# The Jacobi Operator and its Donoghue $m$-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 ,  Mateusz Piorkowski
Mathematical Sciences and Research Institute, UC Berkeley, 17 Gauss Way, CA
94720, USA<EMAIL_ADDRESS>and  Jonathan Stanfill  Department of
Mathematics, Baylor University, Sid Richardson Bldg., 1410 S. 4th Street,
Waco, TX 76706, USA<EMAIL_ADDRESS>http://sites.baylor.edu/jonathan-stanfill/

###### Abstract.

In this paper we construct Donoghue $m$-functions for the Jacobi differential
operator in $L^{2}\big{(}(-1,1);(1-x)^{\alpha}(1+x)^{\beta}dx\big{)}$,
associated to the differential expression

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

whenever at least one endpoint, $x=\pm 1$, is in the limit circle case. In
doing so, we provide a full treatment of the Jacobi operator’s $m$-functions
corresponding to coupled boundary conditions whenever both endpoints are in
the limit circle case, a topic not covered in the literature.

###### Key words and phrases:

Singular Sturm–Liouville operators, Jacobi equation, boundary values, boundary
conditions, Donoghue $m$-functions.

###### 2020 Mathematics Subject Classification:

Primary: 34B20, 34B24, 34L05; Secondary: 47A10, 47E05.

###### Contents

  1. 1 Introduction
  2. 2 Some Background
  3. 3 Donoghue $m$-functions: Two Limit Circle Endpoints
  4. 4 Donoghue $m$-functions: One Limit Circle Endpoint
  5. 5 The Jacobi Operator and its Donoghue $m$-functions
    1. 5.1 The Regular and Limit Circle Case $\alpha,\beta\in(-1,1)$
    2. 5.2 Precisely One Interval Endpoint in the Limit Point Case
  6. A The Hypergeometric and Jacobi Differential Equations
  7. B Connection Formulas
  8. C Behavior of $y_{j,\alpha,\beta,\mp 1}(z,x)$, $j=1,2$, near $x=\pm 1$

##  1\. Introduction

This paper should be regarded as a sequel to the recent [27] in which the
Donoghue $m$-function was derived for singular Sturm–Liouville operators. To
illustrate the theory, we now apply it to a representative example, the Jacobi
differential operator associated with
$L^{2}\big{(}(-1,1);(1-x)^{\alpha}(1+x)^{\beta}dx\big{)}$-realizations of the
the differential expression,

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

whenever at least one endpoint, $x=\pm 1$, is in the limit circle case (see,
e.g. [1, Ch. 22], [4], [9], [20, Sect. 23], [38, Ch. 4], [44, Sects. VII.6.1,
XIV.2], [62, Ch. 18], [71, Ch. IV]). In particular, this provides a full
treatment of $m$-functions corresponding to coupled boundary conditions
whenever both endpoints are in the limit circle case, a new result.

To set the stage we briefly discuss abstract Donoghue $m$-functions following
[29], [25], [27], and [28]. Given a self-adjoint extension $A$ of a densely
defined, closed, symmetric operator ${\hskip
2.27626pt\overset{\textbf{\Large.}}{A}}$ in ${\mathcal{H}}$ (a complex,
separable Hilbert space) with equal deficiency indices and the deficiency
subspace ${\mathcal{N}}_{i}$ of ${\hskip
2.27626pt\overset{\textbf{\Large.}}{A}}$ in ${\mathcal{H}}$, with

${\mathcal{N}}_{i}=\ker\big{(}\big{(}{\hskip
2.27626pt\overset{\textbf{\Large.}}{A}}\big{)}{\raisebox{2.84526pt}{${}^{*}$}}-iI_{{\mathcal{H}}}\big{)},\quad\dim\,({\mathcal{N}}_{i})=k\in{\mathbb{N}}\cup\\{\infty\\},$
(1.2)

the Donoghue $m$-operator
$M_{A,{\mathcal{N}}_{i}}^{Do}(\,\cdot\,)\in{\mathcal{B}}({\mathcal{N}}_{i})$
associated with the pair $(A,{\mathcal{N}}_{i})$ is given by

$\displaystyle\begin{split}M_{A,{\mathcal{N}}_{i}}^{Do}(z)&=P_{{\mathcal{N}}_{i}}(zA+I_{\mathcal{H}})(A-zI_{{\mathcal{H}}})^{-1}P_{{\mathcal{N}}_{i}}\big{|}_{{\mathcal{N}}_{i}}\\\
&=zI_{{\mathcal{N}}_{i}}+(z^{2}+1)P_{{\mathcal{N}}_{i}}(A-zI_{{\mathcal{H}}