  6. 6 Precisely One Interval Endpoint in the Limit Point Case
  7. 7 Both Endpoints in the Limit Point Case
  8. 8 More on the Nevanlinna–Herglotz property of $m$-functions  and on Jacobi Polynomials
  9. A The Hypergeometric and Jacobi Differential Equations
  10. B Connection Formulas
  11. C Behavior of $y_{j,\alpha,\beta,\mp 1}(z,x)$, $j=1,2$, near $x=\pm 1$
  12. D Laguerre $m$-function as a limit of the Jacobi $m$-function
  13. E Special Cases
    1. E.1 The Case of Gegenbauer or Ultraspherical Polynomials
    2. E.2 The Case of Chebyshev Polynomials
    3. E.3 The Case of Zernike Polynomials

##  1\. Introduction

Despite the prominence of the Jacobi differential expression

$\displaystyle\begin{split}\tau_{\alpha,\beta}=-(1-x)^{-\alpha}(1+x)^{-\beta}(d/dx)\big{(}(1-x)^{\alpha+1}(1+x)^{\beta+1}\big{)}(d/dx),&\\\
\alpha,\beta\in{\mathbb{R}},\;x\in(-1,1),&\end{split}$ (1.1)

in the orthogonal polynomial community (see, e.g., [1, Ch. 22], [3], [4, Ch.
6], [6, Ch. 4], [16, Ch. X], [36, Ch. 4], [49, Ch. V], [53, Ch. 18], [58, Ch.
IV]), the actual spectral and Weyl–Titchmarsh–Kodaira theory for self-adjoint
realizations associated with (LABEL:1.1) has received relatively little
attention, with the notable exception of the papers by Bush, Frymark, and Liaw
[10] discussing the case $\alpha,\beta\in(0,1)$, as well as Frymark and Liaw
[20] and Frymark [19] discussing the case $\alpha,\beta\in[0,1)$. This paper
aims at filling this gap and presents a discussion for the general case
$\alpha,\beta\in{\mathbb{R}}$. In particular, we derive the
Weyl–Titchmarsh–Kodaira $m$-functions (for separated boundary conditions) and
the Green’s function induced $M$-functions (for coupled boundary conditions),
for all self-adjoint realizations associated with (LABEL:1.1). We note that in
the context of Donoghue $m$-functions, the general case was recently studied
in [30] when at least one endpoint was in the limit circle case (see also [26]
for background). In particular, the Donoghue $m$-functions associated with all
coupled boundary conditions were explicitly constructed in [30], which, to the
best of our knowledge, was the first complete study of coupled boundary
conditions in this setting.

We now turn to the basic structure of the present paper. Based on [25] and
[29, Ch. 13], we recall the basics of Weyl–Titchmarsh–Kodaira theory in
Section 2, and explicitly treat the corresponding $m$ and $M$ functions for
separated and coupled boundary conditions, respectively, in Section 3. In
particular, the basics of the Green’s function induced $M$-function in the
presence of coupled boundary conditions is recalled in the last part of
Section 3. The minimal and maximal Jacobi operators and associated boundary
values employing principal and nonprincipal solutions are discussed in Section
4.

Our main results on the explicit construction of Jacobi operator $m$ and
$M$-functions are presented in Section 5–7. Section 5 treats the regular and
limit circle cases of $\tau_{\alpha,\beta}$ corresponding to
$\alpha,\beta\in(-1,1)$, for both separated and coupled boundary conditions;
the case where $\tau_{\alpha,\beta}$ is in the limit point case at precisely
one of the interval endpoints $\pm 1$ is treated in Section 6; the case where
$\tau_{\alpha,\beta}$ is in the limit point case at $-1$ and $1$ is presented
in Section 7. In Section 8 we revisit situations where the
Weyl–Titchmarsh–Kodaira $m$-function possesses the Nevanlinna–Herglotz
property with particular emphasis on the case supporting Jacobi polynomials as
eigenfunctions.

Appendix A on solutions of the hypergeometric and Jacobi differential
equations, Appendix B on connection formulas, and Appendix C on the behavior
of solutions near the endpoints $\pm 1$, complement the bulk of this paper.
Moreover, Appendix D addresses how to obtain Weyl–Titchmarsh–Kodaira
$m$-functions for the Laguerre operator as a limit of the
Weyl–Titchmarsh–Kodaira $m$-functions constructed in Sections 5–6 (see also
Section 8). Appendix E provides a brief discussion of special cases of the
Jacobi differential expression, including the Gegenbauer (ultraspherical),
Chebyshev, and radial part of the Zernike differential expressions. Some
material in these appendices was also used in [30].

Finally, we briefly comment on some of the basic notation used throughout this
paper. If $T$ is a linear operator mapping (a subspace of) a Hilbert space
into another, then $\operatorname{dom}(T)$ and $\ker(T)$ denote the domain and
kernel (i.e., null space) of $T$. The spectrum and resolvent set of a closed
linear operator in a Hilbert space will be denoted by $\sigma(\cdot)$ and
$\rho(\cdot)$, respectively. We also use the shorthand notation
${\mathbb{N}}_{0}={\mathbb{N}}\cup\\{0\\}$ and
${\mathbb{C}}_{\pm}=\\{z\in{\mathbb{C}}\,|\,\pm\text{\rm Im}(z)>0\\}$.

##  2\. The Basics of Weyl–Titchmarsh–Kodaira Theory

In this section, following [25] and [29, Ch. 13], we summarize the singular
Weyl–Titchmarsh–Kodaira theory as needed to treat the Jacobi operator in the
remainder of this paper.

Throughout this section we make the following assumptions:

######  Hypothesis 2.1.

Let $(a,b)\subseteq{\mathbb{R}}$ and suppose that $p,q,r$ are $($Lebesgue $)$
measurable functions on $(a,b)$ such that the following items $(i)$–$(iii)$
hold:
$(i)$ $r>0$ a.e. on $(a,b)$, $r\in{L^{1}_{loc}((a,b);dx)}$.
$(ii)$ $p>0$ a.e. on $(a,b)$, $1/p\in{L^{1}_{loc}((a,b);dx)}$.
$(iii)$ $q$ is real-valued a.e. on $(a,b)$, $q\in{L^{1}_{loc}((a,b);dx)}$.

Given Hypothesis 2.1, we study Sturm–Liouville operators associated with the
general, three-coefficient differential expression

$\tau=\frac{1}{r(x)}\left[-\frac{d}{dx}p(x)\frac{d}{dx}+q(x)\right]\,\text{
for a.e.~{}$x\in(a,b)\subseteq{\mathbb{R}}$,}$ (2.1)

and introduce maximal and minimal operators in ${L^{2}((a,b);rdx)}$ associated
with $\tau$ in the usual manner as follows.

######  Definition 2.2.

Assume Hypothesis 2.1. Given $\tau$ as in (2.1), the maximal operator
$T_{max}$ in ${L^{2}((a,b);rdx)}$ associated with $\tau$ is defined by

$\displaystyle T_{max}f=\tau f,$ $\displaystyle
f\in\operatorname{dom}(T_{max})=\big{\\{}g\in{L^{2}((a,b);rdx)}\,\big{|}\,g,g^{[1]}\in{AC_{loc}((a,b))};$
(2.2) $\displaystyle\hskip 179.25244pt\tau g\in{L^{2}((a,b);rdx)}\big{\\}}.$

The preminimal operator $\overset{\textbf{\Large.}}{T}_{min}$ in
${L^{2}((a,b);rdx)}$ associated with $\tau$ is defined by

$\displaystyle\overset{\textbf{\Large.}}{T}_{min}f=\tau f,$ $\displaystyle
f\in\operatorname{dom}\big{(}\overset{\textbf{\Large.}}{T}_{min}\big{)}=\big{\\{}g\in{L^{2}((a,b);rdx)}\,\big{|}\,g,g^{[1]}\in{AC_{loc}((a,b))};$
(2.3) $\displaystyle\hskip
92.47145pt\operatorname{supp}\,(g)\subset(a,b)\text{ is compact; }\tau
g\in{L^{2}((a,b);rdx)}\big{\\}}.$

One can prove that $\overset{\textbf{\Large.}}{T}_{min}$ is closable, and one
then defines the minimal operator $T_{min}$ as the closure of
$\overset{\textbf{\Large.}}{T}_{min}$.

The following facts then are well known:

$\big{(}\overset{\textbf{\Large.}}{T}_{min}\big{)}^{*}=T_{max},$ (2.4)

and hence $T_{max}$ is closed and
$T_{min}=\overline{\overset{\textbf{\Large.}}{T}_{min}}$ is given by

$\displaystyle T_{min}f=\tau f,$ $\displaystyle
f\in\operatorname{dom}(T_{min})=\big{\\{}g\in{L^{2}((a,b);rdx)}\,\big{|}\,g,g^{[1]}\in{AC_{loc}((a,b))};$
(2.5) $\displaystyle\qquad\text{for all
}h\in\operatorname{dom}(T_{max}),\,W(h,g)(a)=0=W(h,g)(b);\,\tau
g\in{L^{2}((a,b);rdx)}\big{\\}}$
$\displaystyle\quad=\big{\\{}g\in\operatorname{dom}(T_{max})\,\big{|}\,W(h,g)(a)=0=W(h,g)(b)\,\text{for
all }h\in\operatorname{dom}(T_{max})\big{\\}}.$

Moreover, $\overset{\textbf{\Large.}}{T}_{min}$ is essentially self-adjoint if
and only if $T_{max}$ is symmetric, and then
$\overline{\overset{\textbf{\Large.}}{T}_{min}}=T_{min}=T_{max}$.

Here the Wronskian of $f$ and $g$, for $f,g\in{AC_{loc}((a,b))}$, is defined
by

$W(f,g)(x)=f(x)g^{[1]}(x)-f^{[1]}(x)g(x),\quad x\in(a,b),\\\ $

with

$y^{[1]}(x)=p(x)y^{\prime}(x),\quad x\in(a,b),$ (2.6)

denoting the first quasi-derivative of a function $y\in AC_{loc}((a,b))$.

The celebrated Weyl alternative then can be stated as follows:

######  Theorem 2.3 (Weyl’s Alternative).

Assume Hypothesis 2.1. Then the following alternative holds: Either
$(i)$ for every $z\in{\mathbb{C}}$, all solutions $u$ of $(\tau-z)u=0$ are in
${L^{2}((a,b);rdx)}$ near $b$ $($resp., near $a$$)$,
or,
$(ii)$ for every $z\in{\mathbb{C}}$, there exists at least one solution $u$ of
$(\tau-z)u=0$ which is not in ${L^{2}((a,b);rdx)}$ near $b$ $($resp., near
$a$$)$. In this case, for each $z\in{\mathbb{C}}\backslash{\mathbb{R}}$, there
exists precisely one solution $u_{b}$ $($resp., $u_{a}$$)$ of $(\tau-z)u=0$
$($up to constant multiples$)$ which lies in ${L^{2}((a,b);rdx)}$ near $b$
$($resp., near $a$$)$.

This yields the limit circle/limit point classification of $\tau$ at an
interval endpoint and links self-adjointness of $T_{min}$ (resp., $T_{max}$)
and the limit point property of $\tau$ at both endpoints as follows.

######  Definition 2.4.

Assume Hypothesis 2.1.
In case $(i)$ in Theorem 2.3, $\tau$ is said to be in the limit circle case at
$b$ $($resp., at $a$$)$. $($Frequently, $\tau$ is then called quasi-regular at
$b$ $($resp., $a$$)$.$)$
In case $(ii)$ in Theorem 2.3, $\tau$ is said to be in the limit point case at
$b$ $($resp., at $a$$)$.
If $\tau$ is in the limit circle case at $a$ and $b$ then $\tau$ is also
called quasi-regular on $(a,b)$.

######  Theorem 2.5.

Assume Hypothesis 2.1, then the following items $(i)$ and $(ii)$ hold:
$(i)$ If $\tau$ is in the limit point case at $a$ $($resp., $b$$)$, then

$W(f,g)(a)=0\,\text{$($resp., $W(f,g)(b)=0$$)$ for all
$f,g\in\operatorname{dom}(T_{max})$.}$ (2.7)

$(ii)$ Let $T_{min}=\overline{\overset{\textbf{\Large.}}{T}_{min}}$. Then

$\displaystyle\begin{split}n_{\pm}(T_{min})&=\dim(\ker(T_{max}\mp iI))\\\
&=\begin{cases}2&\text{if $\tau$ is in the limit circle case at $a$ and
$b$,}\\\ 1&\text{if $\tau$ is in the limit circle case at $a$}\\\ &\text{and
in the limit point case at $b$, or vice versa,}\\\ 0&\text{if $\tau$ is in the
limit point case at $a$ and $b$}.\end{cases}\end{split}$ (2.8)

In particular, $T_{min}=T_{max}$ is self-adjoint $($i.e.,
$\overset{\textbf{\Large.}}{T}_{min,}$ is essentially self-adjoint $)$ if and
only if $\tau$ is in the limit point case at $a$ and $b$.

One now recalls the following fundamental result characterizing all self-
adjoint extensions of $T_{min}$:

######  Theorem 2.6.

Assume Hypothesis 2.1 and that $\tau$ is in the limit circle case at $a$ and
$b$ $($i.e., $\tau$ is quasi-regular on $(a,b)$$)$. In addition, assume that
$v_{j}\in\operatorname{dom}(T_{max})$, $j=1,2$, satisfy

$W(\overline{v_{1}},v_{2})(a)=W(\overline{v_{1}},v_{2})(b)=1,\quad
W(\overline{v_{j}},v_{j})(a)=W(\overline{v_{j}},v_{j})(b)=0,\;j=1,2.$ (2.9)

$($E.g., real-valued solutions $v_{j}$, $j=1,2$, of $(\tau-\lambda)u=0$ with
$\lambda\in{\mathbb{R}}$, such that $W(v_{1},v_{2})=1$.$)$ For
$g\in\operatorname{dom}(T_{max})$ we introduce the generalized boundary values

$\displaystyle\begin{split}\widetilde{g}_{1}(a)&=-W(v_{2},g)(a),\quad\widetilde{g}_{1}(b)=-W(v_{2},g)(b),\\\
\widetilde{g}_{2}(a)&=W(v_{1},g)(a),\quad\;\,\,\,\widetilde{g}_{2}(b)=W(v_{1},g)(b).\end{split}$
(2.10)

Then the following items $(i)$–$(iii)$ hold:
$(i)$ All self-adjoint extensions $T_{\gamma,\delta}$ of $T_{min}$ with
separated boundary conditions are of the form

$\displaystyle T_{\gamma,\delta}f=\tau f,\quad\gamma,\delta\in[0,\pi),$
$\displaystyle
f\in\operatorname{dom}(T_{\gamma,\delta})=\big{\\{}g\in\operatorname{dom}(T_{max})\,\big{|}\,\widetilde{g}_{1}(a)\cos(\gamma)+\widetilde{g}_{2}(a)\sin(\gamma)=0;$
(2.11) $\displaystyle\hskip
156.49014pt\,\widetilde{g}_{1}(b)\cos(\delta)+\widetilde{g}_{2}(b)\sin(\delta)=0\big{\\}}.$

$(ii)$ All self-adjoint extensions $T_{\eta,R}$ of $T_{min}$ with coupled
boundary conditions are of the type

$\displaystyle\begin{split}&T_{\eta,R}f=\tau f,\\\
&f\in\operatorname{dom}(T_{\eta,R})=\bigg{\\{}g\in\operatorname{dom}(T_{max})\,\bigg{|}\begin{pmatrix}\widetilde{g}_{1}(b)\\\
\widetilde{g}_{2}(b)\end{pmatrix}=e^{i\eta}R\begin{pmatrix}\widetilde{g}_{1}(a)\\\
\widetilde{g}_{2}(a)\end{pmatrix}\bigg{\\}},\end{split}$ (2.12)

where $\eta\in[0,\pi)$, and $R$ is a real $2\times 2$ matrix with $\det(R)=1$
$($i.e., $R\in SL(2,{\mathbb{R}})$$)$.
$(iii)$ Every self-adjoint extension of $T_{min}$ is either of type $(i)$
$($i.e., separated $)$ or of type $(ii)$ $($i.e., coupled $)$.

######  Remark 2.7.

$(i)$ If $\tau$ is in the limit point case at one endpoint, say, at the
endpoint $b$, one omits the corresponding boundary condition involving
$\delta\in[0,\pi)$ at $b$ in (2.11) to obtain all self-adjoint extensions
$T_{\gamma}$ of $T_{min}$, indexed by $\gamma\in[0,\pi)$. In the case where
$\tau$ is in the limit point case at both endpoints, all boundary values and
boundary conditions become superfluous as in this case $T_{min}=T_{max}$ is
self-adjoint.
$(ii)$ Assume the special case where $\tau$ is regular on the finite interval
$[a,b]$, that is, suppose that Hypothesis 2.1 is replaced by the more
stringent set of assumptions:
Hypothesis ($\tau$ regular on $[a,b]$.)
Let $(a,b)\subset{\mathbb{R}}$ be a finite interval and suppose that $p,q,r$
are $($Lebesgue $)$ measurable functions on $(a,b)$ such that the following
items $(i)$–$(iii)$ hold:
$(i^{\prime})$ $r>0$ a.e. on $(a,b)$, $r\in L^{1}((a,b);dx)$.
$(ii^{\prime})$ $p>0$ a.e. on $(a,b)$, $1/p\in L^{1}((a,b);dx)$.
$(iii^{\prime})$ $q$ is real-valued a.e. on $(a,b)$, $q\in L^{1}((a,b);dx)$.
In this case one chooses $v_{j}\in\operatorname{dom}(T_{max})$, $j=1,2$, such
that

$\displaystyle v_{1}(x)=\begin{cases}\theta_{0}(\lambda,x,a),&\text{for $x$
near a},\\\ \theta_{0}(\lambda,x,b),&\text{for $x$ near b},\end{cases}\quad
v_{2}(x)=\begin{cases}\phi_{0}(\lambda,x,a),&\text{for $x$ near a},\\\
\phi_{0}(\lambda,x,b),&\text{for $x$ near b},\end{cases}$ (2.13)

where $\phi_{0}(\lambda,\,\cdot\,,d)$, $\theta_{0}(\lambda,\,\cdot\,,d)$,
$d\in\\{a,b\\}$, are real-valued solutions of $(\tau-\lambda)u=0$,
$\lambda\in{\mathbb{R}}$, satisfying the boundary conditions

$\displaystyle\begin{split}&\phi_{0}(\lambda,a,a)=\theta_{0}^{[1]}(\lambda,a,a)=0,\quad\theta_{0}(\lambda,a,a)=\phi_{0}^{[1]}(\lambda,a,a)=1,\\\
&\phi_{0}(\lambda,b,b)=\theta_{0}^{[1]}(\lambda,b,b)=0,\quad\;\theta_{0}(\lambda,b,b)=\phi_{0}^{[1]}(\lambda,b,b)=1.\end{split}$
(2.14)

Then one verifies that

$\displaystyle\widetilde{g}_{1}(a)=g(a),\quad\widetilde{g}_{1}(b)=g(b),\quad\widetilde{g}_{2}(a)=g^{[1]}(a),\quad\widetilde{g}_{2}(b)=g^{[1]}(b),$
(2.15)

and hence Theorem 2.6 in the special regular case recovers the well-known
situation of separated self-adjoint boundary conditions for three-coefficient
regular Sturm–Liouville operators in ${L^{2}((a,b);rdx)}$.
$(iii)$ In connection with (2.10), an explicit calculation demonstrates that
for $g,h\in\operatorname{dom}(T_{max})$,

$W(g,h)(d)=\widetilde{g}_{1}(d)\widetilde{h}_{2}(d)-\widetilde{g}_{2}(d)\widetilde{h}_{1}(d),\quad
d\in\\{a,b\\},$ (2.16)

where either side in (2.16) is finite for $d\in\\{a,b\\}$. Of course, for
(2.16) to hold at $d\in\\{a,b\\}$, it suffices that $g$ and $h$ lie locally in
$\operatorname{dom}(T_{max})$ near $x=d$.
$(iv)$ Clearly, $\widetilde{g}_{1},\widetilde{g}_{2}$ depend on the choice of
$v_{j}$, $j=1,2$, and a more precise notation would indicate this as
$\widetilde{g}_{1,v_{2}},\widetilde{g}_{2,v_{1}}$, etc. $\diamond$

In the special case where $T_{min}$ is bounded from below, one can further
analyze the generalized boundary values (2.10) in the singular context by
invoking principal and nonprincipal solutions of $\tau u=\lambda u$ for
appropriate $\lambda\in{\mathbb{R}}$. This leads to natural analogs of (2.15)
also in the singular case, and we will turn to this topic next.

We start by reviewing some oscillation theory with particular emphasis on
principal and nonprincipal solutions, a notion originally due to Leighton and
Morse [47], Rellich [55], [56], and Hartman and Wintner [35, Appendix] (see
also [11], [13, Sects. 13.6, 13.9, 13.0], [34, Ch. XI], [52], [63, Chs. 4,
6–8]).

######  Definition 2.8.

Assume Hypothesis 2.1.
$(i)$ Fix $c\in(a,b)$ and $\lambda\in{\mathbb{R}}$. Then $\tau-\lambda$ is
called nonoscillatory at $a$ $($resp., $b$$)$, if every real-valued solution
$u(\lambda,\,\cdot\,)$ of $\tau u=\lambda u$ has finitely many zeros in
$(a,c)$ $($resp., $(c,b)$$)$. Otherwise, $\tau-\lambda$ is called oscillatory
at $a$ $($resp., $b$$)$.
$(ii)$ Let $\lambda_{0}\in{\mathbb{R}}$. Then $T_{min}$ is called bounded from
below by $\lambda_{0}$, and one writes $T_{min}\geqslant\lambda_{0}I$, if

$(u,[T_{min}-\lambda_{0}I]u)_{L^{2}((a,b);rdx)}\geqslant 0,\quad
u\in\operatorname{dom}(T_{min}).$ (2.17)

The following is a key result.

######  Theorem 2.9.

Assume Hypothesis 2.1. Then the following items $(i)$–$(iii)$ are equivalent:
$(i)$ $T_{min}$ $($and hence any symmetric extension of $T_{min})$ is bounded
from below.
$(ii)$ There exists a $\nu_{0}\in{\mathbb{R}}$ such that for all
$\lambda<\nu_{0}$, $\tau-\lambda$ is nonoscillatory at $a$ and $b$.
$(iii)$ For fixed $c,d\in(a,b)$, $c\leqslant d$, there exists a
$\nu_{0}\in{\mathbb{R}}$ such that for all $\lambda<\nu_{0}$, $\tau u=\lambda
u$ has $($real-valued $)$ nonvanishing solutions $u_{a}(\lambda,\,\cdot\,)\neq
0$, $\widehat{u}_{a}(\lambda,\,\cdot\,)\neq 0$ in the neighborhood $(a,c]$ of
$a$, and $($real-valued $)$ nonvanishing solutions
$u_{b}(\lambda,\,\cdot\,)\neq 0$, $\widehat{u}_{b}(\lambda,\,\cdot\,)\neq 0$
in the neighborhood $[d,b)$ of $b$, such that

$\displaystyle
W(\widehat{u}_{a}(\lambda,\,\cdot\,),u_{a}(\lambda,\,\cdot\,))=1,\quad
u_{a}(\lambda,x)=o(\widehat{u}_{a}(\lambda,x))\text{ as $x\downarrow a$,}$
(2.18) $\displaystyle
W(\widehat{u}_{b}(\lambda,\,\cdot\,),u_{b}(\lambda,\,\cdot\,))\,=1,\quad
u_{b}(\lambda,x)\,=o(\widehat{u}_{b}(\lambda,x))\text{ as $x\uparrow b$,}$
(2.19)
$\displaystyle\int_{a}^{c}dx\,p(x)^{-1}u_{a}(\lambda,x)^{-2}=\int_{d}^{b}dx\,p(x)^{-1}u_{b}(\lambda,x)^{-2}=\infty,$
(2.20)
$\displaystyle\int_{a}^{c}dx\,p(x)^{-1}{\widehat{u}_{a}(\lambda,x)}^{-2}<\infty,\quad\int_{d}^{b}dx\,p(x)^{-1}{\widehat{u}_{b}(\lambda,x)}^{-2}<\infty.$
(2.21)

######  Definition 2.10.

Assume Hypothesis 2.1, suppose that $T_{min}$ is bounded from below, and let
$\lambda\in{\mathbb{R}}$. Then $u_{a}(\lambda,\,\cdot\,)$ $($resp.,
$u_{b}(\lambda,\,\cdot\,)$$)$ in Theorem 2.9 $(iii)$ is called a principal
$($or minimal $)$ solution of $\tau u=\lambda u$ at $a$ $($resp., $b$$)$. A
real-valued solution $\widetilde{\widetilde{u}}_{a}(\lambda,\,\cdot\,)$
$($resp., $\widetilde{\widetilde{u}}_{b}(\lambda,\,\cdot\,)$$)$ of $\tau
u=\lambda u$ linearly independent of $u_{a}(\lambda,\,\cdot\,)$ $($resp.,
$u_{b}(\lambda,\,\cdot\,)$$)$ is called nonprincipal at $a$ $($resp., $b$$)$.

Principal and nonprincipal solutions are well-defined due to Lemma 2.11 below.

######  Lemma 2.11.

Assume Hypothesis 2.1 and suppose that $T_{min}$ is bounded from below. Then
$u_{a}(\lambda,\,\cdot\,)$ and $u_{b}(\lambda,\,\cdot\,)$ in Theorem 2.9
$(iii)$ are unique up to $($nonvanish- ing $)$ real constant multiples.
Moreover, $u_{a}(\lambda,\,\cdot\,)$ and $u_{b}(\lambda,\,\cdot\,)$ are
minimal solutions of $\tau u=\lambda u$ in the sense that

$\displaystyle u(\lambda,x)^{-1}u_{a}(\lambda,x)$ $\displaystyle=o(1)\text{ as
$x\downarrow a$,}$ (2.22) $\displaystyle u(\lambda,x)^{-1}u_{b}(\lambda,x)$
$\displaystyle=o(1)\text{ as $x\uparrow b$,}$ (2.23)

for any other solution $u(\lambda,\,\cdot\,)$ of $\tau u=\lambda u$ $($which
is nonvanishing near $a$, resp., $b$$)$ with
$W(u_{a}(\lambda,\,\cdot\,),u(\lambda,\,\cdot\,))\neq 0$, respectively,
$W(u_{b}(\lambda,\,\cdot\,),u(\lambda,\,\cdot\,))\neq 0$.

Given these oscillation theoretic preparations, one can now revisit and
complement Theorem 2.6 as follows:

######  Theorem 2.12.

Assume Hypothesis 2.1 and that $\tau$ is in the limit circle case at $a$ and
$b$ $($i.e., $\tau$ is quasi-regular on $(a,b)$$)$. In addition, assume that
$T_{min}\geqslant\lambda_{0}I$ for some $\lambda_{0}\in{\mathbb{R}}$, and
denote by $u_{a}(\lambda_{0},\,\cdot\,)$ and
$\widehat{u}_{a}(\lambda_{0},\,\cdot\,)$ $($resp.,
$u_{b}(\lambda_{0},\,\cdot\,)$ and $\widehat{u}_{b}(\lambda_{0},\,\cdot\,)$$)$
principal and nonprincipal solutions of $\tau u=\lambda_{0}u$ at $a$ $($resp.,
$b$$)$, satisfying

$W(\widehat{u}_{a}(\lambda_{0},\,\cdot\,),u_{a}(\lambda_{0},\,\cdot\,))=W(\widehat{u}_{b}(\lambda_{0},\,\cdot\,),u_{b}(\lambda_{0},\,\cdot\,))=1.$
(2.24)

Introducing $v_{j}\in\operatorname{dom}(T_{max})$, $j=1,2$, via

$\displaystyle v_{1}(x)=\begin{cases}\widehat{u}_{a}(\lambda_{0},x),&\text{for
$x$ near a},\\\ \widehat{u}_{b}(\lambda_{0},x),&\text{for $x$ near
b},\end{cases}\quad v_{2}(x)=\begin{cases}u_{a}(\lambda_{0},x),&\text{for $x$
near a},\\\ u_{b}(\lambda_{0},x),&\text{for $x$ near b},\end{cases}$ (2.25)

one obtains for all $g\in\operatorname{dom}(T_{max})$,

$\displaystyle\begin{split}\widetilde{g}(a)&=-W(v_{2},g)(a)=\widetilde{g}_{1}(a)=-W(u_{a}(\lambda_{0},\,\cdot\,),g)(a)\\\
&=\lim_{x\downarrow a}\frac{g(x)}{\widehat{u}_{a}(\lambda_{0},x)},\\\
\widetilde{g}(b)&=-W(v_{2},g)(b)=\widetilde{g}_{1}(b)=-W(u_{b}(\lambda_{0},\,\cdot\,),g)(b)\\\
&=\lim_{x\uparrow b}\frac{g(x)}{\widehat{u}_{b}(\lambda_{0},x)},\end{split}$
(2.26)
$\displaystyle\begin{split}{\widetilde{g}}^{\,\prime}(a)&=W(v_{1},g)(a)=\widetilde{g}_{2}(a)=W(\widehat{u}_{a}(\lambda_{0},\,\cdot\,),g)(a)\\\
&=\lim_{x\downarrow
a}\frac{g(x)-\widetilde{g}(a)\widehat{u}_{a}(\lambda_{0},x)}{u_{a}(\lambda_{0},x)},\\\
{\widetilde{g}}^{\,\prime}(b)&=W(v_{1},g)(b)=\widetilde{g}_{2}(b)=W(\widehat{u}_{b}(\lambda_{0},\,\cdot\,),g)(b)\\\
&=\lim_{x\uparrow
b}\frac{g(x)-\widetilde{g}(b)\widehat{u}_{b}(\lambda_{0},x)}{u_{b}(\lambda_{0},x)}.\end{split}$
(2.27)

In particular, the limits on the right-hand sides in (2.26), (2.27) exist and
all self-adjoint extensions $T_{\gamma,\delta}$ of $T_{min}$ with separated
boundary conditions are of the form

$\displaystyle T_{\gamma,\delta}f=\tau f,\quad\gamma,\delta\in[0,\pi),$
$\displaystyle
f\in\operatorname{dom}(T_{\gamma,\delta})=\big{\\{}g\in\operatorname{dom}(T_{max})\,\big{|}\,\sin(\gamma){\widetilde{g}}^{\,\prime}(a)+\cos(\gamma)\widetilde{g}(a)=0;$
(2.28) $\displaystyle\hskip
156.49014pt\,\sin(\delta){\widetilde{g}}^{\,\prime}(b)+\cos(\delta)\widetilde{g}(b)=0\big{\\}},$

whereas self-adjoint extensions $T_{\eta,R}$ of $T_{min}$ with coupled
boundary conditions are of the form

$\displaystyle\begin{split}&T_{\eta,R}f=\tau f,\quad\eta\in[0,\pi),\;R\in
SL(2,{\mathbb{R}}),\\\
&f\in\operatorname{dom}(T_{\eta,R})=\bigg{\\{}g\in\operatorname{dom}(T_{max})\,\bigg{|}\begin{pmatrix}\widetilde{g}(b)\\\
\widetilde{g}^{\,\prime}(b)\end{pmatrix}=e^{i\eta}R\begin{pmatrix}\widetilde{g}(a)\\\
\widetilde{g}^{\,\prime}(a)\end{pmatrix}\bigg{\\}}.\end{split}$ (2.29)

Moreover, $\sigma(T_{\gamma,\delta})$ is simple.

The Friedrichs extension $T_{F}$ of $T_{min}$ now permits a particularly
simple characterization in terms of the generalized boundary values
$\widetilde{g}(a),\widetilde{g}(b)$ as derived by Kalf [38] and subsequently
by Niessen and Zettl [52] (see also [56], [57] and the extensive literature
cited in [25], [29, Ch. 13]):

######  Theorem 2.13.

Assume Hypothesis 2.1 and that $\tau$ is in the limit circle case at $a$ and
$b$ $($i.e., $\tau$ is quasi-regular on $(a,b)$$)$. In addition, assume that
$T_{min}\geqslant\lambda_{0}I$ for some $\lambda_{0}\in{\mathbb{R}}$. Then the
Friedrichs extension $T_{F}$ of $T_{min}$ is characterized by

$\displaystyle T_{F}f=\tau f,\quad
f\in\operatorname{dom}(T_{F})=\big{\\{}g\in\operatorname{dom}(T_{max})\,\big{|}\,\widetilde{g}(a)=\widetilde{g}(b)=0\big{\\}}.$
(2.30)

In particular, $T_{F}=T_{0,0}$.

######  Remark 2.14.

$(i)$ As in (2.16), one readily verifies for
$g,h\in\operatorname{dom}(T_{max})$,

$W(g,h)(d)=\widetilde{g}(d){\widetilde{h}}^{\,\prime}(d)-{\widetilde{g}}^{\,\prime}(d)\widetilde{h}(d),\quad
d\in\\{a,b\\},$ (2.31)

where either side in (2.31) has a finite limit as $d\in\\{a,b\\}$. Moreover,
if $\tau$ is regular at an endpoint, then the generalized boundary values in
(2.26), (2.27) reduce to the canonical ones in (2.15) as long as principal and
nonprincipal solutions chosen are appropriately normalized such that (2.13)
and (2.25) coincide. In addition (and now independently of the chosen
normalization of principal and nonprincipal solutions), in the regular context
(2.31) becomes for $g,h\in\operatorname{dom}(T_{max})$,

$W(g,h)(d)=\widetilde{g}(d){\widetilde{h}}^{\,\prime}(d)-{\widetilde{g}}^{\,\prime}(d)\widetilde{h}(d)=g(d)h^{[1]}(d)-g^{[1]}(d)h(d),\quad
d\in\\{a,b\\}.$ (2.32)

$(ii)$ While the generalized boundary values at the endpoint $d\in\\{a,b\\}$
clearly depend on the choice of nonprincipal solution
$\widehat{u}_{d}(\lambda_{0},\,\cdot\,)$ of $\tau u=\lambda_{0}u$ at $d$, the
Friedrichs boundary conditions $\widetilde{g}(a)=\widetilde{g}(b)=0$ are
independent of the choice of this nonprincipal solution.
$(iii)$ As always in this context, if $\tau$ is in the limit point case at one
(or both) interval endpoints, the corresponding boundary conditions at that
endpoint are dropped and only a separated boundary condition at the other end
point (if the latter is a limit circle endpoint for $\tau$), has to be imposed
in Theorems 2.12 and 2.13. In the case where $\tau$ is in the limit point case
at both endpoints, all boundary values and boundary conditions become
superfluous as $T_{min}=T_{max}$ is self-adjoint. $\diamond$

All results surveyed in this section can be found in [25] and [29, Ch. 13]
which contain very detailed lists of references to the basics of
Weyl–Titchmarsh theory. Here we just mention a few additional and classical
sources such as [2, Sect. 129], [7, Ch. 6], [10], [12, Chs. 8, 9], [13, Sects.
13.6, 13.9, 13.0], [19], [37, Ch. III], [51, Ch. V], [52], [54, Ch. 6], [60,
Ch. 9], [61, Sect. 8.3], [62, Ch. 13], [63, Chs. 4, 6–8].

##  3\. The (Singular) Weyl–Titchmarsh–Kodaira $m$-Function

In this section we recall the construction of the (singular)
Weyl–Titchmarsh–Kodaira $m$ and $M$-functions for separated and coupled
boundary conditions, respectively, primarily following the detailed treatment
in [29, Ch. 13] (see also [25]) and demonstrate that the generalized boundary
values in (2.26), (2.27) naturally fit into this framework. For simplicity we
single out the left endpoint $a$ in the following as the endpoint $b$ can be
treated in precisely the same manner.

######  Hypothesis 3.1.

In addition to Hypothesis 2.1, let $T_{\gamma,\delta}$,
$\gamma,\delta\in[0,\pi)$, be any self-adjoint extension of $T_{min}$ in
$L^{2}((a,b);rdx)$ with separated boundary conditions as in (2.11) $($if
$\tau$ is in the limit circle case at $a$ and/or $b$$)$, and suppose that for
some $($and hence for all $)$ $x_{0}\in(a,b)$, the self-adjoint operator
$T_{\gamma,0,a,x_{0}}$ in $L^{2}((a,x_{0});rdx)$, associated with
$\tau|_{(a,x_{0})}$ and a Dirichlet boundary condition at $x_{0}$ $($i.e.,
$g(x_{0})=0$, $g\in\operatorname{dom}(T_{max,a,x_{0}})$, with
$T_{max,a,x_{0}}$ the maximal operator associated with $\tau|_{(a,x_{0})}$ in
$L^{2}((a,x_{0});rdx)$$)$, has purely discrete spectrum.

It is known (see, e.g., [14], [29, Sect. 13.2], [42]) that Hypothesis 3.1 is
equivalent to the existence of an entire solution
$\varphi_{\gamma}(z,\,\cdot\,)$ of $\tau u=zu$, $z\in{\mathbb{C}}$, that is
real-valued for $z\in{\mathbb{R}}$, and lies in
$\operatorname{dom}(T_{\gamma,\delta})$ near the point $a$. In particular,
$\varphi_{\gamma}(z,\,\cdot\,)$ satisfies the boundary condition indexed by
$\gamma$ at the left endpoint $a$ if $\tau$ is in the limit circle case at
$a$, and $\varphi_{\gamma}(z,\,\cdot\,)\in L^{2}((a,x_{0});rdx)$ if $\tau$ is
in the limit point case at $a$. In addition, it is known that a second,
linearly independent entire solution $\vartheta_{\gamma}(z,\,\cdot\,)$ of
$\tau u=zu$ exists, with $\vartheta_{\gamma}(z,\,\cdot\,)$ real-valued for
$z\in{\mathbb{R}}$, satisfying

$W(\vartheta_{\gamma}(z,\,\cdot\,),\varphi_{\gamma}(z,\,\cdot\,))(x)=1,\quad\gamma\in[0,\pi),\;z\in{\mathbb{C}},\;x\in(a,b).$
(3.1)

We note that $\varphi_{\gamma}(z,\,\cdot\,)$ is unique up to a nonvanishing
entire factor (real on the real line) with respect to $z\in{\mathbb{C}}$.
Hence, if $\tau$ is in the limit circle case at $a$ we will always normalize
$\varphi_{\gamma}(z,\,\cdot\,)$ and $\vartheta_{\gamma}(z,\,\cdot\,)$ such
that

$\displaystyle\begin{split}\widetilde{\varphi}_{\gamma}(z,a)&=-\sin(\gamma),\quad\widetilde{\varphi}_{\gamma}^{\,\prime}(z,a)=\cos(\gamma),\\\
\widetilde{\vartheta}_{\gamma}(z,a)&=\cos(\gamma),\quad\widetilde{\vartheta}_{\gamma}^{\,\prime}(z,a)=\sin(\gamma);\quad\gamma\in[0,\pi),\;z\in{\mathbb{C}}.\end{split}$
(3.2)

In addition to the entire fundamental system
$\varphi_{\gamma}(z,\,\cdot\,),\vartheta_{\gamma}(z,\,\cdot\,)$ of $\tau
u=zu$, we also mention the standard entire fundamental system
$\theta_{0}(z,\,\cdot\,,x_{0}),\phi_{0}(z,\,\cdot\,,x_{0})$ of $\tau u=zu$
normalized at $x_{0}\in(a,b)$ in the usual manner,

$\displaystyle\theta_{0}(z,x_{0},x_{0})=1,\quad\theta_{0}^{[1]}(z,x_{0},x_{0})=0,\quad\phi_{0}(z,x_{0},x_{0})=0,\quad\phi_{0}^{[1]}(z,x_{0},x_{0})=1,$
$\displaystyle\hskip 284.52756ptz\in{\mathbb{C}},$ (3.3)

and the Weyl–Titchmarsh solutions $\psi_{\gamma,0,-}(z,\,\cdot\,,x_{0})$ and
$\psi_{0,\delta,+}(z,\,\cdot\,,x_{0})$ of $\tau u=zu$ given by

$\displaystyle\begin{split}\psi_{\gamma,0,-}(z,x,x_{0})=\theta_{0}(z,x,x_{0})+m_{\gamma,0,-}(z,x_{0})\phi_{0}(z,x,x_{0}),&\\\
\gamma\in[0,\pi),\;z\in\rho(T_{\gamma,0,a,x_{0}}),\;x\in(a,b),&\end{split}$
(3.4)
$\displaystyle\begin{split}\psi_{0,\delta,+}(z,x,x_{0})=\theta_{0}(z,x,x_{0})+m_{0,\delta,+}(z,x_{0})\phi_{0}(z,x,x_{0}),&\\\
\gamma\in[0,\pi),\;z\in\rho(T_{0,\delta,x_{0},b}),\;x\in(a,b),&\end{split}$
(3.5)

where $T_{0,\delta,x_{0},b}$ in $L^{2}((x_{0},b);rdx)$ is the self-adjoint
operator associated with $\tau|_{(x_{0},b)}$ and a Dirichlet boundary
condition at $x_{0}$ (i.e., $g(x_{0})=0$,
$g\in\operatorname{dom}(T_{max,x_{0},b})$, with $T_{max,x_{0},b}$ the maximal
operator associated with $\tau|_{(x_{0},b)}$ in $L^{2}((x_{0},b);rdx)$). In
particular, $\psi_{\gamma,0,-}(z,\,\cdot\,,x_{0})$ satisfies the boundary
condition indexed by $\gamma$ at the left endpoint $a$; in addition,
$\psi_{\gamma,0,-}(z,\,\cdot\,,x_{0})\in L^{2}((a,x_{0});rdx)$ if $\tau$ is in
the limit circle case at $a$. Similarly,
$\psi_{\gamma,0,-}(z,\,\cdot\,,x_{0})$ satisfies
$\psi_{\gamma,0,-}(z,\,\cdot\,,x_{0})\in L^{2}((a,x_{0});rdx)$ if $\tau$ is in
the limit point case at $a$. Analogously,
$\psi_{0,\delta,+}(z,\,\cdot\,,x_{0})$ satisfies the boundary condition
indexed by $\delta$ at the right endpoint $b$ and the condition
$\psi_{0,\delta,+}(z,\,\cdot\,,x_{0})\in L^{2}((x_{0},b);rdx)$ if $\tau$ is in
the limit circle case at $b$. Moreover,
$\psi_{0,\delta,+}(z,\,\cdot\,,x_{0})\in L^{2}((x_{0},b);rdx)$ if $\tau$ is in
the limit point case at $b$. Thus, $m_{\gamma,0,-}(\,\cdot\,,x_{0})$ is
analytic on ${\mathbb{C}}\backslash{\mathbb{R}}$, meromorphic on
${\mathbb{C}}$, with simple poles on the real axis precisely at the simple
eigenvalues of $T_{\gamma,0,a,x_{0}}$, and $\varphi_{\gamma}(z,\,\cdot\,)$ is
a $z$-dependent multiple of $\psi_{\gamma,0,-}(z,\,\cdot\,,x_{0})$,

$\varphi_{\gamma}(z,\,\cdot\,)=C_{\gamma,-}(z,x_{0})\psi_{\gamma,0,-}(z,\,\cdot\,,x_{0})$
(3.6)

(with $C_{\gamma,-}(z,x_{0})$ entire with respect to $z$), real-valued for
$z\in{\mathbb{R}}$, and having simple zeros at the simple poles of
$m_{\gamma,0,-}(\,\cdot\,,x_{0})$. In addition, one readily confirms that

$m_{\gamma,0,-}(z,x_{0})=\psi_{\gamma,0,-}^{[1]}(z,x_{0},x_{0})=\varphi_{\gamma}^{[1]}(z,x_{0})/\varphi_{\gamma}(z,x_{0}),\quad
z\in\rho(T_{\gamma,0,a,x_{0}}).$ (3.7)

Next, we rewrite $\psi_{0,\delta,+}(z,\,\cdot\,,x_{0})$ in terms of the entire
fundamental system
$\varphi_{\gamma}(z,\,\cdot\,),\vartheta_{\gamma}(z,\,\cdot\,)$ and thereby
introduce the singular Weyl–Titchmarsh–Kodaira solution,
$\psi_{\gamma,\delta}(z,\,\cdot\,)$, and the Weyl–Titchmarsh–Kodaira
$m$-function, $m_{\gamma,\delta}(\,\cdot\,)$, via

$\displaystyle\begin{split}\psi_{\gamma,\delta}(z,x)&=C_{\gamma,\delta}(z,x_{0})\psi_{0,\delta,+}(z,x,x_{0})\\\
&=\vartheta_{\gamma}(z,x)+m_{\gamma,\delta}(z)\varphi_{\gamma}(z,x),\quad
z\in\rho(T_{\gamma,\delta}),\end{split}$ (3.8)

where $C_{\gamma,\delta}(\,\cdot\,,x_{0})$ is analytic and nonvanishing on
${\mathbb{C}}\backslash{\mathbb{R}}$. In analogy to (3.7) one obtains

$m_{0,\delta,+}(z,x_{0})=\psi_{0,\delta,+}^{[1]}(z,x_{0},x_{0})=\psi_{\gamma,\delta}^{[1]}(z,x_{0})/\psi_{\gamma,\delta}(z,x_{0}),\quad
z\in\rho(T_{0,\delta,x_{0},b}).$ (3.9)

In particular,

$m_{0,\delta,+}(z,x_{0})=\frac{\vartheta_{\gamma}^{[1]}(z,x_{0})+m_{\gamma,\delta}(z)\varphi_{\gamma}^{[1]}(z,x_{0})}{\vartheta_{\gamma}(z,x_{0})+m_{\gamma,\delta}(z)\varphi_{\gamma}(z,x_{0})},\quad
z\in{\mathbb{C}}\rho(T_{0,\delta,x_{0},b}),$ (3.10)

and thus,

$\displaystyle m_{\gamma,\delta}(z)$
$\displaystyle=\frac{-\vartheta_{\gamma}^{[1]}(z,x_{0})+m_{0,\delta,+}(z,x_{0})\vartheta_{\gamma}(z,x_{0})}{\varphi_{\gamma}^{[1]}(z,x_{0})-m_{0,\delta,+}(z,x_{0})\varphi_{\gamma}(z,x_{0})}$
$\displaystyle=\frac{1}{\varphi_{\gamma}(z,x_{0})}\frac{-\vartheta_{\gamma}^{[1]}(z,x_{0})+m_{0,\delta,+}(z,x_{0})\vartheta_{\gamma}(z,x_{0})}{m_{\gamma,0,-}(z,x_{0})-m_{0,\delta,+}(z,x_{0})}$
$\displaystyle=\frac{1}{[\varphi_{\gamma}(z,x_{0})]^{2}}\frac{1}{m_{\gamma,0,-}(z,x_{0})-m_{0,\delta,+}(z,x_{0})}-\frac{\vartheta_{\gamma}(z,x_{0})}{\varphi_{\gamma}(z,x_{0})},$
(3.11) $\displaystyle\hskip
140.84105pt\gamma,\delta\in[0,\pi),\;z\in\rho(T_{\gamma,\delta}),$

employing (3.1).

######  Remark 3.2.

In the case where $\tau_{\alpha,\beta}$ is in the limit circle case at $a$,
(3.2) uniquely determines $\varphi_{\gamma}(z,\,\cdot\,)$ and
$\vartheta_{\gamma}(z,\,\cdot\,)$. If $\tau_{\alpha,\beta}$ is in the limit
point case at $a$, equation (3.1) alone cannot determine
$\varphi_{\gamma}(z,\,\cdot\,)$ and $\vartheta_{\gamma}(z,\,\cdot\,)$
uniquely. In particular, $\varphi_{\gamma}$ is only determined up to a
nonvanishing, entire, multiplicative factor and
$\vartheta_{\gamma}(z,\,\cdot\,)$ can contain an additive term of the type
$c_{\gamma}(\,\cdot\,)\varphi_{\gamma}(z,\,\cdot\,)$, with
$c_{\gamma}(\,\cdot\,)$ entire. This will influence the actual form of
$m_{\gamma,\delta}(\,\cdot\,)$ in (3.8). However, since
$c_{\gamma}(\,\cdot\,)$ and $\varphi_{\gamma}(\,\cdot\,,x)$, $x\in(a,b)$, are
entire, the multiplicative ambiguity in $\varphi_{\gamma}(z,\,\cdot\,)$, as
well as the additive term
$c_{\gamma}(\,\cdot\,)\varphi_{\gamma}(z,\,\cdot\,)$, cannot affect the
measure $d\rho_{\gamma,\delta}$ obtained via the (generalized) Stieltjes
inversion formula from $m_{\gamma,\delta}$. Thus, from a spectral theoretic
point of view, the ambiguities in the choices of
$\varphi_{\gamma}(z,\,\cdot\,)$ and $\vartheta_{\gamma}(z,\,\cdot\,)$ can
safely be ignored and hence in a concrete example, the actual choice of
$\varphi_{\gamma}(z,\,\cdot\,)$ and $\vartheta_{\gamma}(z,\,\cdot\,)$ is
largely a matter of convenience. $\diamond$

Since $-m_{\gamma,0,-}(\,\cdot\,,x_{0})$ and $m_{0,\delta,+}(\,\cdot\,,x_{0})$
are Nevanlinna–Herglotz functions and $\vartheta_{\gamma}(z,\,\cdot\,)$ and
$\varphi_{\gamma}(z,\,\cdot\,)$ are entire with respect to $z$ and real-valued
for $z\in{\mathbb{R}}$, it follows that $m_{\gamma,\delta}(\,\cdot\,)$ is
analytic on ${\mathbb{C}}\backslash{\mathbb{R}}$ and that

$m_{\gamma,\delta}(z)=\overline{m_{\gamma,\delta}(\overline{z})},\quad
z\in{\mathbb{C}}_{+}.$ (3.12)

Since the Green’s function $G_{\gamma,\delta}(z,\,\cdot\,,\,\cdot\,)$ of
$T_{\gamma,\delta}$ (i.e., the integral kernel of its resolvent) is of the
form

$G_{\gamma,\delta}(z,x,x^{\prime})=\begin{cases}\varphi_{\gamma}(z,x)\psi_{\gamma,\delta}(z,x^{\prime}),&a<x\leqslant
x^{\prime}<b,\\\
\varphi_{\gamma}(z,x^{\prime})\psi_{\gamma,\delta}(z,x),&a<x^{\prime}\leqslant
x<b,\end{cases}\quad z\in\rho(T_{\gamma,\delta}),$ (3.13)

one obtains

$\displaystyle\begin{split}G_{\gamma,\delta}(z,x_{0},x_{0})&=\varphi_{\gamma}(z,x_{0})[\vartheta_{\gamma}(z,x_{0})+m_{\gamma,\delta}(z)\varphi_{\gamma}(z,x_{0})]\\\
&=[m_{\gamma,0,-}(z,x_{0})-m_{0,\delta,+}(z,x_{0})]^{-1},\quad
z\in\rho(T_{\gamma,\delta}),\end{split}$ (3.14)

confirming the generally known fact that all diagonal Green’s functions such
as $G_{\gamma,\delta}(z,x_{0},x_{0})$ possess the Nevanlinna–Herglotz
property. Equivalently, one obtains

$m_{\gamma,\delta}(z)=[\varphi_{\gamma}(z,x_{0})]^{-2}G_{\gamma,\delta}(z,x_{0},x_{0})-[\vartheta_{\gamma}(z,x_{0})/\varphi_{\gamma}(z,x_{0})],\quad
z\in\rho(T_{\gamma,\delta}),$ (3.15)

illustrating that generally, $m_{\gamma,\delta}$ is a generalized
Nevanlinna–Herglotz function. One notes that
$\varphi_{\gamma}(\,\cdot\,,x_{0})$ has a discrete set of real zeros
(accumulating at $+\infty$) precisely at the eigenvalues of
$T_{\gamma,0,a,x_{0}}$, but since by definition $m_{\gamma,\delta}(\,\cdot\,)$
is independent of $x_{0}$, a small change in $x_{0}$ removes a possible
ambiguity when applying an analog of the Stieltjes inversion formula to
$m_{\gamma,\delta}$ to derive the spectral function $\rho_{\gamma,\delta}$
associated with $T_{\gamma,\delta}$. Even though $m_{\gamma,\delta}$, and
hence, $\rho_{\gamma,\delta}$, is nonunique, the measure equivalence class
generated by the spectral function $\rho_{\gamma,\delta}$ is unique and hence
the spectrum (and it’s subdivisions) are related to the singularity structure
of $\text{\rm Im}(m_{\gamma,\delta}(\,\cdot\,))$ on the real line.

One also notices that if $\tau$ is in the limit circle case at $a$, the
normalization chosen in (3.2), combined with (3.8) readily implies

$m_{\gamma,\delta}(z)=\cos(\gamma)\widetilde{\psi}_{\gamma,\delta}^{\,\prime}(z,a)-\sin(\gamma)\widetilde{\psi}_{\gamma,\delta}(z,a),\quad
z\in\rho(T_{\gamma,\delta}),$ (3.16)

a result familiar from the special case where $a$ is a regular endpoint
(employing the fact (2.15)), illustrating once more that the generalized
boundary values (2.26), (2.27), in the context of a singular endpoint $a$, are
natural extensions of the familiar boundary values in the case of regular
endpoints.

Moreover, if $\tau$ is in the limit circle case at $b$, then
$\psi_{\gamma,\delta}(z,\,\cdot\,)$, like
$\psi_{0,\delta,+}(z,\,\cdot\,,x_{0})$, satisfies the boundary condition
indexed by $\delta\in[0,\pi)$ at $b$,

$\sin(\delta)\widetilde{\psi}_{\gamma,\delta}^{\prime}(z,b)+\cos(\delta)\widetilde{\psi}_{\gamma,\delta}(z,b)=0,\quad
z\in{\mathbb{C}}\backslash{\mathbb{R}},$ (3.17)

and hence

$\displaystyle
m_{\gamma,\delta}(z)=-\frac{\cos(\delta)\widetilde{\vartheta}_{\gamma}(z,b)+\sin(\delta)\widetilde{\vartheta}_{\gamma}^{\,\prime}(z,b)}{\cos(\delta)\widetilde{\varphi}_{\gamma}(z,b)+\sin(\delta)\widetilde{\varphi}_{\gamma}^{\,\prime}(z,b)},\quad
z\in{\mathbb{C}}\backslash{\mathbb{R}}.$ (3.18)

In addition, given that $m_{\gamma,0,-}(z,x_{0})$ is meromorphic on
${\mathbb{C}}$ by hypothesis, one infers that

$m_{\gamma,\delta}(\,\cdot\,)$ is meromorphic on ${\mathbb{C}}$ if and only if
$m_{0,\delta,+}(\,\cdot\,,x_{0})$ is. (3.19)

In this case (cf. (3.11)) all poles of $m_{\gamma,\delta}(\,\cdot\,)$ are
simple and hence

$\lambda_{0}\in\sigma_{p}(T_{\gamma,\delta})$ if and only if
$\varphi_{\gamma}(\lambda_{0},\,\cdot\,)\in L^{2}((a,b);rdx)$ (3.20)

and then

$-{\rm
Res}_{z=\lambda_{0}}(m_{\gamma,\delta}(\,\cdot\,))=\|\varphi_{\gamma}(\lambda_{0},\,\cdot\,)\|_{L^{2}((a,b);rdx)}^{-2},$
(3.21)

in accordance with the first order pole behavior of
$G_{\gamma,\delta}(z,x_{0},x_{0})$ at $z=\lambda_{0}$,

$\displaystyle
G_{\gamma,\delta}(z,x,x^{\prime})\underset{z\to\lambda_{0}}{=}-\frac{{\rm
Res}_{z=\lambda_{0}}(m_{\gamma,\delta}(\,\cdot\,))}{z-\lambda_{0}}\varphi_{\gamma}(\lambda_{0},x)\varphi_{\gamma}(\lambda_{0},x^{\prime})+O(1).$
(3.22)

Fixing the boundary condition at $b$ indexed by $\delta\in[0,\pi)$ (if any),
and varying the boundary condition at the left endpoint $a$ then yields the
standard linear fractional transformation

$\displaystyle\begin{split}m_{\gamma_{2},\delta}(z)=\frac{-\sin(\gamma_{2}-\gamma_{1})+\cos(\gamma_{2}-\gamma_{1})m_{\gamma_{1},\delta}(z)}{\cos(\gamma_{2}-\gamma_{1})+\sin(\gamma_{2}-\gamma_{1})m_{\gamma_{1},\delta}(z)},&\\\
\gamma_{1},\gamma_{2},\delta\in[0,\pi),\;z\in{\mathbb{C}}\backslash{\mathbb{R}}.&\end{split}$
(3.23)

We also note a well-known observation (see, [14], [43]) when $\tau$ is in the
limit circle case at $a$: In this situation Hypothesis 3.1 is always satisfied
and one infers that

$\frac{\text{\rm Im}(m_{\gamma,\delta}(z))}{\text{\rm
Im}(z)}=\int_{a}^{b}r(x)dx\,|\psi_{\gamma,\delta}(z,x)|^{2}>0,\quad
z\in\rho(T_{\gamma,\delta}),$ (3.24)

in particular, in this special case $m_{\gamma,\delta}(\,\cdot\,)$ is actually
a Nevanlinna–Herglotz function.

In case $\tau$ is in the limit circle case at $a$ and in the limit point case
at $b$, one simply drops the $\delta$-dependence of all quantities. Similarly,
the $\gamma$-dependence of all quantities is dropped if $\tau$ is in the limit
point case at $a$.

Next, we briefly recall the case of coupled boundary conditions following [29,
Ch. 13]. Recalling (3.1), (3.2), one obtains for the Green’s function of
$T_{\eta,R}$,

$\displaystyle
G_{\eta,R}(z,x,x^{\prime})=\frac{-e^{i\eta}}{\widetilde{F}_{\eta,R}(z)}$
$\displaystyle\qquad\times\Big{\\{}\big{[}-R_{1,2}{\widetilde{\varphi}_{0}}^{\,\prime}(z,b)+R_{2,2}\widetilde{\varphi}_{0}(z,b)\big{]}\vartheta_{0}(z,x)\vartheta_{0}(z,x^{\prime})$
$\displaystyle\hskip
36.98866pt+\big{[}-R_{1,1}{\widetilde{\vartheta}_{0}}^{\,\prime}(z,b)+R_{2,1}\widetilde{\vartheta}_{0}(z,b)\big{]}\varphi_{0}(z,x)\varphi_{0}(z,x^{\prime})$
$\displaystyle\hskip
36.98866pt+\big{[}e^{-i\eta}+R_{1,2}{\widetilde{\vartheta}_{0}}^{\,\prime}(z,b)-R_{2,2}\widetilde{\vartheta}_{0}(z,b)\big{]}\vartheta_{0}(z,x)\varphi_{0}(z,x^{\prime})$
$\displaystyle\hskip
36.98866pt+\big{[}-e^{-i\eta}+R_{1,1}{\widetilde{\varphi}_{0}}^{\,\prime}(z,b)-R_{2,1}\widetilde{\varphi}_{0}(z,b)\big{]}\varphi_{0}(z,x)\vartheta_{0}(z,x^{\prime})\Big{\\}}$
$\displaystyle\quad+\begin{cases}0,&a<x\leqslant x^{\prime}<b,\\\
[\vartheta_{0}(z,x)\varphi_{0}(z,x^{\prime})-\varphi_{0}(z,x)\vartheta_{0}(z,x^{\prime})],&a<x^{\prime}\leqslant
x<b,\end{cases}$ (3.25) $\displaystyle\hskip 213.39566ptz\in\rho(T_{\eta,R}),$

where

$\displaystyle\widetilde{F}_{\eta,R}(z)=-e^{i\eta}\big{[}R_{1,1}{\widetilde{\varphi}_{0}}^{\,\prime}(z,b)+R_{2,2}\widetilde{\vartheta}_{0}(z,b)-R_{2,1}\widetilde{\varphi}_{0}(z,b)-R_{1,2}{\widetilde{\vartheta}_{0}}^{\,\prime}(z,b)$
$\displaystyle\hskip 69.70915pt-2\cos(\eta)\big{]}.$ (3.26)

The corresponding Green’s function induced $M$-function is then given by

$\displaystyle
M_{\eta,R}(z,a)=\begin{pmatrix}\widetilde{G}_{\eta,R}(z,a,a)&2^{-1}\big{(}\big{[}\widetilde{\partial}_{1}+\widetilde{\partial}_{2}\big{]}G_{\eta,R}\big{)}^{\widetilde{}}(z,a,a)\\\\[2.84526pt]
2^{-1}\big{(}\big{[}\widetilde{\partial}_{1}+\widetilde{\partial}_{2}\big{]}G_{\eta,R}\big{)}^{\widetilde{}}(z,a,a)&\big{(}\widetilde{\partial}_{1}\widetilde{\partial}_{2}G_{\eta,R}\big{)}(z,a,a)\end{pmatrix}$
(3.27)
$\displaystyle\quad=\frac{-e^{i\eta}}{\widetilde{F}_{\eta,R}(z)}\left(\begin{smallmatrix}R_{2,2}\widetilde{\varphi}_{0}(z,b)-R_{1,2}{\widetilde{\varphi}_{0}}^{\,\prime}(z,b)&2^{-1}\big{[}R_{1,1}{\widetilde{\varphi}_{0}}^{\,\prime}(z,b)-R_{2,1}\widetilde{\varphi}_{0}(z,b)\\\\[2.84526pt]
{}&\qquad\,-R_{2,2}\widetilde{\vartheta}_{0}(z,b)+R_{1,2}{\widetilde{\vartheta}_{0}}^{\,\prime}(z,b)\big{]}\\\\[5.69054pt]
2^{-1}\big{[}R_{1,1}{\widetilde{\varphi}_{0}}^{\,\prime}(z,b)-R_{2,1}\widetilde{\varphi}_{0}(z,b)&R_{2,1}\widetilde{\vartheta}_{0}(z,b)-R_{1,1}{\widetilde{\vartheta}_{0}}^{\,\prime}(z,b)\\\\[2.84526pt]
\\\\[2.84526pt]
\qquad\,-R_{2,2}\widetilde{\vartheta}_{0}(z,b)+R_{1,2}{\widetilde{\vartheta}_{0}}^{\,\prime}(z,b)\big{]}&{}\end{smallmatrix}\right),$
$\displaystyle\hskip 239.00298ptz\in{\mathbb{C}}\backslash\sigma(T_{\eta,R}),$
(3.28)

with

$\big{(}\widetilde{\partial}_{1}\widetilde{\partial}_{2}G_{\eta,R}\big{)}(z,a,a)=\big{(}\widetilde{\partial}_{2}\widetilde{\partial}_{1}G_{\eta,R}\big{)}(z,a,a),\quad
z\in{\mathbb{C}}\backslash\sigma(T_{\eta,R}).$ (3.29)

Here we employed the notation,

$\displaystyle\widetilde{G}(z,a,a)=\lim_{x_{1}\downarrow
a}\lim_{x_{2}\downarrow
a}\frac{G(z,x_{1},x_{2})}{\widehat{u}_{a}(\lambda_{0},x_{1})\widehat{u}_{a}(\lambda_{0},x_{2})},$
$\displaystyle\big{(}\big{[}\widetilde{\partial}_{1}+\widetilde{\partial}_{2}\big{]}G\big{)}^{\widetilde{}}(z,a,a)=\lim_{x_{2}\downarrow
a}\lim_{x_{1}\downarrow
a}\bigg{[}\frac{G(z,x_{1},x_{2})-\widetilde{G}(z,a,x_{2})\widehat{u}_{a}(\lambda_{0},x_{1})}{u_{a}(\lambda_{0},x_{1})\widehat{u}_{a}(\lambda_{0},x_{2})}\bigg{]}$
$\displaystyle\hskip 101.00737pt+\lim_{x_{1}\downarrow a}\lim_{x_{2}\downarrow
a}\bigg{[}\frac{G(z,x_{1},x_{2})-\widetilde{G}(z,x_{1},a)\widehat{u}_{a}(\lambda_{0},x_{2})}{\widehat{u}_{a}(\lambda_{0},x_{1})u_{a}(\lambda_{0},x_{2})}\bigg{]}$
$\displaystyle\widetilde{G}(z,a,x_{2})=\lim_{x_{1}\downarrow
a}\frac{G(z,x_{1},x_{2})}{\widehat{u}_{a}(\lambda_{0},x_{1})},\quad\widetilde{G}(z,x_{1},a)=\lim_{x_{2}\downarrow
a}\frac{G(z,x_{1},x_{2})}{\widehat{u}_{a}(\lambda_{0},x_{2})},$ (3.30)
$\displaystyle\big{(}\widetilde{\partial}_{1}\widetilde{\partial}_{2}G\big{)}(z,a,a)$
$\displaystyle\quad=\lim_{x_{1}\downarrow
a}\Bigg{\\{}\frac{\lim_{x_{2}\downarrow
a}\Big{(}\frac{G(z,x_{1},x_{2})-\widetilde{G}(z,x_{1},a)\widehat{u}_{a}(\lambda_{0},x_{2})}{u_{a}(\lambda_{0},x_{2})}\Big{)}}{u_{a}(\lambda_{0},x_{1})}$
$\displaystyle\hskip 48.36958pt-\frac{\lim_{x^{\prime}_{1}\downarrow
a}\lim_{x_{2}\downarrow
a}\Big{(}\frac{G(z,x^{\prime}_{1},x_{2})-\widetilde{G}(z,x^{\prime}_{1},a)\widehat{u}_{a}(\lambda_{0},x_{2})}{\widehat{u}_{a}(\lambda_{0},x^{\prime}_{1})u_{a}(\lambda_{0},x_{2})}\Big{)}\widehat{u}_{a}(\lambda_{0},x_{1})}{u_{a}(\lambda_{0},x_{1})}\Bigg{\\}}$
$\displaystyle\quad=\lim_{x_{2}\downarrow
a}\Bigg{\\{}\frac{\lim_{x_{1}\downarrow
a}\Big{(}\frac{G(z,x_{1},x_{2})-\widetilde{G}(z,a,x_{2})\widehat{u}_{a}(\lambda_{0},x_{1})}{u_{a}(\lambda_{0},x_{1})}\Big{)}}{u_{a}(\lambda_{0},x_{2})}$
$\displaystyle\hskip 48.36958pt-\frac{\lim_{x^{\prime}_{2}\downarrow
a}\lim_{x_{1}\downarrow
a}\Big{(}\frac{G(z,x_{1},x^{\prime}_{2})-\widetilde{G}(z,a,x^{\prime}_{2})\widehat{u}_{a}(\lambda_{0},x_{1})}{u_{a}(\lambda_{0},x_{1})\widehat{u}_{a}(\lambda_{0},x^{\prime}_{2})}\Big{)}\widehat{u}_{a}(\lambda_{0},x_{2})}{u_{a}(\lambda_{0},x_{2})}\Bigg{\\}}$
$\displaystyle\quad=\big{(}\widetilde{\partial}_{2}\widetilde{\partial}_{1}G\big{)}(z,a,a),$

where $G$ represents $G_{\eta,R}$. The actual computations are greatly
simplified since by (3.25), $G_{\eta,R}$ is a finite sum of products of
solutions of $\tau u=zu$ and hence one can employ the facts,

$\displaystyle[f(x_{1})g(x_{2})]^{\;\widetilde{}}\,\big{|}_{x_{1}=a,\,x_{2}=a}=\widetilde{f}(a)\widetilde{g}(a),$
$\displaystyle\widetilde{\partial}_{1}[f(x_{1})g(x_{2})]^{\;\widetilde{}}\,\big{|}_{x_{1}=a,\,x_{2}=a}={\widetilde{f}}^{\,\prime}(a)\widetilde{g}(a),$
$\displaystyle\widetilde{\partial}_{2}[f(x_{1})g(x_{2})]^{\;\widetilde{}}\,\big{|}_{x_{1}=a,\,x_{2}=a}=\widetilde{f}(a){\widetilde{g}}^{\,\prime}(a),$
(3.31)
$\displaystyle\widetilde{\partial}_{1}\widetilde{\partial}_{2}[f(x_{1})g(x_{2})]\big{|}_{x_{1}=a,\,x_{2}=a}={\widetilde{f}}^{\,\prime}(a){\widetilde{g}}^{\,\prime}(a)=\widetilde{\partial}_{2}\widetilde{\partial}_{1}[f(x_{1})g(x_{2})]\big{|}_{x_{1}=a,\,x_{2}=a},$
$\displaystyle\hskip 231.88979ptf,g\in\operatorname{dom}(T_{max}).$

For basic literature on the notion of Weyl–Titchmarsh–Kodaira $m$-functions
see [14], [21], [23], [24], [25], [32], [29, Ch. 13], [39], [42], [43], and
the references cited therein.

##  4\. The Jacobi Operator and its Boundary Values

We now turn to the principal topic of this paper, the Jacobi differential
expression

$\displaystyle\begin{split}\tau_{\alpha,\beta}=-(1-x)^{-\alpha}(1+x)^{-\beta}(d/dx)\big{(}(1-x)^{\alpha+1}(1+x)^{\beta+1}\big{)}(d/dx),&\\\
\alpha,\beta\in{\mathbb{R}},\;x\in(-1,1),\end{split}$ (4.1)

that is, in connection with Sections 2, 3 one now has

$\displaystyle a=-1,\quad b=1,$ $\displaystyle
p(x)=p_{\alpha,\beta}(x)=(1-x)^{\alpha+1}(1+x)^{\beta+1},\quad
q(x)=q_{\alpha,\beta}(x)=0,$ (4.2) $\displaystyle
r(x)=r_{\alpha,\beta}(x)=(1-x)^{\alpha}(1+x)^{\beta},\quad\alpha,\beta\in{\mathbb{R}},\;x\in(-1,1)$

(see, e.g. [1, Ch. 22], [5], [8], [17, Sect. 23], [36, Ch. 4], [44, Sects.
VII.6.1, XIV.2], [53, Ch. 18], [58, Ch. IV]).

$L^{2}$-realizations of $\tau_{\alpha,\beta}$ are thus most naturally
associated with the Hilbert space $L^{2}((-1,1);r_{\alpha,\beta}dx)$. However,
occasionally the weight function is absorbed into the differential expression,
as in (4.4) below, leading to the unweighted Hilbert space $L^{2}((-1,1);dx)$
(cf. [13, p. 1510–1520], [17, Sect. 24], [33]). Indeed, utilizing the unitary
map

$U_{\alpha,\beta}\colon\begin{cases}L^{2}((-1,1);dx)\to
L^{2}\big{(}(-1,1);r_{\alpha,\beta}dx\big{)},\\\
f\mapsto(1-\,\cdot\,)^{-\alpha/2}(1+\,\cdot\,)^{-\beta/2}f\end{cases}$ (4.3)

one confirms that

$\displaystyle(1-x)^{\alpha/2}(1+x)^{\beta/2}\tau_{\alpha,\beta}(1-x)^{-\alpha/2}(1+x)^{-\beta/2}$
$\displaystyle\quad=-\frac{d}{dx}\big{(}1-x^{2}\big{)}\frac{d}{dx}+\frac{\alpha^{2}}{4}\frac{1+x}{1-x}+\frac{\beta^{2}}{4}\frac{1-x}{1+x}-\frac{\alpha+\beta+\alpha\beta}{2},$
(4.4) $\displaystyle\hskip
168.72462pt\alpha,\beta\in{\mathbb{R}},\;x\in(-1,1).$

For more recent developments see, for instance, [18], [19], [20], [41], [45].

For later convenience we also mention the following relations that hold for
all $\alpha,\beta\in{\mathbb{R}}$,

$\displaystyle(1+x)^{-\beta}\tau_{\alpha,-\beta}(1+x)^{\beta}=\tau_{\alpha,\beta}+(1+\alpha)\beta,$
$\displaystyle(1-x)^{-\alpha}\tau_{-\alpha,\beta}(1-x)^{\alpha}=\tau_{\alpha,\beta}+(1+\beta)\alpha,$
(4.5)
$\displaystyle(1-x)^{-\alpha}(1+x)^{-\beta}\tau_{-\alpha,-\beta}(1-x)^{\alpha}(1+x)^{\beta}=\tau_{\alpha,\beta}+\alpha+\beta,$

where $(1+x)^{\pm\beta}$ and $(1-x)^{\pm\alpha}$ are regarded as operators of
multiplication.

To decide the limit point/limit circle classification of $\tau_{\alpha,\beta}$
at the interval endpoints $\pm 1$, it suffices to note that if $y_{1}$ is a
given solution of $\tau y=0$, then a 2nd linearly independent solution $y_{2}$
of $\tau y=0$ is obtained via the standard formula

$y_{2}(x)=y_{1}(x)\int_{c}^{x}dx^{\prime}\,p(x^{\prime})^{-1}y_{1}(x^{\prime})^{-2},\quad
c,x\in(a,b).$ (4.6)

Returning to the concrete Jacobi case at hand, one notices that

$\displaystyle y_{1}(x)=1,\quad x\in(-1,1),$ $\displaystyle
y_{2}(x)=\int_{0}^{x}dx^{\prime}\,(1-x^{\prime})^{-1-\alpha}(1+x^{\prime})^{-1-\beta}$
(4.7)
$\displaystyle\quad=\begin{cases}2^{-1-\alpha}\beta^{-1}(1+x)^{-\beta}[1+O(1+x)]+O(1),&\alpha\in{\mathbb{R}},\,\beta\in{\mathbb{R}}\backslash\\{0\\},\;\text{as
$x\downarrow-1$},\\\ -2^{-1-\alpha}\text{\rm
ln}(1+x)+O(1),&\alpha\in{\mathbb{R}},\,\beta=0,\;\text{as $x\downarrow-1$},\\\
2^{-1-\beta}\alpha^{-1}(1-x)^{-\alpha}[1+O(1-x)]+O(1),&\alpha\in{\mathbb{R}}\backslash\\{0\\},\,\beta\in{\mathbb{R}},\;\text{as
$x\uparrow 1$},\\\ -2^{-1-\beta}\text{\rm
ln}(1-x)+O(1),&\alpha=0,\,\beta\in{\mathbb{R}},\;\text{as $x\uparrow
1$}.\end{cases}$

Thus, an application of Theorem 2.3, Definition 2.4, and Remark 2.7 $(ii)$
implies the classification,

$\tau_{\alpha,\beta}\,\text{ is }\begin{cases}\text{regular at $-1$ if and
only if $\alpha\in{\mathbb{R}}$, $\beta\in(-1,0)$,}\\\ \text{in the limit
circle case (and not regular) at $-1$ if and only if}\\\
\quad\text{$\alpha\in{\mathbb{R}}$, $\beta\in[0,1)$,}\\\ \text{in the limit
point case at $-1$ if and only if $\alpha\in{\mathbb{R}}$,
$\beta\in{\mathbb{R}}\backslash(-1,1)$,}\\\ \text{regular at $1$ if and only
if $\alpha\in(-1,0)$, $\beta\in{\mathbb{R}}$,}\\\ \text{in the limit circle
case (and not regular) at $1$ if and only if}\\\ \quad\text{$\alpha\in[0,1)$,
$\beta\in{\mathbb{R}}$,}\\\ \text{in the limit point case at $1$ if and only
if $\alpha\in{\mathbb{R}}\backslash(-1,1)$,
$\beta\in{\mathbb{R}}$.}\end{cases}$ (4.8)

The maximal and preminimal operators, $T_{max,\alpha,\beta}$ and
$\overset{\textbf{\Large.}}{T}_{min,\alpha,\beta}$, associated to
$\tau_{\alpha,\beta}$ in $L^{2}((-1,1);r_{\alpha,\beta}dx)$ are then given by

$\displaystyle T_{max,\alpha,\beta}f=\tau_{\alpha,\beta}f,$ $\displaystyle
f\in\operatorname{dom}(T_{max,\alpha,\beta})=\big{\\{}g\in
L^{2}((-1,1);r_{\alpha,\beta}dx)\,\big{|}\,g,g^{[1]}\in AC_{loc}((-1,1));$
$\displaystyle\hskip 193.47882pt\tau_{\alpha,\beta}g\in
L^{2}((-1,1);r_{\alpha,\beta}dx)\big{\\}},$ (4.9)

and

$\displaystyle\overset{\textbf{\Large.}}{T}_{min,\alpha,\beta}f=\tau_{\alpha,\beta}f,$
$\displaystyle
f\in\operatorname{dom}\big{(}\overset{\textbf{\Large.}}{T}_{min,\alpha,\beta}\big{)}=\big{\\{}g\in
L^{2}((-1,1);r_{\alpha,\beta}dx)\,\big{|}\,g,g^{[1]}\in AC_{loc}((-1,1));$
(4.10) $\displaystyle\hskip
58.32823pt\operatorname{supp}\,(g)\subset(-1,1)\text{ is compact;
}\tau_{\alpha,\beta}g\in L^{2}((-1,1);r_{\alpha,\beta}dx)\big{\\}}.$

The fact (4.7) naturally leads to principal and nonprincipal solutions $u_{\pm
1,\alpha,\beta}(0,x)$ and $\widehat{u}_{\pm 1,\alpha,\beta}(0,x)$ of
$\tau_{\alpha,\beta}y=0$ near $\pm 1$ as follows:

$\displaystyle\begin{split}u_{-1,\alpha,\beta}(0,x)&=\begin{cases}-2^{-\alpha-1}\beta^{-1}(1+x)^{-\beta}[1+O(1+x)],&\beta\in(-\infty,0),\\\
1,&\beta\in[0,\infty),\end{cases}\\\
\widehat{u}_{-1,\alpha,\beta}(0,x)&=\begin{cases}1,&\beta\in(-\infty,0),\\\
-2^{-\alpha-1}\text{\rm ln}((1+x)/2),&\beta=0,\\\
2^{-\alpha-1}\beta^{-1}(1+x)^{-\beta}[1+O(1+x)],&\beta\in(0,\infty);\end{cases}\end{split}\quad\alpha\in{\mathbb{R}},$
(4.11)

and

$\displaystyle\begin{split}u_{1,\alpha,\beta}(0,x)&=\begin{cases}2^{-\beta-1}\alpha^{-1}(1-x)^{-\alpha}[1+O(1-x)],&\alpha\in(-\infty,0),\\\
1,&\alpha\in[0,\infty),\end{cases}\\\
\widehat{u}_{1,\alpha,\beta}(0,x)&=\begin{cases}1,&\alpha\in(-\infty,0),\\\
2^{-\beta-1}\text{\rm ln}((1-x)/2),&\alpha=0,\\\
-2^{-\beta-1}\alpha^{-1}(1-x)^{-\alpha}[1+O(1-x)],&\alpha\in(0,\infty);\end{cases}\end{split}\quad\beta\in{\mathbb{R}}.$
(4.12)

Combining the fact (4.8) with Theorem 2.5,
$\overset{\textbf{\Large.}}{T}_{min,\alpha,\beta}$ is essentially self-adjoint
in $L^{2}((-1,1);r_{\alpha,\beta}dx)$ if and only if
$\alpha,\beta\in{\mathbb{R}}\backslash(-1,1)$. Thus, boundary values for
$T_{max,\alpha,\beta}$ at $-1$ exist if and only if $\alpha\in{\mathbb{R}}$,
$\beta\in(-1,1)$, and similarly, boundary values for $T_{max,\alpha,\beta}$ at
$1$ exist if and only if $\alpha\in(-1,1)$, $\beta\in{\mathbb{R}}$.

Employing the principal and nonprincipal solutions (4.11), (4.12) at $\pm 1$,
according to (2.26), (2.27), generalized boundary values for
$g\in\operatorname{dom}(T_{max,\alpha,\beta})$ are of the form

$\displaystyle\begin{split}\widetilde{g}(-1)&=\begin{cases}g(-1),&\beta\in(-1,0),\\\
-2^{\alpha+1}\lim_{x\downarrow-1}g(x)/\text{\rm ln}((1+x)/2),&\beta=0,\\\
\beta
2^{\alpha+1}\lim_{x\downarrow-1}(1+x)^{\beta}g(x),&\beta\in(0,1),\end{cases}\\\
{\widetilde{g}}^{\,\prime}(-1)&=\begin{cases}g^{[1]}(-1),&\beta\in(-1,0),\\\
\lim_{x\downarrow-1}\big{[}g(x)+\widetilde{g}(-1)2^{-\alpha-1}\text{\rm
ln}((1+x)/2)\big{]},&\beta=0,\\\
\lim_{x\downarrow-1}\big{[}g(x)-\widetilde{g}(-1)2^{-\alpha-1}\beta^{-1}(1+x)^{-\beta}\big{]},&\beta\in(0,1);\end{cases}\end{split}\quad\alpha\in{\mathbb{R}},$
(4.13)
$\displaystyle\begin{split}\widetilde{g}(1)&=\begin{cases}g(1),&\alpha\in(-1,0),\\\
2^{\beta+1}\lim_{x\uparrow 1}g(x)/\text{\rm ln}((1-x)/2),&\alpha=0,\\\ -\alpha
2^{\beta+1}\lim_{x\uparrow 1}(1-x)^{\alpha}g(x),&\alpha\in(0,1),\end{cases}\\\
{\widetilde{g}}^{\,\prime}(1)&=\begin{cases}g^{[1]}(1),&\alpha\in(-1,0),\\\
\lim_{x\uparrow 1}\big{[}g(x)-\widetilde{g}(1)2^{-\beta-1}\text{\rm
ln}((1-x)/2)\big{]},&\alpha=0,\\\ \lim_{x\uparrow
1}\big{[}g(x)+\widetilde{g}(1)2^{-\beta-1}\alpha^{-1}(1-x)^{-\alpha}\big{]},&\alpha\in(0,1);\end{cases}\end{split}\quad\beta\in{\mathbb{R}}.$
(4.14)

As a result (cf., (2.5), (2.10), (2.26), (2.27)), the minimal operator
$T_{min,\alpha,\beta}$ associated to $\tau_{\alpha,\beta}$, that is,
$T_{min,\alpha,\beta}=\overline{\overset{\textbf{\Large.}}{T}_{min,\alpha,\beta}}$,
is thus given by

$\displaystyle T_{min,\alpha,\beta}f=\tau_{\alpha,\beta}f,$ $\displaystyle
f\in\operatorname{dom}(T_{min,\alpha,\beta})=\big{\\{}g\in
L^{2}((-1,1);r_{\alpha,\beta}dx)\,\big{|}\,g,g^{[1]}\in AC_{loc}((-1,1));$
(4.15) $\displaystyle\hskip
31.2982pt\widetilde{g}(-1)={\widetilde{g}}^{\,\prime}(-1)=\widetilde{g}(1)={\widetilde{g}}^{\,\prime}(1)=0;\,\tau_{\alpha,\beta}g\in
L^{2}((-1,1);r_{\alpha,\beta}dx)\big{\\}}.$

All self-adjoint extensions of $T_{min,\alpha,\beta}$ are thus either of the
type (2.28) in the case of separated boundary conditions, or of the type
(2.29) in the case of coupled boundary conditions. In particular, the
Friedrichs extension $T_{F,\alpha,\beta}$ of $T_{min,\alpha,\beta}$ is given
as in (2.30).

We note that self-adjoint extensions of $T_{min,\alpha,\beta}$ in the limit
circle case at both endpoints $x=\pm 1$, that is, $\alpha,\beta\in[0,1)$, were
studied in [19] and [20], in particular, the self-adjoint extension containing
Jacobi polynomials in its domain was characterized. In addition, [19] and [20]
discussed the domains of powers of self-adjoint Jacobi operators. Self-adjoint
extensions in the case $\alpha,\beta\in(0,1)$ are also studied in [10]; the
case $\alpha,\beta\in(-1,\infty)$ was also treated by Grünewald [33].

For a detailed treatment of solutions of the Jacobi differential equation and
the associated hypergeometric differential equations we refer to Appendices
A–C.

##  5\. The Regular and Limit Circle Case $\alpha,\beta\in(-1,1)$

In this section we compute the Weyl–Titchmarsh–Kodaira and Green’s function
induced $m$ (resp., $M$) functions when the Jacobi differential expression
$\tau_{\alpha,\beta}$ is either in the regular or limit circle case at $\pm
1$.

We begin by determining the solutions
$\varphi_{\gamma,\alpha,\beta}(z,\,\cdot\,)$ and
$\vartheta_{\gamma,\alpha,\beta}(z,\,\cdot\,)$ of
$\tau_{\alpha,\beta}u=zu,z\in{\mathbb{C}}$, that are subject to the conditions

$\displaystyle\begin{split}\widetilde{\varphi}_{\gamma,\alpha,\beta}(z,-1)&=-\sin(\gamma),\quad\widetilde{\varphi}_{\gamma,\alpha,\beta}^{\,\prime}(z,-1)=\cos(\gamma),\\\
\widetilde{\vartheta}_{\gamma,\alpha,\beta}(z,-1)&=\cos(\gamma),\quad\;\;\,\,\widetilde{\vartheta}_{\gamma,\alpha,\beta}^{\,\prime}(z,-1)=\sin(\gamma);\quad\gamma\in[0,\pi).\end{split}$
(5.1)

In particular, one can write (see Appendix A for the solutions
$y_{j,\alpha,\beta,-1}(z,\,\cdot\,)$, $j=1,2$)

$\displaystyle\varphi_{\gamma,\alpha,\beta}(z,x)$
$\displaystyle=c_{\gamma,\varphi,1}(z)y_{1,\alpha,\beta,-1}(z,x)+c_{\gamma,\varphi,2}(z)y_{2,\alpha,\beta,-1}(z,x),$
(5.2) $\displaystyle\vartheta_{\gamma,\alpha,\beta}(z,x)$
$\displaystyle=c_{\gamma,\vartheta,1}(z)y_{1,\alpha,\beta,-1}(z,x)+c_{\gamma,\vartheta,2}(z)y_{2,\alpha,\beta,-1}(z,x);\
\ z\in{\mathbb{C}},\ x\in(-1,1),$

where the coefficients
$c_{\gamma,\varphi,1}(z),c_{\gamma,\varphi,2}(z),c_{\gamma,\vartheta,1}(z),c_{\gamma,\vartheta,2}(z)\in{\mathbb{C}}$
are given by

$\displaystyle\begin{split}c_{\gamma,\varphi,1}(z)&=\frac{-\sin(\gamma)\widetilde{y}_{2,\alpha,\beta,-1}^{\,\prime}(z,-1)-\cos(\gamma)\widetilde{y}_{2,\alpha,\beta,-1}(z,-1)}{\widetilde{y}_{1,\alpha,\beta,-1}(z,-1)\widetilde{y}_{2,\alpha,\beta,-1}^{\,\prime}(z,-1)-\widetilde{y}_{1,\alpha,\beta,-1}^{\,\prime}(z,-1)\widetilde{y}_{2,\alpha,\beta,-1}(z,-1)},\\\\[2.84526pt]
c_{\gamma,\varphi,2}(z)&=\frac{\cos(\gamma)\widetilde{y}_{1,\alpha,\beta,-1}(z,-1)+\sin(\gamma)\widetilde{y}_{1,\alpha,\beta,-1}^{\,\prime}(z,-1)}{\widetilde{y}_{1,\alpha,\beta,-1}(z,-1)\widetilde{y}_{2,\alpha,\beta,-1}^{\,\prime}(z,-1)-\widetilde{y}_{1,\alpha,\beta,-1}^{\,\prime}(z,-1)\widetilde{y}_{2,\alpha,\beta,-1}(z,-1)},\\\\[2.84526pt]
c_{\gamma,\vartheta,1}(z)&=\frac{\cos(\gamma)\widetilde{y}_{2,\alpha,\beta,-1}^{\,\prime}(z,-1)-\sin(\gamma)\widetilde{y}_{2,\alpha,\beta,-1}(z,-1)}{\widetilde{y}_{1,\alpha,\beta,-1}(z,-1)\widetilde{y}_{2,\alpha,\beta,-1}^{\,\prime}(z,-1)-\widetilde{y}_{1,\alpha,\beta,-1}^{\,\prime}(z,-1)\widetilde{y}_{2,\alpha,\beta,-1}(z,-1)},\\\\[2.84526pt]
c_{\gamma,\vartheta,2}(z)&=\frac{\sin(\gamma)\widetilde{y}_{1,\alpha,\beta,-1}(z,-1)-\cos(\gamma)\widetilde{y}_{1,\alpha,\beta,-1}^{\,\prime}(z,-1)}{\widetilde{y}_{1,\alpha,\beta,-1}(z,-1)\widetilde{y}_{2,\alpha,\beta,-1}^{\,\prime}(z,-1)-\widetilde{y}_{1,\alpha,\beta,-1}^{\,\prime}(z,-1)\widetilde{y}_{2,\alpha,\beta,-1}(z,-1)}.\end{split}$
(5.3)

###  5.1. The Regular and Limit Circle Case for Separated Boundary Conditions

Given these preparations, one obtains the principal result on $m$-functions
corresponding to separated boundary conditions of this section:

######  Theorem 5.1.

Let $T_{\gamma,\delta,\alpha,\beta}$, $\gamma,\delta\in[0,\pi)$, be the self-
adjoint operator associated with $\tau_{\alpha,\beta}$ applying $\gamma$
-boundary conditions at $x=-1$ and $\delta$-boundary conditions at $x=1$ in
the regular or limit circle case $\alpha,\beta\in(-1,1)$. Then the associated
Weyl–Titchmarsh–Kodaira $m$-function is of the form

$\displaystyle m_{\gamma,\delta,\alpha,\beta}(z)=$
$\displaystyle\begin{cases}\big{[}\beta
2^{\alpha+1}\cos(\gamma)(\cos(\delta)\widetilde{y}_{1,\alpha,\beta,-1}(z,1)+\sin(\delta)\widetilde{y}_{1,\alpha,\beta,-1}^{\,\prime}(z,1))\\\
\quad-\sin(\gamma)(\cos(\delta)\widetilde{y}_{2,\alpha,\beta,-1}(z,1)+\sin(\delta)\widetilde{y}_{2,\alpha,\beta,-1}^{\,\prime}(z,1))\big{]}\\\
\times\big{[}\beta
2^{\alpha+1}\sin(\gamma)(\cos(\delta)\widetilde{y}_{1,\alpha,\beta,-1}(z,1)+\sin(\delta)\widetilde{y}_{1,\alpha,\beta,-1}^{\,\prime}(z,1))\\\
\quad+\cos(\gamma)(\cos(\delta)\widetilde{y}_{2,\alpha,\beta,-1}(z,1)+\sin(\delta)\widetilde{y}_{2,\alpha,\beta,-1}^{\,\prime}(z,1))\big{]}^{-1},\\\
\hskip 199.16928pt\beta\in(-1,0),\\\\[2.84526pt]
\big{[}-2^{\alpha+1}\sin(\gamma)(\cos(\delta)\widetilde{y}_{1,\alpha,\beta,-1}(z,1)+\sin(\delta)\widetilde{y}_{1,\alpha,\beta,-1}^{\,\prime}(z,1))\\\
\quad+\cos(\gamma)(\cos(\delta)\widetilde{y}_{2,\alpha,\beta,-1}(z,1)+\sin(\delta)\widetilde{y}_{2,\alpha,\beta,-1}^{\,\prime}(z,1))\big{]}\\\
\times\big{[}2^{\alpha+1}\cos(\gamma)(\cos(\delta)\widetilde{y}_{1,\alpha,\beta,-1}(z,1)+\sin(\delta)\widetilde{y}_{1,\alpha,\beta,-1}^{\,\prime}(z,1))\\\
\quad+\sin(\gamma)(\cos(\delta)\widetilde{y}_{2,\alpha,\beta,-1}(z,1)+\sin(\delta)\widetilde{y}_{2,\alpha,\beta,-1}^{\,\prime}(z,1))\big{]}^{-1},\\\\[2.84526pt]
\hskip 221.93158pt\beta=0,\\\\[2.84526pt] \big{[}\beta
2^{\alpha+1}\sin(\gamma)(\cos(\delta)\widetilde{y}_{1,\alpha,\beta,-1}(z,1)+\sin(\delta)\widetilde{y}_{1,\alpha,\beta,-1}^{\,\prime}(z,1))\\\
\quad+\cos(\gamma)(\cos(\delta)\widetilde{y}_{2,\alpha,\beta,-1}(z,1)+\sin(\delta)\widetilde{y}_{2,\alpha,\beta,-1}^{\,\prime}(z,1))\big{]}\\\
\times\big{[}-\beta
2^{\alpha+1}\cos(\gamma)(\cos(\delta)\widetilde{y}_{1,\alpha,\beta,-1}(z,1)+\sin(\delta)\widetilde{y}_{1,\alpha,\beta,-1}^{\,\prime}(z,1))\\\
\quad+\sin(\gamma)(\cos(\delta)\widetilde{y}_{2,\alpha,\beta,-1}(z,1)+\sin(\delta)\widetilde{y}_{2,\alpha,\beta,-1}^{\,\prime}(z,1))\big{]}^{-1},\\\\[2.84526pt]
\hskip 209.12791pt\beta\in(0,1);\end{cases}$ $\displaystyle\hskip
85.35826pt\alpha\in(-1,1),\ \gamma,\delta\in[0,\pi),\
z\in\rho(T_{\gamma,\delta,\alpha,\beta}).$ (5.4)

The $($necessarily simple $)$ poles of $m_{\gamma,\delta,\alpha,\beta}$ occur
precisely at the $($necessarily simple $)$ eigenvalues of
$T_{\gamma,\delta,\alpha,\beta}$.

###### Proof.

Substituting the boundary values (C.13) into (5.3) and combining this with
equation (5.2) yields $\varphi_{\gamma,\alpha,\beta}(z,x)$ and
$\vartheta_{\gamma,\alpha,\beta}(z,x)$, $\gamma\in[0,\pi)$. Thus, (3.18)
implies the form of the general $m$-function as displayed in (5.4). ∎

Theorem 5.1 permits one to efficiently compute any $m$-function for
$\alpha,\beta\in(-1,1)$ by substituting the appropriate boundary values
(C.14)–(C.16) into (5.4). In particular, the special Legendre case,
$\alpha=\beta=0$, to be reconsidered in Example 5.3 $\mathbf{(II)}$ below, has
frequently been discussed in the literature, see, for instance, [25] and the
extensive list of references cited therein.

######  Remark 5.2.

We mention a few other special cases of interest other than Legendre. The
Gegenbauer, or ultraspherical, equation (see, e.g., [1, Ch. 22], [53, Ch. 18],
[58, Ch. IV]) can be realized by choosing the parameters
$\alpha=\beta=\mu-1/2$, noting at the endpoints $x=\pm 1$, $\tau_{\mu}$ is
regular for $\mu\in(-1/2,1/2)$, in the limit circle case for
$\mu\in[1/2,3/2)$, and in the limit point case for
$\mu\in{\mathbb{R}}\backslash(-1/2,3/2)$. In particular, choosing $\mu=1/2$ we
arrive at the Legendre equation once again. After the appropriate change in
parameter, (5.4) describes the form of the $m$-function for the new parameter
$\mu\in(-1/2,3/2)$. For $\mu\in{\mathbb{R}}\backslash(-1/2,3/2)$, see Theorem
7.1.

The Chebyshev equations of the first and second kinds are two more important
special cases covered in Example 5.3 $\mathbf{(I)}$ and $\mathbf{(III)}$
respectively. The Chebyshev equation of the first kind is realized by choosing
$\mu=0$ in the Gegenbauer equation, or $\alpha=\beta=-1/2$ in the Jacobi
equation (see, e.g., [1, Ch. 22], [53, Ch. 18], [58, Ch. IV]), whereas the
Chebyshev equation of the second kind is realized by choosing $\mu=1$ in the
Gegenbauer equation, or $\alpha=\beta=1/2$ in the Jacobi equation (see, e.g.,
[1, Ch. 22], [53, Ch. 18], [58, Ch. IV]).

See Appendix E for more details. $\diamond$

In the following we employ the abbreviation

$\sigma_{\alpha,\beta}(z)=\big{[}(1+\alpha+\beta)^{2}+4z\big{]}^{1/2},\quad\alpha,\beta\in{\mathbb{R}},\;z\in{\mathbb{C}}.$
(5.5)

The following example provides an illustration of Theorem 5.1 while applying
(C.14)–(C.16), (5.1)–(5.4). Throughout, we recall that the $($necessarily
simple $)$ poles of $m_{\gamma,\delta,\alpha,\beta}$ occur precisely at the
$($necessarily simple $)$ eigenvalues of $T_{\gamma,\delta,\alpha,\beta}$.

######  Example 5.3.

When boundary conditions are chosen such that the Jacobi polynomials,
$P_{n}^{\alpha,\beta}(\,\cdot\,)$ $($see (LABEL:A.31)$)$, satisfy them for the
given choice of parameters $\alpha,\beta$, one verifies the spectrum is given
by $\\{n(n+1+\alpha+\beta)\\}_{n\in{\mathbb{N}}_{0}}$ in accordance with
Remark A.1 $($see $\mathbf{(I)}$–$\mathbf{(III)}$, $\mathbf{(VII)}$,
$\mathbf{(VIII)}$ below$)$.
$\mathbf{(I)}$ The case $\alpha,\beta\in(-1,0)$$:$
The Neumann extension $\gamma=\delta=\pi/2$,
$T_{N,\alpha,\beta}=T_{\frac{\pi}{2},\frac{\pi}{2},\alpha,\beta}$, yields for
$z\in\rho(T_{N,\alpha,\beta})$,

$\displaystyle
m_{\frac{\pi}{2},\frac{\pi}{2},\alpha,\beta}(z)=\frac{-\widetilde{y}_{2,\alpha,\beta,-1}^{\,\prime}(z,1)}{\beta
2^{\alpha+1}\widetilde{y}_{1,\alpha,\beta,-1}^{\,\prime}(z,1)}$
$\displaystyle\quad=\frac{-\Gamma(1-\beta)\Gamma([1+\alpha+\beta+\sigma_{\alpha,\beta}(z)]/2)\Gamma([1+\alpha+\beta-\sigma_{\alpha,\beta}(z)]/2)}{\beta
2^{1+\alpha+\beta}\Gamma(1+\beta)\Gamma([1+\alpha-\beta+\sigma_{\alpha,\beta}(z)]/2)\Gamma([1+\alpha-\beta-\sigma_{\alpha,\beta}(z)]/2)},$
$\displaystyle\quad\sigma(T_{N,\alpha,\beta})=\\{n(n+1+\alpha+\beta)\\}_{n\in{\mathbb{N}}_{0}};\quad\alpha,\beta\in(-1,0).$
(5.6)

$\mathbf{(II)}$ The case $\alpha=\beta=0$ $($Legendre$)$$:$
The Friedrichs extension, $\gamma=\delta=0$, $T_{F,Leg}=T_{0,0,0,0}$, yields
for $z\in\rho(T_{F,Leg})$,

$\displaystyle m_{0,0,Leg}(z)$
$\displaystyle=\frac{\widetilde{y}_{2,0,0,-1}(z,1)}{2\widetilde{y}_{1,0,0,-1}(z,1)}$
$\displaystyle=-\gamma_{E}-\\{\psi([1+\sigma_{0,0}(z)]/2)+\psi([1-\sigma_{0,0}(z)]/2)\\}/2$
$\displaystyle=-\gamma_{E}-\psi([1+\sigma_{0,0}(z)]/2)+(\pi/2)\cot(\pi[1-\sigma_{0,0}(z)]/2),$
$\displaystyle\sigma(T_{F,Leg})$
$\displaystyle=\\{n(n+1)\\}_{n\in{\mathbb{N}}_{0}};\quad\alpha=\beta=0,$ (5.7)

in agreement with [25, eq. (6.46)]. Here we used the reflection formula $($cf.
[1, eq. 6.3.6]$)$

$\displaystyle\psi(1-z)-\psi(z)=\pi\cot(\pi z).$ (5.8)

$\mathbf{(III)}$ The case $\alpha,\beta\in(0,1)$$:$
The Friedrichs extension, $\gamma=\delta=0$,
$T_{F,\alpha,\beta}=T_{0,0,\alpha,\beta}$, yields for
$z\in\rho(T_{F,\alpha,\beta})$,

$\displaystyle
m_{0,0,\alpha,\beta}(z)=\frac{\widetilde{y}_{2,\alpha,\beta,-1}(z,1)}{-\beta
2^{\alpha+1}\widetilde{y}_{1,\alpha,\beta,-1}(z,1)}$
$\displaystyle\quad=\frac{-\Gamma(1-\beta)\Gamma([1+\alpha+\beta+\sigma_{\alpha,\beta}(z)]/2)\Gamma([1+\alpha+\beta-\sigma_{\alpha,\beta}(z)]/2)}{\beta
2^{1+\alpha+\beta}\Gamma(1+\beta)\Gamma([1+\alpha-\beta+\sigma_{\alpha,\beta}(z)]/2)\Gamma([1+\alpha-\beta-\sigma_{\alpha,\beta}(z)]/2)},$
$\displaystyle\ \
\sigma(T_{F,\alpha,\beta})=\\{n(n+1+\alpha+\beta)\\}_{n\in{\mathbb{N}}_{0}};\quad\alpha,\beta\in(0,1).$
(5.9)

$\mathbf{(IV)}$ The case $\alpha\in(0,1),\;\beta\in(-1,0)$$:$
The Friedrichs extension, $\gamma=\delta=0$,
$T_{F,\alpha,\beta}=T_{0,0,\alpha,\beta}$, yields for
$z\in\rho(T_{F,\alpha,\beta})$,

$\displaystyle m_{0,0,\alpha,\beta}(z)=\frac{\beta
2^{\alpha+1}\widetilde{y}_{1,\alpha,\beta,-1}(z,1)}{\widetilde{y}_{2,\alpha,\beta,-1}(z,1)}$
$\displaystyle\quad=\frac{\beta
2^{1+\alpha+\beta}\Gamma(1+\beta)\Gamma([1+\alpha-\beta+\sigma_{\alpha,\beta}(z)]/2)\Gamma([1+\alpha-\beta-\sigma_{\alpha,\beta}(z)]/2)}{\Gamma(1-\beta)\Gamma([1+\alpha+\beta+\sigma_{\alpha,\beta}(z)]/2)\Gamma([1+\alpha+\beta-\sigma_{\alpha,\beta}(z)]/2)},$
$\displaystyle\ \
\sigma(T_{F,\alpha,\beta})=\\{(n-\beta)(n+1+\alpha)\\}_{n\in{\mathbb{N}}_{0}};\quad\alpha\in(0,1),\;\beta\in(-1,0).$
(5.10)

$\mathbf{(V)}$ The case $\alpha\in(-1,0),\;\beta\in(0,1)$$:$
The Friedrichs extension, $\gamma=\delta=0$,
$T_{F,\alpha,\beta}=T_{0,0,\alpha,\beta}$, yields for
$z\in\rho(T_{F,\alpha,\beta})$,

$\displaystyle
m_{0,0,\alpha,\beta}(z)=\frac{\widetilde{y}_{2,\alpha,\beta,-1}(z,1)}{-\beta
2^{\alpha+1}\widetilde{y}_{1,\alpha,\beta,-1}(z,1)}$
$\displaystyle\quad=\frac{-\Gamma(1-\beta)\Gamma([1-\alpha+\beta+\sigma_{\alpha,\beta}(z)]/2)\Gamma([1-\alpha+\beta-\sigma_{\alpha,\beta}(z)]/2)}{\beta
2^{1+\alpha+\beta}\Gamma(1+\beta)\Gamma([1-\alpha-\beta+\sigma_{\alpha,\beta}(z)]/2)\Gamma([1-\alpha-\beta-\sigma_{\alpha,\beta}(z)]/2)},$
$\displaystyle\ \
\sigma(T_{F,\alpha,\beta})=\\{(n-\alpha)(n+1+\beta)\\}_{n\in{\mathbb{N}}_{0}};\quad\alpha\in(-1,0);\;\beta\in(0,1).$
(5.11)

$\mathbf{(VI)}$ The case $\alpha=0,\;\beta\in(-1,0)$$:$
The Friedrichs extension, $\gamma=\delta=0$, $T_{F,0,\beta}=T_{0,0,0,\beta}$,
yields for $z\in\rho(T_{F,0,\beta})$,

$\displaystyle m_{0,0,0,\beta}(z)$ $\displaystyle=\frac{\beta
2\widetilde{y}_{1,0,\beta,-1}(z,1)}{\widetilde{y}_{2,0,\beta,-1}(z,1)}$
$\displaystyle=\frac{\beta
2^{1+\beta}\Gamma(1+\beta)\Gamma([1-\beta+\sigma_{0,\beta}(z)]/2)\Gamma([1-\beta-\sigma_{0,\beta}(z)]/2)}{\Gamma(1-\beta)\Gamma([1+\beta+\sigma_{0,\beta}(z)]/2)\Gamma([1+\beta-\sigma_{0,\beta}(z)]/2)},$
$\displaystyle\sigma(T_{F,0,\beta})$
$\displaystyle=\\{(n-\beta)(n+1)\\}_{n\in{\mathbb{N}}_{0}};\quad\alpha=0,\;\beta\in(-1,0).$
(5.12)

$\mathbf{(VII)}$ The case $\alpha=0,\;\beta\in(0,1)$$:$
The Friedrichs extension, $\gamma=\delta=0$, $T_{F,0,\beta}=T_{0,0,0,\beta}$,
yields for $z\in\rho(T_{F,0,\beta})$,

$\displaystyle m_{0,0,0,\beta}(z)$
$\displaystyle=\frac{\widetilde{y}_{2,0,\beta,-1}(z,1)}{-\beta
2\widetilde{y}_{1,0,\beta,-1}(z,1)}$
$\displaystyle=\frac{-\Gamma(1-\beta)\Gamma([1+\beta+\sigma_{0,\beta}(z)]/2)\Gamma([1+\beta-\sigma_{0,\beta}(z)]/2)}{\beta
2^{1+\beta}\Gamma(1+\beta)\Gamma([1-\beta+\sigma_{0,\beta}(z)]/2)\Gamma([1-\beta-\sigma_{0,\beta}(z)]/2)},$
$\displaystyle\sigma(T_{F,0,\beta})$
$\displaystyle=\\{n(n+1+\beta)\\}_{n\in{\mathbb{N}}_{0}};\quad\alpha=0,\;\beta\in(0,1).$
(5.13)

$\mathbf{(VIII)}$ The case $\alpha\in(0,1),\;\beta=0$ :
The Friedrichs extension, $\gamma=\delta=0$,
$T_{F,\alpha,0}=T_{0,0,\alpha,0}$, yields for $z\in\rho(T_{F,\alpha,0})$,

$\displaystyle m_{0,0,\alpha,0}(z)$
$\displaystyle=\frac{\widetilde{y}_{2,\alpha,0,-1}(z,1)}{2^{\alpha+1}\widetilde{y}_{1,\alpha,0,-1}(z,1)}$
$\displaystyle=-2^{-1-\alpha}[2\gamma_{E}+\psi([1+\alpha+\sigma_{\alpha,0}(z)]/2)+\psi([1+\alpha-\sigma_{\alpha,0}(z)]/2)],$
$\displaystyle\sigma(T_{F,\alpha,0})$
$\displaystyle=\\{n(n+1+\alpha)\\}_{n\in{\mathbb{N}}_{0}};\quad\alpha\in(0,1);\;\beta=0.$
(5.14)

$\mathbf{(IX)}$ The case $\alpha\in(-1,0),\;\beta=0$ :
The Friedrichs extension, $\gamma=\delta=0$,
$T_{F,\alpha,0}=T_{0,0,\alpha,0}$, yields for $z\in\rho(T_{F,\alpha,0})$,

$\displaystyle m_{0,0,\alpha,0}(z)$
$\displaystyle=\frac{\widetilde{y}_{2,\alpha,\beta,-1}(z,1)}{2^{\alpha+1}\widetilde{y}_{1,\alpha,\beta,-1}(z,1)}$
$\displaystyle=-2^{-1-\alpha}[2\gamma_{E}+\psi([1-\alpha+\sigma_{\alpha,0}(z)]/2)+\psi([1-\alpha-\sigma_{\alpha,0}(z)]/2)],$
$\displaystyle\sigma(T_{F,\alpha,0})$
$\displaystyle=\\{(n-\alpha)(n+1)\\}_{n\in{\mathbb{N}}_{0}};\quad\alpha\in(-1,0),\;\beta=0.$
(5.15)

We note that the relations in (4.5) can be extended to the Hilbert space
setting using the unitary operators $U_{\pm\beta}$ and $V_{\pm\alpha}$:

$\displaystyle\begin{split}&U_{\pm\beta}\colon\begin{cases}L^{2}((-1,1);r_{\alpha,\pm\beta}dx)\rightarrow
L^{2}((-1,1);r_{\alpha,\mp\beta}dx),\\\
f\mapsto(1+\,\cdot\,)^{\pm\beta}f,\end{cases}\\\\[2.84526pt]
&V_{\pm\alpha}\colon\begin{cases}L^{2}((-1,1);r_{\pm\alpha,\beta}dx)\rightarrow
L^{2}((-1,1);r_{\mp\alpha,\beta}dx),\\\
f\mapsto(1+\,\cdot\,)^{\pm\alpha}f,\end{cases}\end{split}$ (5.16)

resulting in

$\displaystyle\begin{split}&(1+x)^{-\beta}T_{\gamma,\delta,\alpha,-\beta}(1+x)^{\beta}=T_{\gamma,\delta,\alpha,\beta}+(1+\alpha)\beta,\\\
&(1-x)^{-\alpha}T_{\gamma,\delta,-\alpha,\beta}(1-x)^{\alpha}=T_{\gamma,\delta,\alpha,\beta}+(1+\beta)\alpha,\\\
&(1-x)^{-\alpha}(1+x)^{-\beta}T_{\gamma,\delta,-\alpha,-\beta}(1-x)^{\alpha}(1+x)^{\beta}=T_{\gamma,\delta,\alpha,\beta}+\alpha+\beta;\\\
&\hskip 221.93158pt\gamma,\delta\in\\{0,\pi/2\\}.\end{split}$ (5.17)

These relations can be used to explain the patterns found in Example 5.3.

###  5.2. The Regular and Limit Circle Case for Coupled Boundary Conditions

We now turn to the case of coupled boundary conditions as described in (2.29).
Following [29, Ch. 13] (see also [28], [27]) and utilizing generalized
boundary values, we introduce the $2\times 2$ matrix-valued Green’s function
induced $M$-function corresponding to $T_{\eta,R,\alpha,\beta}$ via

$\displaystyle M_{\eta,R,\alpha,\beta}(z,-1)$
$\displaystyle\quad=\footnotesize\begin{pmatrix}\widetilde{G}_{\eta,R,\alpha,\beta}(z,-1,-1)&2^{-1}\big{(}\big{[}\widetilde{\partial}_{1}+\widetilde{\partial}_{2}\big{]}G_{\eta,R,\alpha,\beta}\big{)}^{\widetilde{}}(z,-1,-1)\\\\[2.84526pt]
2^{-1}\big{(}\big{[}\widetilde{\partial}_{1}+\widetilde{\partial}_{2}\big{]}G_{\eta,R,\alpha,\beta}\big{)}^{\widetilde{}}(z,-1,-1)&\big{(}\widetilde{\partial}_{1}\widetilde{\partial}_{2}G_{\eta,R,\alpha,\beta}\big{)}(z,-1,-1)\end{pmatrix}$
$\displaystyle\quad=\footnotesize\begin{pmatrix}\begin{matrix}R_{2,2}\widetilde{\varphi}_{0,\alpha,\beta}(z,1)-R_{1,2}\widetilde{\varphi}^{\,\prime}_{0,\alpha,\beta}(z,1)\\\
\quad\end{matrix}&\begin{matrix}2^{-1}\big{[}R_{1,1}\widetilde{\varphi}^{\,\prime}_{0,\alpha,\beta}(z,1)-R_{2,1}\widetilde{\varphi}_{0,\alpha,\beta}(z,1)\\\
\quad-
R_{2,2}\widetilde{\vartheta}_{0,\alpha,\beta}(z,1)+R_{1,2}\widetilde{\vartheta}^{\,\prime}_{0,\alpha,\beta}(z,1)\big{]}\end{matrix}\\\\[11.38109pt]
\begin{matrix}2^{-1}\big{[}R_{1,1}\widetilde{\varphi}^{\,\prime}_{0,\alpha,\beta}(z,1)-R_{2,1}\widetilde{\varphi}_{0,\alpha,\beta}(z,1)\\\
\quad-
R_{2,2}\widetilde{\vartheta}_{0,\alpha,\beta}(z,1)+R_{1,2}\widetilde{\vartheta}^{\,\prime}_{0,\alpha,\beta}(z,1)\big{]}\end{matrix}&\begin{matrix}R_{2,1}\widetilde{\vartheta}_{0,\alpha,\beta}(z,1)-R_{1,1}\widetilde{\vartheta}^{\,\prime}_{0,\alpha,\beta}(z,1)\\\
\quad\end{matrix}\end{pmatrix}$
$\displaystyle\qquad\times\big{[}-e^{i\eta}\big{]}\big{[}\widetilde{F}_{\eta,R,\alpha,\beta}(z)\big{]}^{-1},\quad\alpha,\beta\in(-1,1),\
z\in\rho(T_{\eta,R,\alpha,\beta}),$ (5.18)

where $\eta\in[0,\pi)$, $R\in SL(2,{\mathbb{R}})$, and the Green’s function,
$G_{\eta,R,\alpha,\beta}(z,\,\cdot\,,\,\cdot\,)$, of $T_{\eta,R,\alpha,\beta}$
is given by

$\displaystyle
G_{\eta,R,\alpha,\beta}(z,x,x^{\prime})=\dfrac{-e^{i\eta}}{\widetilde{F}_{\eta,R,\alpha,\beta}(z)}$
$\displaystyle\qquad\times\Big{\\{}[-R_{1,2}\widetilde{\varphi}^{\,\prime}_{0,\alpha,\beta}(z,1)+R_{2,2}\widetilde{\varphi}_{0,\alpha,\beta}(z,1)]\vartheta_{0,\alpha,\beta}(z,x)\vartheta_{0,\alpha,\beta}(z,x^{\prime})$
$\displaystyle\hskip
34.14322pt+[-R_{1,1}\widetilde{\vartheta}^{\,\prime}_{0,\alpha,\beta}(z,1)+R_{2,1}\widetilde{\vartheta}_{0,\alpha,\beta}(z,1)]\varphi_{0,\alpha,\beta}(z,x)\varphi_{0,\alpha,\beta}(z,x^{\prime})$
$\displaystyle\hskip
34.14322pt+[e^{-i\eta}+R_{1,2}\widetilde{\vartheta}^{\,\prime}_{0,\alpha,\beta}(z,1)-R_{2,2}\widetilde{\vartheta}_{0,\alpha,\beta}(z,1)]\vartheta_{0,\alpha,\beta}(z,x)\varphi_{0,\alpha,\beta}(z,x^{\prime})$
$\displaystyle\hskip
34.14322pt+[-e^{-i\eta}+R_{1,1}\widetilde{\varphi}^{\,\prime}_{0,\alpha,\beta}(z,1)-R_{2,1}\widetilde{\varphi}_{0,\alpha,\beta}(z,1)]\varphi_{0,\alpha,\beta}(z,x)\vartheta_{0,\alpha,\beta}(z,x^{\prime})\Big{\\}}$
$\displaystyle\quad+\begin{cases}0,&a<x\leqslant x^{\prime}<b,\\\
[\vartheta_{0,\alpha,\beta}(z,x)\varphi_{0,\alpha,\beta}(z,x^{\prime})-\varphi_{0,\alpha,\beta}(z,x)\vartheta_{0,\alpha,\beta}(z,x^{\prime})],&a<x^{\prime}\leqslant
x<b,\end{cases}$ $\displaystyle\hskip 256.0748ptz\in\rho(T_{\eta,R}),$ (5.19)

with

$\displaystyle\begin{split}\widetilde{F}_{\eta,R,\alpha,\beta}(z)&=-e^{i\eta}[R_{1,1}\widetilde{\varphi}^{\,\prime}_{0,\alpha,\beta}(z,1)+R_{2,2}\widetilde{\vartheta}_{0,\alpha,\beta}(z,1)-R_{2,1}\widetilde{\varphi}_{0,\alpha,\beta}(z,1)\\\
&\hskip 36.98866pt-
R_{1,2}\widetilde{\vartheta}^{\,\prime}_{0,\alpha,\beta}(z,1)-2\cos(\eta)],\\\
\varphi_{0,\alpha,\beta}(z,x)&=\begin{cases}-\beta^{-1}2^{-\alpha-1}y_{2,\alpha,\beta,-1}(z,x),&\beta\in(-1,0),\\\
y_{1,\alpha,0,-1}(z,x),&\beta=0,\\\
y_{1,\alpha,\beta,-1}(z,x),&\beta\in(0,1),\end{cases}\\\
\vartheta_{0,\alpha,\beta}(z,x)&=\begin{cases}y_{1,\alpha,\beta,-1}(z,x),&\beta\in(-1,0),\\\
-2^{-\alpha-1}y_{2,\alpha,0,-1}(z,x),&\beta=0,\\\
\beta^{-1}2^{-\alpha-1}y_{2,\alpha,\beta,-1}(z,x),&\beta\in(0,1),\end{cases}\end{split}\qquad\alpha\in{\mathbb{R}}.$
(5.20)

We note that $M_{\eta,R,\alpha,\beta}(\,\cdot\,,-1)$ is a $2\times 2$ matrix-
valued Nevanlinna–Herglotz function as shown in [28].

Next, we discuss in detail two important cases of coupled boundary conditions:
the general $\eta$-periodic and Krein–von Neumann extensions.

In the $\eta$-periodic case where $R=I_{2}$, $\eta\in[0,\pi),$ (see [27]) one
has from (5.18) and (5.20),

$\displaystyle M_{\eta,I_{2},\alpha,\beta}(z,-1)$
$\displaystyle=\footnotesize\begin{pmatrix}\widetilde{\varphi}_{0,\alpha,\beta}(z,1)&2^{-1}\big{[}\widetilde{\varphi}^{\,\prime}_{0,\alpha,\beta}(z,1)-\widetilde{\vartheta}_{0,\alpha,\beta}(z,1)\big{]}\\\
2^{-1}\big{[}\widetilde{\varphi}^{\,\prime}_{0,\alpha,\beta}(z,1)-\widetilde{\vartheta}_{0,\alpha,\beta}(z,1)\big{]}&-\widetilde{\vartheta}^{\,\prime}_{0,\alpha,\beta}(z,1)\end{pmatrix}$
$\displaystyle\quad\;\times[\widetilde{\varphi}^{\,\prime}_{0,\alpha,\beta}(z,1)+\widetilde{\vartheta}_{0,\alpha,\beta}(z,1)-2\cos(\eta)]^{-1},$
(5.21) $\displaystyle\hskip
18.49428pt\alpha,\beta\in(-1,1),\,\eta\in[0,\pi),\,z\in\rho(T_{\eta,I_{2},\alpha,\beta}).$

The principal result on $\eta$-periodic Green’s function induced $M$-functions
of this section can then be written as follows:

######  Theorem 5.4.

Let $T_{\eta,I_{2},\alpha,\beta}$, be the self-adjoint operator associated
with $\tau_{\alpha,\beta}$ with $\eta$-periodic coupled boundary conditions,
$\eta\in[0,\pi)$, $\alpha,\beta\in(-1,1)$. Then the associated Green’s
function induced $M$-function is of the form

$\displaystyle M_{\eta,I_{2},\alpha,\beta}(z,-1)$
$\displaystyle=\begin{cases}\footnotesize\begin{pmatrix}-\beta^{-1}2^{-\alpha-1}\widetilde{y}_{2,\alpha,\beta,-1}(z,1)&\begin{matrix}2^{-1}\big{[}-\beta^{-1}2^{-\alpha-1}\widetilde{y}^{\,\prime}_{2,\alpha,\beta,-1}(z,1)\\\
\hfill-\widetilde{y}_{1,\alpha,\beta,-1}(z,1)\big{]}\end{matrix}\\\
\begin{matrix}2^{-1}\big{[}-\beta^{-1}2^{-\alpha-1}\widetilde{y}^{\,\prime}_{2,\alpha,\beta,-1}(z,1)\\\
\hfill-\widetilde{y}_{1,\alpha,\beta,-1}(z,1)\big{]}\end{matrix}&-\widetilde{y}^{\,\prime}_{1,\alpha,\beta,-1}(z,1)\end{pmatrix}\\\
\ \
\times[-\beta^{-1}2^{-\alpha-1}\widetilde{y}^{\,\prime}_{2,\alpha,\beta,-1}(z,1)+\widetilde{y}_{1,\alpha,\beta,-1}(z,1)-2\cos(\eta)]^{-1},&\beta\in(-1,0),\\\\[8.53581pt]
\footnotesize\begin{pmatrix}\widetilde{y}_{1,\alpha,0,-1}(z,1)&\begin{matrix}2^{-1}\big{[}2^{-\alpha-1}\widetilde{y}_{2,\alpha,0,-1}(z,1)\\\
\hfill+\widetilde{y}^{\,\prime}_{1,\alpha,0,-1}(z,1)\big{]}\end{matrix}\\\
\begin{matrix}2^{-1}\big{[}2^{-\alpha-1}\widetilde{y}_{2,\alpha,0,-1}(z,1)\\\
\hfill+\widetilde{y}^{\,\prime}_{1,\alpha,0,-1}(z,1)\big{]}\end{matrix}&2^{-\alpha-1}\widetilde{y}^{\,\prime}_{2,\alpha,0,-1}(z,1)\end{pmatrix}\\\
\quad\
\times[\widetilde{y}^{\,\prime}_{1,\alpha,0,-1}(z,1)-2^{-\alpha-1}\widetilde{y}_{2,\alpha,0,-1}(z,1)-2\cos(\eta)]^{-1},&\beta=0,\\\\[8.53581pt]
\footnotesize\begin{pmatrix}\widetilde{y}_{1,\alpha,\beta,-1}(z,1)&\begin{matrix}2^{-1}\big{[}-\beta^{-1}2^{-\alpha-1}\widetilde{y}_{2,\alpha,\beta,-1}(z,1)\\\
\hfill+\widetilde{y}^{\,\prime}_{1,\alpha,\beta,-1}(z,1)\big{]}\end{matrix}\\\
\begin{matrix}2^{-1}\big{[}-\beta^{-1}2^{-\alpha-1}\widetilde{y}_{2,\alpha,\beta,-1}(z,1)\\\
\hfill+\widetilde{y}^{\,\prime}_{1,\alpha,\beta,-1}(z,1)\big{]}\end{matrix}&-\beta^{-1}2^{-\alpha-1}\widetilde{y}^{\,\prime}_{2,\alpha,\beta,-1}(z,1)\end{pmatrix}\\\
\quad\
\times[\widetilde{y}^{\,\prime}_{1,\alpha,\beta,-1}(z,1)+\beta^{-1}2^{-\alpha-1}\widetilde{y}_{2,\alpha,\beta,-1}(z,1)-2\cos(\eta)]^{-1},&\beta\in(0,1);\end{cases}$
$\displaystyle\hskip
142.26378pt\alpha\in(-1,1),\,\eta\in[0,\pi),\,z\in\rho(T_{\eta,I_{2},\alpha,\beta}).$
(5.22)

The poles of $M_{\eta,I_{2},\alpha,\beta}(\,\cdot\,,-1)$ occur precisely at
the eigenvalues of $T_{\eta,I_{2},\alpha,\beta}$.

Hence, utilizing the boundary values given by (C.14)–(C.16) in (5.22) yields
the $2\times 2$ matrix-valued $M$-function
$M_{\eta,I_{2},\alpha,\beta}(\,\cdot\,,-1)$.

The following example provides an explicit illustration of Theorem 5.4 in the
singular case $\alpha,\beta\in(0,1)$:

######  Example 5.5 ($\eta$-periodic extension).

Let $\alpha,\beta\in(0,1)$. Substituting the boundary values (C.14) and (C.15)
into (5.22) yields the $\eta$-periodic $M$-function

$\displaystyle M_{\eta,I_{2},\alpha,\beta}(z,-1)$
$\displaystyle\quad=\footnotesize\begin{pmatrix}\dfrac{-2^{1+\alpha+\beta}\Gamma(1+\alpha)\Gamma(1+\beta)}{\Gamma(a_{\alpha,\beta,\sigma_{\alpha,\beta}(z)})\Gamma(a_{\alpha,\beta,-\sigma_{\alpha,\beta}(z)})}&\begin{matrix}2^{-1}\Bigg{[}\dfrac{-\Gamma(1+\alpha)\Gamma(-\beta)}{\Gamma(a_{\alpha,-\beta,\sigma_{\alpha,\beta}(z)})\Gamma(a_{\alpha,-\beta,-\sigma_{\alpha,\beta}(z)})}\\\
\hfill+\dfrac{\Gamma(1+\beta)\Gamma(-\alpha)}{\Gamma(a_{-\alpha,\beta,\sigma_{\alpha,\beta}(z)})\Gamma(a_{-\alpha,\beta,-\sigma_{\alpha,\beta}(z)})}\Bigg{]}\end{matrix}\\\
\begin{matrix}2^{-1}\Bigg{[}\dfrac{-\Gamma(1+\alpha)\Gamma(-\beta)}{\Gamma(a_{\alpha,-\beta,\sigma_{\alpha,\beta}(z)})\Gamma(a_{\alpha,-\beta,-\sigma_{\alpha,\beta}(z)})}\\\
\hfill+\dfrac{\Gamma(1+\beta)\Gamma(-\alpha)}{\Gamma(a_{-\alpha,\beta,\sigma_{\alpha,\beta}(z)})\Gamma(a_{-\alpha,\beta,-\sigma_{\alpha,\beta}(z)})}\Bigg{]}\end{matrix}&\dfrac{2^{-1-\alpha-\beta}\Gamma(-\beta)\Gamma(-\alpha)}{\Gamma(a_{-\alpha,-\beta,\sigma_{\alpha,\beta}(z)})\Gamma(a_{-\alpha,-\beta,-\sigma_{\alpha,\beta}(z)})}\end{pmatrix}$
$\displaystyle\hskip
28.45274pt\times\Bigg{[}-2\cos(\eta)+\dfrac{\Gamma(1+\beta)\Gamma(-\alpha)}{\Gamma(a_{-\alpha,\beta,\sigma_{\alpha,\beta}(z)})\Gamma(a_{-\alpha,\beta,-\sigma_{\alpha,\beta}(z)})}$
(5.23) $\displaystyle\hskip
96.73918pt+\dfrac{\Gamma(1+\alpha)\Gamma(-\beta)}{\Gamma(a_{\alpha,-\beta,\sigma_{\alpha,\beta}(z)})\Gamma(a_{\alpha,-\beta,-\sigma_{\alpha,\beta}(z)})}\Bigg{]}^{-1},$
$\displaystyle\hskip 98.16191pt\alpha,\beta\in(0,1),\
\eta\in[0,\pi),\,z\in\rho(T_{\eta,I_{2},\alpha,\beta}),$

where we abbreviated

$\displaystyle
a_{\mu,\nu,\pm\sigma}=[1+\mu+\nu\pm\sigma]/2,\quad\mu,\nu,\sigma\in{\mathbb{C}}.$
(5.24)

As a final concrete example of coupled boundary conditions, we now consider
the Krein–von Neumann extension following Example 4.3 in [22]:

######  Example 5.6 (Krein–von Neumann extension).

For $\alpha,\beta\in(-1,1)$, the following five cases are associated with a
strictly positive minimal operator $T_{min,\alpha,\beta}$ and we now provide
the corresponding $R_{K,\alpha,\beta}$ that yield the Krein–von Neumann
extension $T_{0,R_{K},\alpha,\beta}$ of $T_{min,\alpha,\beta}$:

$\displaystyle T_{0,R_{K},\alpha,\beta}f=\tau_{\alpha,\beta}f,$ (5.25)
$\displaystyle
f\in\operatorname{dom}(T_{0,R_{K},\alpha,\beta})=\Bigg{\\{}g\in\operatorname{dom}(T_{max,\alpha,\beta})\,\Bigg{|}\Bigg{(}\begin{matrix}\widetilde{g}(1)\\\
{\widetilde{g}}^{\,\prime}(1)\end{matrix}\Bigg{)}=R_{K,\alpha,\beta}\Bigg{(}\begin{matrix}\widetilde{g}(-1)\\\
{\widetilde{g}}^{\,\prime}(-1)\end{matrix}\Bigg{)}\Bigg{\\}},$ $\displaystyle
R_{K,\alpha,\beta}=\begin{cases}\begin{pmatrix}1&2^{-\alpha-\beta-1}\dfrac{\Gamma(-\alpha)\Gamma(-\beta)}{\Gamma(-\alpha-\beta)}\\\
0&1\end{pmatrix},&\alpha,\beta\in(-1,0),\\\\[19.91692pt]
\begin{pmatrix}-2^{-\alpha-\beta-1}\dfrac{\Gamma(-\alpha)\Gamma(-\beta)}{\Gamma(-\alpha-\beta)}&1\\\
-1&0\end{pmatrix},&\alpha\in(-1,0),\;\beta\in(0,1),\\\\[19.91692pt]
\begin{pmatrix}0&-1\\\
1&2^{-\alpha-\beta-1}\dfrac{\Gamma(-\alpha)\Gamma(-\beta)}{\Gamma(-\alpha-\beta)}\end{pmatrix},&\alpha\in(0,1),\;\beta\in(-1,0),\\\\[19.91692pt]
\begin{pmatrix}0&-1\\\
1&-2^{-\beta-1}[\gamma_{E}+\psi(-\beta)]\end{pmatrix},&\alpha=0,\;\beta\in(-1,0)\\\\[19.91692pt]
\begin{pmatrix}2^{-\alpha-1}[\gamma_{E}+\psi(-\alpha)]&1\\\
-1&0\end{pmatrix},&\alpha\in(-1,0),\;\beta=0,\end{cases}$ (5.26)

where we interpret $1/\Gamma(0)=0$.

For the remaining four cases given by all combinations of $\alpha=0,\
\beta=0,\ \alpha\in(0,1),$ and $\beta\in(0,1)$, we note that [22, Theorem 3.5]
is not applicable as the minimal operator, $T_{\alpha,\beta,min}$, is not
strictly positive. This is easily seen by considering the spectrum explicitly
given in Example 5.3 $(ii),\ (iii),\ (vii)$ and $(viii)$. In particular,
$0\in\sigma(T_{F,\alpha,\beta}),\ \alpha,\beta\in[0,1),$ and hence
$T_{min,\alpha,\beta}\geqslant 0$ is nonnegative, but not strictly positive
when $\alpha,\beta\in[0,1)$. We omit further details at this point.

Finally, from (5.18) and (5.26), one can compute the Green’s function induced
$M$-function for the Krein extension. We provide the explicit form for
$\alpha,\beta\in(-1,0)$ next, noting that the remaining cases can be treated
analogously using the appropriate generalized boundary values at each endpoint
corresponding to the ranges chosen for $\alpha$ and $\beta$. Abbreviating
$a_{\mu,\nu,\pm\sigma}=[1+\mu+\nu\pm\sigma]/2$,
$\mu,\nu,\sigma\in{\mathbb{C}}$ as in (5.24), one finds

$\displaystyle M_{0,R_{K},\alpha,\beta}(z,-1)$
$\displaystyle\quad=\footnotesize\begin{pmatrix}\begin{matrix}-\beta^{-1}2^{-\alpha-1}\widetilde{y}_{2,\alpha,\beta,-1}(z,1)\\\
\quad\end{matrix}&\begin{matrix}2^{-1}\Bigg{[}-\beta^{-1}2^{-\alpha-1}\widetilde{y}_{2,\alpha,\beta,-1}^{\,\prime}(z,1)\hfill\\\
+2^{-2\alpha-\beta-2}\dfrac{\Gamma(-\alpha)\Gamma(-\beta)}{\beta\Gamma(-\alpha-\beta)}\widetilde{y}_{2,\alpha,\beta,-1}(z,1)\\\
\hfill-\widetilde{y}_{1,\alpha,\beta,-1}(z,1)\Bigg{]}\end{matrix}\\\\[11.38109pt]
\begin{matrix}2^{-1}\Bigg{[}-\beta^{-1}2^{-\alpha-1}\widetilde{y}_{2,\alpha,\beta,-1}^{\,\prime}(z,1)\hfill\\\
+2^{-2\alpha-\beta-2}\dfrac{\Gamma(-\alpha)\Gamma(-\beta)}{\beta\Gamma(-\alpha-\beta)}\widetilde{y}_{2,\alpha,\beta,-1}(z,1)\\\
\hfill-\widetilde{y}_{1,\alpha,\beta,-1}(z,1)\Bigg{]}\end{matrix}&\begin{matrix}2^{-\alpha-\beta-1}\dfrac{\Gamma(-\alpha)\Gamma(-\beta)}{\Gamma(-\alpha-\beta)}\widetilde{y}_{1,\alpha,\beta,-1}(z,1)\\\
\hfill-\widetilde{y}_{1,\alpha,\beta,-1}^{\,\prime}(z,1)\\\
\quad\end{matrix}\end{pmatrix}$
$\displaystyle\qquad\quad\times\Bigg{[}\beta^{-1}2^{-\alpha-1}\widetilde{y}_{2,\alpha,\beta,-1}^{\,\prime}(z,1)-\widetilde{y}_{1,\alpha,\beta,-1}(z,1)$
$\displaystyle\hskip
85.35826pt+2^{-\alpha-\beta-1}\dfrac{\Gamma(-\alpha)\Gamma(-\beta)}{\Gamma(-\alpha-\beta)}\widetilde{y}_{1,\alpha,\beta,-1}^{\,\prime}(z,1)+2\Bigg{]}^{-1}$
$\displaystyle\quad=\begin{pmatrix}M_{1,1}&M_{1,2}\\\
M_{2,1}&M_{2,2}\end{pmatrix}$
$\displaystyle\qquad\quad\times\Bigg{[}\dfrac{\Gamma(1+\alpha)\Gamma(1-\beta)}{\beta\Gamma(a_{\alpha,-\beta,\sigma_{\alpha,\beta}(z)})\Gamma(a_{\alpha,-\beta,-\sigma_{\alpha,\beta}(z)})}-\dfrac{\Gamma(1+\beta)\Gamma(-\alpha)}{\Gamma(a_{-\alpha,\beta,\sigma_{\alpha,\beta}(z)})\Gamma(a_{-\alpha,\beta,-\sigma_{\alpha,\beta}(z)})}$
$\displaystyle\hskip
128.0374pt+\dfrac{\Gamma(-\alpha)\Gamma(-\beta)\Gamma(1+\alpha)\Gamma(1+\beta)}{\Gamma(-\alpha-\beta)\Gamma(a_{\alpha,\beta,\sigma_{\alpha,\beta}(z)})\Gamma(a_{\alpha,\beta,-\sigma_{\alpha,\beta}(z)})}+2\Bigg{]}^{-1},$
$\displaystyle\hskip 170.71652pt\alpha,\beta\in(-1,0),\
z\in\rho(T_{0,R_{K},\alpha,\beta}),$ (5.27)

where

$\displaystyle\begin{split}M_{1,1}&=\dfrac{2^{-\alpha-\beta-1}\Gamma(-\beta)\Gamma(-\alpha)}{\Gamma(a_{-\alpha,-\beta,\sigma_{\alpha,\beta}(z)})\Gamma(a_{-\alpha,-\beta,-\sigma_{\alpha,\beta}(z)})},\\\\[5.69054pt]
M_{1,2}&=M_{2,1}=2^{-1}\Bigg{[}\dfrac{\Gamma(1+\alpha)\Gamma(-\beta)}{\Gamma(a_{\alpha,-\beta,\sigma_{\alpha,\beta}(z)})\Gamma(a_{\alpha,-\beta,-\sigma_{\alpha,\beta}(z)})}\\\\[2.84526pt]
&\hskip
65.44142pt-\dfrac{2^{-2\alpha-2\beta-2}(\Gamma(-\alpha)\Gamma(-\beta))^{2}}{\Gamma(-\alpha-\beta)\Gamma(a_{-\alpha,-\beta,\sigma_{\alpha,\beta}(z)})\Gamma(a_{-\alpha,-\beta,-\sigma_{\alpha,\beta}(z)})}\\\\[2.84526pt]
&\hskip
65.44142pt-\dfrac{\Gamma(1+\beta)\Gamma(-\alpha)}{\Gamma(a_{-\alpha,\beta,\sigma_{\alpha,\beta}(z)})\Gamma(a_{-\alpha,\beta,-\sigma_{\alpha,\beta}(z)})}\Bigg{]},\\\\[5.69054pt]
M_{2,2}&=\dfrac{2^{-\alpha-\beta-1}\Gamma(-\alpha)^{2}\Gamma(-\beta)\Gamma(1+\beta)}{\Gamma(-\alpha-\beta)\Gamma(a_{-\alpha,\beta,\sigma_{\alpha,\beta}(z)})\Gamma(a_{-\alpha,\beta,-\sigma_{\alpha,\beta}(z)})}\\\\[2.84526pt]
&\quad\,-\dfrac{2^{1+\alpha+\beta}\Gamma(1+\alpha)\Gamma(1+\beta)}{\Gamma(a_{\alpha,\beta,\sigma_{\alpha,\beta}(z)})\Gamma(a_{\alpha,\beta,-\sigma_{\alpha,\beta}(z)})},\\\
&\hskip 18.49428pt\alpha,\beta\in(-1,0),\
z\in\rho(T_{0,R_{K},\alpha,\beta}).\end{split}$ (5.28)

##  6\. Precisely One Interval Endpoint in the Limit Point Case

In this section we determine the Weyl–Titchmarsh–Kodaira $m$-function in all
situations where precisely one interval endpoint is in the limit point case.
We will focus on the case when $\alpha\in(-\infty,-1]$ or
$\alpha\in[1,\infty)$, so that the right endpoint $x=1$ represents the limit
point case. The converse situation can be obtained by reflection with respect
to the origin (i.e., considering the transform $(-1,1)\ni
x\mapsto-x\in(-1,1)$).

We start with the case $\alpha\in[1,\infty)$, $\beta\in(-1,1)$ so that the
left endpoint $-1$ is regular or limit circle and the right endpoint $1$ is in
the limit point case. In accordance with Section 3, we single out the left
endpoint $x=-1$ and determine the Weyl–Titchmarsh–Kodaira solution and
$m$-function via

$\displaystyle\begin{split}\psi_{\gamma,\alpha,\beta}(z,\,\cdot\,)=\vartheta_{\gamma,\alpha,\beta}(z,\,\cdot\,)+m_{\gamma,\alpha,\beta}(z)\varphi_{\gamma,\alpha,\beta}(z,\,\cdot\,)\in
L^{2}((c,1);r_{\alpha,\beta}dx),&\\\
z\in\rho(T_{\gamma,\alpha,\beta}),\;\alpha\in[1,\infty),\;\beta\in(-1,1),\;c\in(-1,1).&\end{split}$
(6.1)

To simplify matters we next focus on the Friedrichs extension, $\gamma=0$,
noting that for $\gamma\in[0,\pi)$, the general
$m_{\gamma,\alpha,\beta}$-function can be found by employing the linear
fractional transformation (LABEL:3.23), choosing $\gamma_{1}=0$ and
$\gamma_{2}=\gamma$, fixing the limit point endpoint $x=1$ (dropping the
parameter $\delta\in[0,\pi)$ as there is no boundary condition to be imposed
at the limit point endpoint $x=1$). We provide a summary at the end of this
section in Theorem 6.1.
$\mathbf{(I)}$ The Case $\alpha\in[1,\infty)$ and $\beta\in(-1,0)$:
Consider the Friedrichs extension $\gamma=0$,
$T_{F,\alpha,\beta}=T_{0,\alpha,\beta}$, $\alpha\in[1,\infty)$,
$\beta\in(-1,0)$. Then

$\displaystyle\varphi_{0,\alpha,\beta}(z,x)=-2^{-\alpha-1}\beta^{-1}y_{2,\alpha,\beta,-1}(z,x),\quad\vartheta_{0,\alpha,\beta}(z,x)=y_{1,\alpha,\beta,-1}(z,x),$
(6.2)

and the requirement $\psi_{0,\alpha,\beta}(z,\,\cdot\,)\in
L^{2}\big{(}(c,1);r_{\alpha,\beta}dx\big{)}$, $c\in(-1,1)$,
$z\in{\mathbb{C}}\backslash{\mathbb{R}}$, can be recast as

$\big{[}y_{1,\alpha,\beta,-1}(z,\,\cdot\,)-2^{-\alpha-1}\beta^{-1}y_{2,\alpha,\beta,-1}(z,\,\cdot\,)m_{0,\alpha,\beta}(z)\big{]}\in
L^{2}\big{(}(c,1);r_{\alpha,\beta}dx\big{)}.$ (6.3)

Using the limiting behavior of $y_{j,\alpha,\beta,-1}(z,x),\ j=1,2$, near
$x=1$ in (C.1), (C.4), (C.5), (C.8), and observing that only the part
$(1-x)^{\alpha}$ in $r_{\alpha,\beta}(x)$ matters for $x$ in a neighborhood of
$1$, one concludes that the square integrability in (6.3) reduces to
integrability of

$\displaystyle(1-x)^{-\alpha}\bigg{[}\frac{2^{\alpha}\Gamma(1+\beta)\Gamma(\alpha)}{\Gamma([1+\alpha+\beta+\sigma_{\alpha,\beta}(z)]/2)\Gamma([1+\alpha+\beta-\sigma_{\alpha,\beta}(z)]/2)}$
(6.4) $\displaystyle\hskip
48.36958pt-\frac{2^{-\beta-1}\beta^{-1}\Gamma(1-\beta)\Gamma(\alpha)}{\Gamma([1+\alpha-\beta+\sigma_{\alpha,\beta}(z)]/2)\Gamma([1+\alpha-\beta-\sigma_{\alpha,\beta}(z)]/2)}m_{0,\alpha,\beta}(z)\bigg{]}^{2},$

near $x=1$ which, in turn, happens if and only if the expression in brackets
vanishes. Thus,

$\displaystyle
m_{0,\alpha\,\beta}(z)=2^{1+\alpha+\beta}\beta\frac{\Gamma(1+\beta)}{\Gamma(1-\beta)}$
$\displaystyle\hskip
51.21504pt\times\frac{\Gamma([1+\alpha-\beta+\sigma_{\alpha,\beta}(z)]/2)\Gamma([1+\alpha-\beta-\sigma_{\alpha,\beta}(z)]/2)}{\Gamma([1+\alpha+\beta+\sigma_{\alpha,\beta}(z)]/2)\Gamma([1+\alpha+\beta-\sigma_{\alpha,\beta}(z)]/2)},$
(6.5) $\displaystyle\hskip
133.72786ptz\in\rho(T_{F,\alpha,\beta}),\;\alpha\in[1,\infty),\;\beta\in(-1,0),$
$\displaystyle\sigma(T_{F,\alpha,\beta})=\\{(n-\beta)(n+1+\alpha)\\}_{n\in{\mathbb{N}}_{0}},\quad\alpha\in[1,\infty),\;\beta\in(-1,0).$

$\mathbf{(II)}$ The Case $\alpha\in[1,\infty)$ and $\beta=0$ :
Consider the Friedrichs extension $\gamma=0$, $T_{F,\alpha,0}=T_{0,\alpha,0}$,
$\alpha\in[1,\infty)$, $\beta=0$. Then

$\displaystyle\varphi_{0,\alpha,0}(z,x)=y_{1,\alpha,0,-1}(z,x),\quad\vartheta_{0,\alpha,0}(z,x)=-2^{-\alpha-1}y_{2,\alpha,0,-1}(z,x),$
(6.6)

and the requirement $\psi_{0,\alpha,0}(z,\,\cdot\,)\in
L^{2}\big{(}(c,1);r_{\alpha,\beta}dx\big{)}$, $c\in(-1,1)$,
$z\in{\mathbb{C}}\backslash{\mathbb{R}}$, can again be recast as

$\big{[}-2^{-\alpha-1}y_{2,\alpha,0,-1}(z,\,\cdot\,)+y_{1,\alpha,0,-1}(z,\,\cdot\,)m_{0,\alpha,0}(z)\big{]}\in
L^{2}\big{(}(c,1);r_{\alpha,\beta}dx\big{)}.$ (6.7)

Using the limiting behavior $y_{j,\alpha,\beta,-1}(z,x),\ j=1,2$, near $x=1$
in (C.1), (C.4), (C.9), and (C.12), one concludes that the square
integrability in (6.7) reduces to integrability of

$\displaystyle\begin{split}&(1-x)^{-\alpha}\bigg{[}\frac{2\gamma_{E}+\psi([1+\alpha+\sigma_{\alpha,0}(z)]/2)+\psi([1+\alpha-\sigma_{\alpha,0}(z)]/2)}{2\Gamma([1+\alpha+\sigma_{\alpha,0}(z)]/2)\Gamma([1+\alpha-\sigma_{\alpha,0}(z)]/2)}\Gamma(\alpha)\\\
&\hskip
45.52458pt+\frac{2^{\alpha}\Gamma(\alpha)}{\Gamma([1+\alpha+\sigma_{\alpha,0}(z)]/2)\Gamma([1+\alpha-\sigma_{\alpha,0}(z)]/2)}m_{0,\alpha,0}(z)\bigg{]}^{2},\end{split}$
(6.8)

near $x=1$, which happens if and only if the expression in brackets vanishes.
Thus,

$\displaystyle
m_{0,\alpha,0}(z)=-2^{-\alpha-1}\\{2\gamma_{E}+\psi([1+\alpha+\sigma_{\alpha,0}(z)]/2)+\psi([1+\alpha-\sigma_{\alpha,0}(z)]/2)\\},$
$\displaystyle\hskip
170.71652ptz\in\rho(T_{F,\alpha,0}),\;\alpha\in[1,\infty),\;\beta=0,$ (6.9)
$\displaystyle\sigma(T_{F,\alpha,0})=\\{n(n+1+\alpha)\\}_{n\in{\mathbb{N}}_{0}},\quad\alpha\in[1,\infty),\;\beta=0.$

$\mathbf{(III)}$ The Case $\alpha\in[1,\infty)$ and $\beta\in(0,1)$:
Finally, consider the Friedrichs extension $\gamma=0$,
$T_{F,\alpha,\beta}=T_{0,\alpha,\beta}$, $\alpha\in[1,\infty)$,
$\beta\in(0,1)$. Then

$\displaystyle\varphi_{0,\alpha,\beta}(z,x)=y_{1,\alpha,\beta,-1}(z,x),\quad\vartheta_{0,\alpha,\beta}(z,x)=2^{-\alpha-1}\beta^{-1}y_{2,\alpha,\beta,-1}(z,x),$
(6.10)

and the requirement $\psi_{0,\alpha,\beta}(z,\,\cdot\,)\in
L^{2}\big{(}(c,1);r_{\alpha,\beta}dx\big{)}$, $c\in(-1,1)$,
$z\in{\mathbb{C}}\backslash{\mathbb{R}}$, can again be recast as

$\displaystyle\big{[}2^{-\alpha-1}\beta^{-1}y_{2,\alpha,\beta,-1}(z,\,\cdot\,)+y_{1,\alpha,\beta,-1}(z,\,\cdot\,)m_{0,\alpha,\beta}(z)\big{]}\in
L^{2}\big{(}(c,1);r_{\alpha,\beta}dx\big{)}.$ (6.11)

Once again, using the limiting behavior of $y_{j,\alpha,\beta,-1}(z,x),\
j=1,2$, near $x=1$ in (C.1), (C.4), (C.5), (C.8), one concludes that the
square integrability in (6.11) reduces to integrability of

$\displaystyle(1-x)^{-\alpha}\bigg{[}\frac{2^{-\beta-1}\beta^{-1}\Gamma(1-\beta)\Gamma(\alpha)}{\Gamma([1+\alpha-\beta+\sigma_{\alpha,\beta}(z)]/2)\Gamma([1+\alpha-\beta-\sigma_{\alpha,\beta}(z)]/2)}$
(6.12) $\displaystyle\hskip
45.52458pt+\frac{2^{\alpha}\Gamma(1+\beta)\Gamma(\alpha)}{\Gamma([1+\alpha+\beta+\sigma_{\alpha,\beta}(z)]/2)\Gamma([1+\alpha+\beta-\sigma_{\alpha,\beta}(z)]/2)}m_{0,\alpha,\beta}(z)\bigg{]}^{2},$

near $x=1$, which happens if and only if the expression in brackets is zero.
Thus,

$\displaystyle m_{0,\alpha\,\beta}(z)$
$\displaystyle=\beta^{-1}2^{-1-\alpha-\beta}\frac{-\Gamma(1-\beta)}{\Gamma(1+\beta)}$
$\displaystyle\quad\times\frac{\Gamma([1+\alpha+\beta+\sigma_{\alpha,\beta}(z)]/2)\Gamma([1+\alpha+\beta-\sigma_{\alpha,\beta}(z)]/2)}{\Gamma([1+\alpha-\beta+\sigma_{\alpha,\beta}(z)]/2)\Gamma((1+\alpha-\beta-\sigma_{\alpha,\beta}(z))/2)},$
(6.13) $\displaystyle\hskip
99.58464ptz\in\rho(T_{F,\alpha,\beta}),\;\alpha\in[1,\infty),\;\beta\in(0,1),$
$\displaystyle\sigma(T_{F,\alpha,\beta})$
$\displaystyle=\\{n(n+1+\alpha+\beta)\\}_{n\in{\mathbb{N}}_{0}},\quad\alpha\in[1,\infty),\;\beta\in(0,1).$

Next, we turn to the remaining case $\alpha\in(-\infty,-1]$, $\beta\in(-1,1)$,
and determine the Weyl–Titchmarsh–Kodaira solution and $m$-function in
accordance with Section 3 via

$\displaystyle\begin{split}\psi_{\gamma,\alpha,\beta}(z,\,\cdot\,)=\vartheta_{\gamma,\alpha,\beta}(z,\,\cdot\,)+m_{\gamma,\alpha,\beta}(z)\varphi_{\gamma,\alpha,\beta}(z,\,\cdot\,)\in
L^{2}((c,1);r_{\alpha,\beta}dx),&\\\
z\in\rho(T_{\gamma,\alpha,\beta}),\;\alpha\in(-\infty.-1],\;\beta\in(-1,1),\;c\in(-1,1).&\end{split}$
(6.14)

To simplify matters we once more focus on the Friedrichs extension, $\gamma=0$
since for $\gamma\in[0,\pi)$ the general $m_{\gamma,\alpha,\beta}$-function
can again be determined by employing the linear fractional transformation
(LABEL:3.23).
$\mathbf{(IV)}$ The Case $\alpha\in(-\infty,-1]$ and $\beta\in(-1,0)$:
Consider the Friedrichs extension $\gamma=0$,
$T_{F,\alpha,\beta}=T_{0,\alpha,\beta}$, $\alpha\in(-\infty,-1]$,
$\beta\in(-1,0)$. Then

$\displaystyle\varphi_{0,\alpha,\beta}(z,x)=-2^{-\alpha-1}\beta^{-1}y_{2,\alpha,\beta,-1}(z,x),\quad\vartheta_{0,\alpha,\beta}(z,x)=y_{1,\alpha,\beta,-1}(z,x),$
(6.15)

and the requirement $\psi_{0,\alpha,\beta}(z,\,\cdot\,)\in
L^{2}\big{(}(c,1);r_{\alpha,\beta}dx\big{)}$, $c\in(-1,1)$,
$z\in{\mathbb{C}}\backslash{\mathbb{R}}$, can be recast as

$\displaystyle\big{[}y_{1,\alpha,\beta,-1}(z,\,\cdot\,)-2^{-\alpha-1}\beta^{-1}y_{2,\alpha,\beta,-1}(z,\,\cdot\,)m_{0,\alpha,\beta}(z)\big{]}\in
L^{2}\big{(}(c,1);r_{\alpha,\beta}dx\big{)}.$ (6.16)

Using again the limiting behavior of $y_{j,\alpha,\beta,-1}(z,x),\ j=1,2$,
near $x=1$ in (C.1), (C.2), (C.5), (C.6), one concludes that the square
integrability in (6.16) reduces to the integrability of

$\displaystyle(1-x)^{\alpha}\bigg{[}\frac{\Gamma(1+\beta)\Gamma(-\alpha)}{\Gamma([1+\beta-\alpha+\sigma_{\alpha,\beta}(z)]/2)\Gamma([1+\beta-\alpha-\sigma_{\alpha,\beta}(z)]/2)}$
(6.17) $\displaystyle\hskip
39.83368pt-\frac{2^{-\alpha-\beta-1}\beta^{-1}\Gamma(1-\beta)\Gamma(-\alpha)}{\Gamma([1-\alpha-\beta+\sigma_{\alpha,\beta}(z)]/2)\Gamma([1-\alpha-\beta-\sigma_{\alpha,\beta}(z)]/2)}m_{0,\alpha,\beta}(z)\bigg{]}^{2},$

near $x=1$, which happens if and only if the expression in brackets is zero.
Thus,

$\displaystyle m_{0,\alpha\,\beta}(z)$
$\displaystyle=2^{1+\alpha+\beta}\beta\frac{\Gamma(1+\beta)}{\Gamma(1-\beta)}$
$\displaystyle\quad\times\frac{\Gamma([1-\alpha-\beta+\sigma_{\alpha,\beta}(z)]/2)\Gamma([1-\alpha-\beta-\sigma_{\alpha,\beta}(z)]/2)}{\Gamma([1+\beta-\alpha+\sigma_{\alpha,\beta}(z)]/2)\Gamma([1+\beta-\alpha-\sigma_{\alpha,\beta}(z)]/2)},$
(6.18) $\displaystyle\hskip 72.55461pt\
z\in\rho(T_{F,\alpha,\beta}),\;\alpha\in(-\infty,-1],\;\beta\in(-1,0),$
$\displaystyle\sigma(T_{F,\alpha,\beta})$
$\displaystyle=\\{(n-\alpha-\beta)(n+1)\\}_{n\in{\mathbb{N}}_{0}},\quad\alpha\in(-\infty,-1],\;\beta\in(-1,0).$

$\mathbf{(V)}$ The Case $\alpha\in(-\infty,-1]$ and $\beta=0$ :
Consider the Friedrichs extension $\gamma=0$, $T_{F,\alpha,0}=T_{0,\alpha,0}$,
$\alpha\in(-\infty,-1]$, $\beta=0$. Then

$\displaystyle\varphi_{0,\alpha,0}(z,x)=y_{1,\alpha,0,-1}(z,x),\quad\vartheta_{0,\alpha,0}(z,x)=-2^{-\alpha-1}y_{2,\alpha,0,-1}(z,x),$
(6.19)

and the requirement $\psi_{0,\alpha,0}(z,\,\cdot\,)\in
L^{2}\big{(}(c,1);r_{\alpha,\beta}dx\big{)}$, $c\in(-1,1)$,
$z\in{\mathbb{C}}\backslash{\mathbb{R}}$, can be recast as

$\displaystyle\big{[}-2^{-\alpha-1}y_{2,\alpha,0,-1}(z,\,\cdot\,)+y_{1,\alpha,0,-1}(z,\,\cdot\,)m_{0,\alpha,0}(z)\big{]}\in
L^{2}\big{(}(c,1);r_{\alpha,\beta}dx\big{)}.$ (6.20)

Using again the limiting behavior of $y_{j,\alpha,\beta,-1}(z,x),\ j=1,2$,
near $x=1$ in (C.1), (C.2), (C.9), and (C.10), one concludes that the square
integrability in (6.20) reduces to integrability of

$\displaystyle\begin{split}&(1-x)^{\alpha}\bigg{[}\frac{(2\gamma_{E}+\psi([1-\alpha+\sigma_{\alpha,0}(z)]/2)+\psi([1-\alpha-\sigma_{\alpha,0}(z)]/2))\Gamma(-\alpha)}{2^{\alpha+1}\Gamma([1-\alpha+\sigma_{\alpha,0}(z)]/2)\Gamma([1-\alpha-\sigma_{\alpha,0}(z)]/2)}\\\
&\hskip
39.83368pt+\frac{\Gamma(-\alpha)}{\Gamma([1-\alpha+\sigma_{\alpha,0}(z)]/2)\Gamma([1-\alpha-\sigma_{\alpha,0}(z)]/2)}m_{0,\alpha,0}(z)\bigg{]}^{2},\end{split}$
(6.21)

near $x=1$, which happens if and only if the expression in brackets is zero.
Thus,

$\displaystyle
m_{0,\alpha,0}(z)=-2^{-\alpha-1}\\{2\gamma_{E}+\psi([1-\alpha+\sigma_{\alpha,0}(z)]/2)+\psi([1-\alpha-\sigma_{\alpha,0}(z)]/2)\\},$
$\displaystyle\hskip
156.49014ptz\in\rho(T_{F,\alpha,0}),\;\alpha\in(-\infty,-1],\;\beta=0,$ (6.22)
$\displaystyle\sigma(T_{F,\alpha,0})=\\{(n-\alpha)(n+1)\\}_{n\in{\mathbb{N}}_{0}},\quad\alpha\in(-\infty,-1],\;\beta=0.$

$\mathbf{(VI)}$ The Case $\alpha\in(-\infty,-1]$ and $\beta\in(0,1)$:
Finally, consider the Friedrichs extension $\gamma=0$,
$T_{F,\alpha,\beta}=T_{0,\alpha,\beta}$, $\alpha\in(-\infty,-1]$,
$\beta\in(0,1)$. Then

$\displaystyle\varphi_{0,\alpha,\beta}(z,x)=y_{1,\alpha,\beta,-1}(z,x),\quad\vartheta_{0,\alpha,\beta}(z,x)=2^{-\alpha-1}\beta^{-1}y_{2,\alpha,\beta,-1}(z,x),$
(6.23)

and the requirement $\psi_{0,\alpha,\beta}(z,\,\cdot\,)\in
L^{2}\big{(}(c,1);r_{\alpha,\beta}dx\big{)}$, $c\in(-1,1)$,
$z\in{\mathbb{C}}\backslash{\mathbb{R}}$, can be recast as

$\displaystyle\big{[}2^{-\alpha-1}\beta^{-1}y_{2,\alpha,\beta,-1}(z,\,\cdot\,)+y_{1,\alpha,\beta,-1}(z,\,\cdot\,)m_{0,\alpha,\beta}(z)\big{]}\in
L^{2}\big{(}(c,1);r_{\alpha,\beta}dx\big{)}.$ (6.24)

Once again using the limiting behavior near $x=1$ of
$y_{j,\alpha,\beta,-1}(z,x),\ j=1,2,$ found in (C.1), (C.2), (C.5), (C.6) one
concludes that the square integrability in (6.24) reduces to integrability of

$\displaystyle\begin{split}&(1-x)^{\alpha}\bigg{[}\frac{2^{-\alpha-\beta-1}\beta^{-1}\Gamma(1-\beta)\Gamma(-\alpha)}{\Gamma([1-\alpha-\beta+\sigma_{\alpha,\beta}(z)]/2)\Gamma([1-\alpha-\beta-\sigma_{\alpha,\beta}(z)]/2)}\\\
&\quad+\frac{\Gamma(1+\beta)\Gamma(-\alpha)}{\Gamma([1+\beta-\alpha+\sigma_{\alpha,\beta}(z)]/2)\Gamma([1+\beta-\alpha-\sigma_{\alpha,\beta}(z)]/2)}m_{0,\alpha,\beta}(z)\bigg{]}^{2},\end{split}$
(6.25)

near $x=1$, which happens if and only if the expression in brackets is zero.
Thus,

$\displaystyle m_{0,\alpha\,\beta}(z)$
$\displaystyle=-\beta^{-1}2^{-1-\alpha-\beta}\frac{\Gamma(1-\beta)}{\Gamma(1+\beta)}$
$\displaystyle\quad\times\frac{\Gamma([1+\beta-\alpha+\sigma_{\alpha,\beta}(z)]/2)\Gamma([1+\beta-\alpha-\sigma_{\alpha,\beta}(z)]/2)}{\Gamma([1-\alpha-\beta+\sigma_{\alpha,\beta}(z)]/2)\Gamma([1-\alpha-\beta-\sigma_{\alpha,\beta}(z)]/2)},$
(6.26) $\displaystyle\hskip
83.93553ptz\in\rho(T_{F,\alpha,\beta}),\;\alpha\in(-\infty,-1],\;\beta\in(0,1),$
$\displaystyle\sigma(T_{F,\alpha,\beta})$
$\displaystyle=\\{(n-\alpha)(n+1+\beta)\\}_{n\in{\mathbb{N}}_{0}},\quad\alpha\in(-\infty,-1],\;\beta\in(0,1).$

We briefly summarize the principal result of this section as follows:

######  Theorem 6.1.

Let $\gamma\in[0,\pi)$, $\alpha\in(-\infty,-1]\cup[1,\infty)$ and
$\beta\in(-1,1)$. Then the Weyl–Titchmarsh–Kodaira $m$-function
$m_{\gamma,\alpha,\beta}$ associated with $T_{\gamma,\alpha,\beta}$ is given
by

$\displaystyle
m_{\gamma,\alpha,\beta}(z)=\frac{-\sin(\gamma)+\cos(\gamma)m_{0,\alpha,\beta}(z)}{\cos(\gamma)+\sin(\gamma)m_{0,\alpha,\beta}(z)},\quad
z\in\rho(T_{\gamma,\alpha,\beta}),$ (6.27)

where $m_{0,\alpha,\beta}$ is given by (6.5), (6.9), (6.13), (6.18), (6.22),
and (6.26), respectively. The $($necessarily simple $)$ poles of
$m_{\gamma,\alpha,\beta}$ occur precisely at the $($necessarily simple $)$
eigenvalues of $T_{\gamma,\alpha,\beta}$.

##  7\. Both Endpoints in the Limit Point Case

In this section we treat the remaining case where $\tau_{\alpha,\beta}$ is in
the limit point case at both interval endpoints $\pm 1$, that is, where
$\alpha,\beta\in(-\infty,-1]\cup[1,\infty)$.

We start by recalling a number of facts implied by the fact that
$\tau_{\alpha,\beta}$ is in the limit point case at $\pm 1$:

$\displaystyle\psi_{\alpha,\beta}(z,\,\cdot\,)$
$\displaystyle=C_{\alpha,\beta}(z,x_{0})\psi_{0,+,\alpha,\beta}(z,\,\cdot\,,x_{0})$
(7.1)
$\displaystyle=\vartheta_{\alpha,\beta}(z,\,\cdot\,)+m_{\alpha,\beta}(z)\varphi_{\alpha,\beta}(z,\,\cdot\,)\in
L^{2}((c,1);r_{\alpha,\beta}dx),\quad c\in(-1,1),$

where

$\varphi_{\alpha,\beta}(z,\,\cdot\,)=C_{\alpha,\beta}(z)\psi_{0,-,\alpha,\beta}(z,\,\cdot\,)\in
L^{2}((-1,d);r_{\alpha,\beta}dx),\quad d\in(-1,1).$ (7.2)

Thus, (C.1)–(C.12), (C.17)–(C.20) imply

$\displaystyle\varphi_{\alpha,\beta}(z,x)=c_{\alpha,\beta}(z)\begin{cases}y_{1,\alpha,\beta,-1}(z,x),&\beta\in[1,\infty),\\\
y_{2,\alpha,\beta,-1}(z,x),&\beta\in(-\infty,-1],\end{cases}\quad x\in(-1,1),$
(7.3) $\displaystyle\hskip
36.98866pt\underset{x\downarrow-1}{=}c_{\alpha,\beta}(z)\begin{cases}1+O(1+x),&\beta\in[1,\infty),\\\
(1+x)^{-\beta}[1+O(1+x)],&\beta\in(-\infty,-1],\end{cases}$ (7.4)
$\displaystyle\psi_{0,+,\alpha,\beta}(z,\,\cdot\,,x_{0})=c_{\alpha,\beta}(z,x_{0})\begin{cases}y_{1,\alpha,\beta,1}(z,x),&\alpha\in[1,\infty),\\\
y_{2,\alpha,\beta,1}(z,x),&\alpha\in(-\infty,-1],\end{cases}$ (7.5)

with
$C_{\alpha,\beta}(\,\cdot\,,x_{0}),C_{\alpha,\beta}(\,\cdot\,),c_{\alpha,\beta}(\,\cdot\,),c_{\alpha,\beta}(\,\cdot\,,x_{0})\in{\mathbb{C}}\backslash\\{0\\}$,
appropriate coefficients.

Since $\vartheta_{\alpha,\beta}(z,\,\cdot\,)$ must satisfy
$W(\vartheta_{\alpha,\beta}(z,\,\cdot\,),\varphi_{\alpha,\beta}(z,\,\cdot\,))=1$,
we conveniently choose $\vartheta_{\alpha,\beta}(z,\,\cdot\,)$ (cf. Remark
3.2) such that

$\vartheta_{\alpha,\beta}(z,x)=d_{\alpha,\beta}(z)\begin{cases}(1+x)^{-\beta}[1+O(1+x)],&\beta\in[1,\infty),\\\
1+O(1+x),&\beta\in(-\infty,-1],\end{cases}$ (7.6)

with $d_{\alpha,\beta}(\,\cdot\,)\in{\mathbb{C}}\backslash\\{0\\}$ an
appropriate coefficient.

Our strategy to find (a multiple of) $m_{\alpha,\beta}(\,\cdot\,)$ is now
rather simple: We will search for “the part of
$\varphi_{\alpha,\beta}(z,\,\cdot\,)$ in
$\psi_{0,+,\alpha,\beta}(z,\,\cdot\,,x_{0})$,” as the former is multiplied by
$m_{\alpha,\beta}(\,\cdot\,)$ according to (7.1). In this context we emphasize
once more that since $\tau_{\alpha,\beta}$ is in the limit point case at $\pm
1$, $m_{\alpha,\beta}(\,\cdot\,)$ is nonunique and can only be characterized
up to inessential $z$-dependent multiples as well as an additive entire term
as discussed in Section 3 (see again Remark 3.2). Since
$\alpha,\beta\in(-\infty,-1]\cup[1,\infty)$, this requires to distinguish the
following eight cases. We once again provide a summary at the end of this
section in Theorem 7.1 and once more employ the abbreviation (5.24),
$a_{\mu,\nu,\pm\sigma}=[1+\mu+\nu\pm\sigma]/2$,
$\mu,\nu,\sigma\in{\mathbb{C}}$.
$\mathbf{(I)}$ The case $\alpha\in(-\infty,-1]$,
$\beta\in(-\infty,-1]\backslash(-{\mathbb{N}})$:
Then an application of (C.17)–(C.20) yields

$\displaystyle\psi_{0,+,\alpha,\beta}(z,\,\cdot\,,x_{0})\underset{x\downarrow-1}{\sim}y_{2,\alpha,\beta,1}(z,\,\cdot\,)$
$\displaystyle\quad\underset{x\downarrow-1}{\sim}\frac{2^{-\alpha}\Gamma(1-\alpha)\Gamma(-\beta)}{\Gamma(a_{-\alpha,-\beta,\sigma_{\alpha,\beta}(z)})\Gamma(a_{-\alpha,-\beta,-\sigma_{\alpha,\beta}(z)})}\bigg{\\{}y_{1,\alpha,\beta,-1}(z,\,\cdot\,)$
(7.7) $\displaystyle\hskip
34.14322pt+\frac{2^{\beta}[\Gamma(\beta)/\Gamma(-\beta)]\Gamma(a_{-\alpha,-\beta,\sigma_{\alpha,\beta}(z)})\Gamma(a_{-\alpha,-\beta,-\sigma_{\alpha,\beta}(z)})}{\Gamma(a_{-\alpha,\beta,\sigma_{\alpha,\beta}(z)})\Gamma(a_{-\alpha,\beta,-\sigma_{\alpha,\beta}(z)})}\underbrace{y_{2,\alpha,\beta,-1}(z,\,\cdot\,)}_{\sim\varphi_{\alpha,\beta}(z,\,\cdot\,)}\bigg{\\}},$
$\displaystyle\hskip
187.78836pt\alpha\in(-\infty,-1],\,\beta\in(-\infty,-1]\backslash(-{\mathbb{N}}),$

implying

$\displaystyle
m_{\alpha,\beta}(z)\sim\frac{\Gamma([1-\alpha-\beta+\sigma_{\alpha,\beta}(z)]/2)\Gamma([1-\alpha-\beta-\sigma_{\alpha,\beta}(z)]/2)}{\Gamma([1-\alpha+\beta+\sigma_{\alpha,\beta}(z)]/2)\Gamma([1-\alpha+\beta-\sigma_{\alpha,\beta}(z)]/2)},$
(7.8)
$\displaystyle\alpha\in(-\infty,-1],\;\beta\in(-\infty,-1]\backslash(-{\mathbb{N}}).$

$\mathbf{(II)}$ The case $\alpha\in[1,\infty)$,
$\beta\in(-\infty,-1]\backslash(-{\mathbb{N}})$:
Then an application of (C.17)–(C.20) yields

$\displaystyle\psi_{0,+,\alpha,\beta}(z,\,\cdot\,,x_{0})\underset{x\downarrow-1}{\sim}y_{1,\alpha,\beta,1}(z,\,\cdot\,)$
$\displaystyle\quad\underset{x\downarrow-1}{\sim}\frac{\Gamma(1+\alpha)\Gamma(-\beta)}{\Gamma(a_{\alpha,-\beta,\sigma_{\alpha,\beta}(z)})\Gamma(a_{\alpha,-\beta,-\sigma_{\alpha,\beta}(z)})}\bigg{\\{}y_{1,\alpha,\beta,-1}(z,\,\cdot\,)$
(7.9) $\displaystyle\hskip
31.2982pt\,+\frac{2^{\beta}[\Gamma(\beta)/\Gamma(-\beta)]\Gamma(a_{\alpha,-\beta,\sigma_{\alpha,\beta}(z)})\Gamma(a_{\alpha,-\beta,-\sigma_{\alpha,\beta}(z)})}{\Gamma(a_{\alpha,\beta,\sigma_{\alpha,\beta}(z)})\Gamma(a_{\alpha,\beta,-\sigma_{\alpha,\beta}(z)})}\underbrace{y_{2,\alpha,\beta,-1}(z,\,\cdot\,)}_{\sim\varphi_{\alpha,\beta}(z,\,\cdot\,)}\bigg{\\}},$
$\displaystyle\hskip
187.78836pt\alpha\in[1,\infty),\;\beta\in(-\infty,-1]\backslash(-{\mathbb{N}}),$

implying

$\displaystyle
m_{\alpha,\beta}(z)\sim\frac{\Gamma([1+\alpha-\beta+\sigma_{\alpha,\beta}(z)]/2)\Gamma([1+\alpha-\beta-\sigma_{\alpha,\beta}(z)]/2)}{\Gamma([1+\alpha+\beta+\sigma_{\alpha,\beta}(z)]/2)\Gamma([1+\alpha+\beta-\sigma_{\alpha,\beta}(z)]/2)},$
(7.10)
$\displaystyle\alpha\in[1,\infty),\;\beta\in(-\infty,-1]\backslash(-{\mathbb{N}}).$

$\mathbf{(III)}$ The case $\alpha\in(-\infty,-1]$,
$\beta\in[1,\infty)\backslash{\mathbb{N}}$:
Then an application of (C.17)–(C.20) yields

$\displaystyle\psi_{0,+,\alpha,\beta}(z,\,\cdot\,,x_{0})\underset{x\downarrow-1}{\sim}y_{2,\alpha,\beta,1}(z,\,\cdot\,)$
$\displaystyle\quad\underset{x\downarrow-1}{\sim}\frac{2^{-\alpha}\Gamma(1-\alpha)\Gamma(\beta)}{\Gamma(a_{-\alpha,+\beta,\sigma_{\alpha,\beta}(z)})\Gamma(a_{-\alpha,+\beta,-\sigma_{\alpha,\beta}(z)})}\bigg{\\{}2^{\beta}y_{2,\alpha,\beta,-1}(z,\,\cdot\,)$
(7.11) $\displaystyle\hskip
32.72049pt+\frac{[\Gamma(-\beta)/\Gamma(\beta)]\Gamma(a_{-\alpha,\beta,\sigma_{\alpha,\beta}(z)})\Gamma(a_{-\alpha,\beta,-\sigma_{\alpha,\beta}(z)})}{\Gamma(a_{-\alpha,-\beta,\sigma_{\alpha,\beta}(z)})\Gamma(a_{-\alpha,-\beta,-\sigma_{\alpha,\beta}(z)})}\underbrace{y_{1,\alpha,\beta,-1}(z,\,\cdot\,)}_{\sim\varphi_{\alpha,\beta}(z,\,\cdot\,)}\bigg{\\}},$
$\displaystyle\hskip
193.47882pt\alpha\in(-\infty,-1],\;\beta\in[1,\infty)\backslash{\mathbb{N}},$

implying

$\displaystyle
m_{\alpha,\beta}(z)\sim\frac{\Gamma([1-\alpha+\beta+\sigma_{\alpha,\beta}(z)]/2)\Gamma([1-\alpha+\beta-\sigma_{\alpha,\beta}(z)]/2)}{\Gamma([1-\alpha-\beta+\sigma_{\alpha,\beta}(z)]/2)\Gamma([1-\alpha-\beta-\sigma_{\alpha,\beta}(z)]/2)},$
(7.12)
$\displaystyle\alpha\in(-\infty,-1],\;\beta\in[1,\infty)\backslash{\mathbb{N}}.$

$\mathbf{(IV)}$ The case $\alpha\in[1,\infty)$,
$\beta\in[1,\infty)\backslash{\mathbb{N}}$:
Then an application of (C.17)–(C.20) yields

$\displaystyle\psi_{0,+,\alpha,\beta}(z,\,\cdot\,,x_{0})\underset{x\downarrow-1}{\sim}y_{1,\alpha,\beta,1}(z,\,\cdot\,)$
$\displaystyle\quad\underset{x\downarrow-1}{\sim}\frac{\Gamma(1+\alpha)\Gamma(\beta)}{\Gamma(a_{\alpha,\beta,\sigma_{\alpha,\beta}(z)})\Gamma(a_{\alpha,\beta,-\sigma_{\alpha,\beta}(z)})}\bigg{\\{}2^{\beta}y_{2,\alpha,\beta,-1}(z,\,\cdot\,)$
(7.13) $\displaystyle\hskip
34.14322pt+\frac{[\Gamma(-\beta)/\Gamma(\beta)]\Gamma(a_{\alpha,\beta,\sigma_{\alpha,\beta}(z)})\Gamma(a_{\alpha,\beta,-\sigma_{\alpha,\beta}(z)})}{\Gamma(a_{\alpha,-\beta,\sigma_{\alpha,\beta}(z)})\Gamma(a_{\alpha,-\beta,-\sigma_{\alpha,\beta}(z)})}\underbrace{y_{1,\alpha,\beta,-1}(z,\,\cdot\,)}_{\sim\varphi_{\alpha,\beta}(z,\,\cdot\,)}\bigg{\\}},$
$\displaystyle\hskip
199.16928pt\alpha\in[1,\infty),\;\beta\in[1,\infty)\backslash{\mathbb{N}},$

implying

$\displaystyle
m_{\alpha,\beta}(z)\sim\frac{\Gamma([1+\alpha+\beta+\sigma_{\alpha,\beta}(z)]/2)\Gamma([1+\alpha+\beta-\sigma_{\alpha,\beta}(z)]/2)}{\Gamma([1+\alpha-\beta+\sigma_{\alpha,\beta}(z)]/2)\Gamma([1+\alpha-\beta-\sigma_{\alpha,\beta}(z)]/2)},$
(7.14)
$\displaystyle\alpha\in[1,\infty),\;\beta\in[1,\infty)\backslash{\mathbb{N}}.$

$\mathbf{(V)}$ The case $\alpha\in[1,\infty)$, $\beta=k\in{\mathbb{N}}$:
Using (B.26) we see that

$\displaystyle\psi_{0,+,\alpha,k}(z,\,\cdot\,,x_{0})\underset{x\downarrow-1}{\sim}y_{1,\alpha,k,1}(z,\,\cdot\,)$
$\displaystyle\quad\underset{x\downarrow-1}{\sim}\frac{\Gamma(1+\alpha)}{\Gamma(a_{\alpha,-k,\sigma_{\alpha,k}(z)})\Gamma(a_{\alpha,-k,-\sigma_{\alpha,k}(z)})k!}\bigg{\\{}y_{2,\alpha,k,-1}(z,\,\cdot\,)$
(7.15) $\displaystyle\hskip
34.14322pt+\big{[}\psi(a_{\alpha,k,\sigma_{\alpha,k}(z)})+\psi(a_{\alpha,k,-\sigma_{\alpha,k}(z)})-\psi(k+1)+\gamma_{E}\big{]}\underbrace{y_{1,\alpha,k,-1}(z,\,\cdot\,)}_{\sim\varphi_{\alpha,\beta}(z,\,\cdot\,)}\bigg{\\}},$
$\displaystyle\hskip 240.42569pt\alpha\in[1,\infty),\;\beta=k\in{\mathbb{N}},$

implying

$\displaystyle m_{\alpha,k}(z)$
$\displaystyle\sim\psi([1+\alpha+k+\sigma_{\alpha,k}(z)]/2])+\psi([1+\alpha+k-\sigma_{\alpha,k}(z)]/2]),$
(7.16) $\displaystyle\hskip
165.02606pt\alpha\in[1,\infty),\;\beta=k\in{\mathbb{N}},$

where we have omitted $\psi(k+1)-\gamma_{E}$ which has no spectral
significance.
$\mathbf{(VI)}$ The case $\alpha\in(-\infty,-1]$, $\beta=k\in{\mathbb{N}}$:
After setting the parameters $\alpha,\,k,\,z$ in (7) to be
$-\alpha,\,k,\,z+(1+k)\alpha$ and multiplying by $(1-x)^{-\alpha}$, we can use
(A.21), together with (A.29), (A.30) to conclude

$\displaystyle\psi_{0,+,\alpha,k}(z,\,\cdot\,,x_{0})\underset{x\downarrow-1}{\sim}y_{2,\alpha,k,1}(z,\,\cdot\,)$
$\displaystyle\quad\underset{x\downarrow-1}{\sim}\frac{\Gamma(1-\alpha)}{\Gamma(a_{-\alpha,-k,\sigma_{\alpha,k}(z)})\Gamma(a_{-\alpha,-k,-\sigma_{\alpha,k}(z)})k!}\bigg{\\{}y_{2,\alpha,k,-1}(z,\,\cdot\,)$
(7.17) $\displaystyle\hskip
34.14322pt+\big{[}\psi(a_{-\alpha,k,\sigma_{\alpha,k}(z)})+\psi(a_{-\alpha,k,-\sigma_{\alpha,k}(z)})-\psi(k+1)+\gamma_{E}\big{]}\underbrace{y_{1,\alpha,k,-1}(z,\,\cdot\,)}_{\sim\varphi_{\alpha,\beta}(z,\,\cdot\,)}\bigg{\\}},$
$\displaystyle\hskip
239.00298pt\alpha\in(-\infty,-1],\;\beta=k\in{\mathbb{N}},$

implying

$\displaystyle m_{\alpha,k}(z)$
$\displaystyle\sim\psi([1-\alpha+k+\sigma_{\alpha,k}(z)]/2])+\psi([1-\alpha+k-\sigma_{\alpha,k}(z)]/2]),$
(7.18) $\displaystyle\hskip
149.37697pt\alpha\in(-\infty,-1],\;\beta=k\in{\mathbb{N}}.$

$\mathbf{(VII)}$ The case $\alpha\in[1,\infty)$,
$\beta=-k,\;k\in{\mathbb{N}}$:
After setting the parameter $z$ in (7) to be $z-(1+\alpha)k$, multiplying by
$(1+x)^{k}$ and using (A.26), (A.27) we see that

$\displaystyle\psi_{0,+,\alpha,-k}(z,\,\cdot\,,x_{0})\underset{x\downarrow-1}{\sim}\overbrace{(1+\,\cdot\,)^{k}y_{1,\alpha,k,1}(z-(1+\alpha)k,\,\cdot\,)}^{=\,2^{k}y_{1,\alpha,-k,1}(z,\,\cdot\,)}$
$\displaystyle\quad\underset{x\downarrow-1}{\sim}\frac{\Gamma(1+\alpha)}{\Gamma(a_{\alpha,-k,\sigma_{\alpha,-k}(z)})\Gamma(a_{\alpha,-k,-\sigma_{\alpha,-k}(z)})k!}\bigg{\\{}y_{1,\alpha,-k,-1}(z,\,\cdot\,)$
(7.19) $\displaystyle\hskip
34.14322pt+\big{[}\psi(a_{\alpha,k,\sigma_{\alpha,-k}(z)})+\psi(a_{\alpha,k,-\sigma_{\alpha,-k}(z)})-\psi(k+1)+\gamma_{E}\big{]}\underbrace{y_{2,\alpha,-k,-1}(z,\,\cdot\,)}_{\sim\varphi_{\alpha,\beta}(z,\,\cdot\,)}\bigg{\\}},$
$\displaystyle\hskip
239.00298pt\alpha\in[1,\infty),\;\beta=-k,\,k\in{\mathbb{N}},$

implying

$\displaystyle m_{\alpha,-k}(z)$
$\displaystyle\sim\psi([1+\alpha+k+\sigma_{\alpha,-k}(z)]/2])+\psi([1+\alpha+k-\sigma_{\alpha,-k}(z)]/2]),$
(7.20) $\displaystyle\hskip
156.49014pt\alpha\in[1,\infty),\;\beta=-k,\;k\in{\mathbb{N}}.$

$\mathbf{(VIII)}$ The case $\alpha\in(-\infty,-1]$,
$\beta=-k,\;k\in{\mathbb{N}}$:
In this case we set $z$ in (7.17) to be equal to $z-(1+\alpha)k$, multiply
with $(1+x)^{k}$ and use (A.26), (A.27) to see

$\displaystyle\psi_{0,+,\alpha,-k}(z,\,\cdot\,,x_{0})\underset{x\downarrow-1}{\sim}\overbrace{(1+\,\cdot\,)^{k}y_{2,\alpha,k,1}(z-(1+\alpha)k,\,\cdot\,)}^{=\,2^{k}y_{2,\alpha,-k,1}(z,\,\cdot\,)}$
$\displaystyle\
\underset{x\downarrow-1}{\sim}\frac{\Gamma(1-\alpha)}{\Gamma(a_{-\alpha,-k,\sigma_{\alpha,-k}(z)})\Gamma(a_{-\alpha,-k,-\sigma_{\alpha,-k}(z)})k!}\bigg{\\{}y_{1,\alpha,-k,-1}(z,\,\cdot\,)$
(7.21) $\displaystyle\hskip
22.76228pt+\big{[}\psi(a_{-\alpha,k,\sigma_{\alpha,-k}(z)})+\psi(a_{-\alpha,k,-\sigma_{\alpha,-k}(z)})-\psi(k+1)+\gamma_{E}\big{]}\underbrace{y_{2,\alpha,-k,-1}(z,\,\cdot\,)}_{\sim\varphi_{\alpha,\beta}(z,\,\cdot\,)}\bigg{\\}},$
$\displaystyle\hskip
226.19931pt\alpha\in(-\infty,-1],\;\beta=-k,\,k\in{\mathbb{N}},$

implying

$\displaystyle m_{\alpha,-k}(z)$
$\displaystyle\sim\psi([1-\alpha+k+\sigma_{\alpha,-k}(z)]/2])+\psi([1-\alpha+k-\sigma_{\alpha,-k}(z)]/2]),$
(7.22) $\displaystyle\hskip
140.84105pt\alpha\in(-\infty,-1],\;\beta=-k,\;k\in{\mathbb{N}}.$

We summarize the results of this section as follows:

######  Theorem 7.1.

Let $\alpha,\beta\in(-\infty,-1]\cup[1,\infty)$. Then the
Weyl–Titchmarsh–Kodaira $m$-function $m_{\alpha,\beta}$ associated with
$T_{\alpha,\beta}$ $($the unique
$L^{2}((-1,1);r_{\alpha,\beta}dx)$-realization of $\tau_{\alpha,\beta}$$)$ is
of the following form$:$
$\mathbf{(I)}$ In the case $\alpha\in(-\infty,-1]$,
$\beta\in(-\infty,-1]\backslash(-{\mathbb{N}})$:

$\displaystyle\begin{split}m_{\alpha,\beta}(z)&\sim\frac{\Gamma([1-\alpha-\beta+\sigma_{\alpha,\beta}(z)]/2)\Gamma([1-\alpha-\beta-\sigma_{\alpha,\beta}(z)]/2)}{\Gamma([1-\alpha+\beta+\sigma_{\alpha,\beta}(z)]/2)\Gamma([1-\alpha+\beta-\sigma_{\alpha,\beta}(z)]/2)},\\\
\sigma(T_{\alpha,\beta})&=\\{(n-\alpha-\beta)(n+1)\\}_{n\in{\mathbb{N}}_{0}}.\end{split}$
(7.23)

$\mathbf{(II)}$ In the case $\alpha\in[1,\infty)$,
$\beta\in(-\infty,-1]\backslash(-{\mathbb{N}})$:

$\displaystyle\begin{split}m_{\alpha,\beta}(z)&\sim\frac{\Gamma([1+\alpha-\beta+\sigma_{\alpha,\beta}(z)]/2)\Gamma([1+\alpha-\beta-\sigma_{\alpha,\beta}(z)]/2)}{\Gamma([1+\alpha+\beta+\sigma_{\alpha,\beta}(z)]/2)\Gamma([1+\alpha+\beta-\sigma_{\alpha,\beta}(z)]/2)},\\\
\sigma(T_{\alpha,\beta})&=\\{(n-\beta)(n+1+\alpha)\\}_{n\in{\mathbb{N}}_{0}}.\end{split}$
(7.24)

$\mathbf{(III)}$ In the case $\alpha\in(-\infty,-1]$,
$\beta\in[1,\infty)\backslash{\mathbb{N}}$:

$\displaystyle\begin{split}m_{\alpha,\beta}(z)&\sim\frac{\Gamma([1-\alpha+\beta+\sigma_{\alpha,\beta}(z)]/2)\Gamma([1-\alpha+\beta-\sigma_{\alpha,\beta}(z)]/2)}{\Gamma([1-\alpha-\beta+\sigma_{\alpha,\beta}(z)]/2)\Gamma([1-\alpha-\beta-\sigma_{\alpha,\beta}(z)]/2)},\\\
\sigma(T_{\alpha,\beta})&=\\{(n-\alpha)(n+1+\beta)\\}_{n\in{\mathbb{N}}_{0}}.\end{split}$
(7.25)

$\mathbf{(IV)}$ In the case $\alpha\in[1,\infty)$,
$\beta\in[1,\infty)\backslash{\mathbb{N}}$:

$\displaystyle\begin{split}m_{\alpha,\beta}(z)&\sim\frac{\Gamma([1+\alpha+\beta+\sigma_{\alpha,\beta}(z)]/2)\Gamma([1+\alpha+\beta-\sigma_{\alpha,\beta}(z)]/2)}{\Gamma([1+\alpha-\beta+\sigma_{\alpha,\beta}(z)]/2)\Gamma([1+\alpha-\beta-\sigma_{\alpha,\beta}(z)]/2)},\\\
\sigma(T_{\alpha,\beta})&=\\{n(n+1+\alpha+\beta)\\}_{n\in{\mathbb{N}}_{0}}.\end{split}$
(7.26)

$\mathbf{(V)}$ In the case $\alpha\in[1,\infty)$, $\beta=k\in{\mathbb{N}}$:

$\displaystyle\begin{split}m_{\alpha,k}(z)&\sim\psi([1+\alpha+k+\sigma_{\alpha,k}(z)]/2])+\psi([1+\alpha+k-\sigma_{\alpha,k}(z)]/2]),\\\
\sigma(T_{\alpha,k})&=\\{n(n+1+\alpha+k)\\}_{n\in{\mathbb{N}}_{0}}.\end{split}$
(7.27)

$\mathbf{(VI)}$ In the case $\alpha\in(-\infty,-1]$, $\beta=k\in{\mathbb{N}}$:

$\displaystyle\begin{split}m_{\alpha,k}(z)&\sim\psi([1-\alpha+k+\sigma_{\alpha,k}(z)]/2])+\psi([1-\alpha+k-\sigma_{\alpha,k}(z)]/2]),\\\
\sigma(T_{\alpha,k})&=\\{(n-\alpha)(n+1+k)\\}_{n\in{\mathbb{N}}_{0}}.\end{split}$
(7.28)

$\mathbf{(VII)}$ In the case $\alpha\in[1,\infty)$,
$\beta=-k,\;k\in{\mathbb{N}}$:

$\displaystyle\begin{split}m_{\alpha,-k}(z)&\sim\psi([1+\alpha+k+\sigma_{\alpha,-k}(z)]/2])+\psi([1+\alpha+k-\sigma_{\alpha,-k}(z)]/2]),\\\
\sigma(T_{\alpha,-k})&=\\{(n+k)(n+1+\alpha)\\}_{n\in{\mathbb{N}}_{0}}.\end{split}$
(7.29)

$\mathbf{(VIII)}$ In the case $\alpha\in(-\infty,-1]$,
$\beta=-k,\;k\in{\mathbb{N}}$:

$\displaystyle\begin{split}m_{\alpha,-k}(z)&\sim\psi([1-\alpha+k+\sigma_{\alpha,-k}(z)]/2])+\psi([1-\alpha+k-\sigma_{\alpha,-k}(z)]/2]),\\\
\sigma(T_{\alpha,-k})&=\\{(n-\alpha+k)(n+1)\\}_{n\in{\mathbb{N}}_{0}}.\end{split}$
(7.30)

The $($necessarily simple $)$ poles of $m_{\alpha,\beta}$ occur precisely at
the $($necessarily simple $)$ eigenvalues of $T_{\alpha,\beta}$.

##  8\. More on the Nevanlinna–Herglotz property of $m$-functions
and on Jacobi Polynomials

In this section we prove the Nevanlinna–Herglotz property for some of the
$m$-functions discussed in Sections 5 and 6 from scratch, in particular, we
will prove the Nevanlinna–Herglotz property in connection with Jacobi
polynomials. For brevity we will focus on the non-degenerate case $\beta\neq
0$ only.

More precisely, we will study the function

$\displaystyle\widehat{m}_{\alpha,\beta}(z)$
$\displaystyle\quad=\frac{-\Gamma(1-\beta)}{2^{1+\alpha+\beta}\beta\Gamma(1+\beta)}\frac{\Gamma([1+\alpha+\beta+\sigma_{\alpha,\beta}(z)]/2)\Gamma([1+\alpha+\beta-\sigma_{\alpha,\beta}(z)]/2)}{\Gamma([1+\alpha-\beta+\sigma_{\alpha,\beta}(z)]/2)\Gamma([1+\alpha-\beta-\sigma_{\alpha,\beta}(z)]/2)},$
$\displaystyle\hskip
102.43008pt\alpha\in{\mathbb{R}},\;\beta\in{\mathbb{R}}\backslash{\mathbb{Z}},\;z\in{\mathbb{C}}\backslash\\{n(n+1+\alpha+\beta)\\}_{n\in{\mathbb{N}}_{0}},$
(8.1)

where we again employed the abbreviation (5.5),
$\sigma_{\alpha,\beta}(z)=\big{[}(1+\alpha+\beta)^{2}+4z\big{]}^{1/2}$,
$\alpha,\beta\in{\mathbb{R}},\;z\in{\mathbb{C}}$.

With the notation used in Sections 5 and 6, one has the following connections
between $\widehat{m}_{\alpha,\beta}$ and Jacobi $m$-functions discussed in
Sections 5 and 6,

$\displaystyle\widehat{m}_{\alpha,\beta}(z)=\begin{cases}m_{0,0,\alpha,\beta}(z),&\alpha\in(0,1),\
\beta\in(0,1),\\\
m_{\frac{\pi}{2},\frac{\pi}{2},\alpha,\beta}(z),&\alpha\in(-1,0),\
\beta\in(-1,0),\\\ m_{0,\frac{\pi}{2},\alpha,\beta}(z),&\alpha\in(-1,0),\
\beta\in(0,1),\\\ m_{\frac{\pi}{2},0,\alpha,\beta}(z),&\alpha\in(0,1),\
\beta\in(-1,0),\\\ m_{0,\alpha,\beta}(z),&\alpha\geqslant 1,\
\beta\in(0,1),\\\ m_{\frac{\pi}{2},\alpha,\beta}(z),&\alpha\geqslant 1,\
\beta\in(-1,0).\end{cases}$ (8.2)

We recall that a Nevanlinna–Herglotz (in short, an N–H) function
$m(\,\cdot\,)$ is defined as analytic in ${\mathbb{C}}_{+}$ such that
$\text{\rm Im}(m(z))>0$, $z\in{\mathbb{C}}_{+}$. In addition, $m$ is extended
to ${\mathbb{C}}_{-}$ by reflection, that is, one defines

$m(z)=\overline{m(\overline{z})},\quad z\in{\mathbb{C}}_{-}$ (8.3)

(which, in general, does not describe the analytic continuation of $m$ to
${\mathbb{C}}_{-}$). If $m$ and $n$ are N–H functions, then so are $m+n$ and
$m(n)$, in particular, as ${\mathbb{C}}_{+}\ni z\mapsto-1/z$ is N–H, and
hence,

$m\,\text{ is N--H if and only if $-1/m$ is.}$ (8.4)

Any poles and isolated zeros of $m$ are simple and located on the real axis,
the residues at poles being strictly negative. If $m$ is N–H, then $m$ has the
Nevanlinna, respectively, Riesz–Herglotz representation, that is, there exists
a nonnegative measure $d\omega$ on ${\mathbb{R}}$ satisfying

$\int_{{\mathbb{R}}}\frac{d\omega(\lambda)}{1+\lambda^{2}}<\infty$ (8.5)

such that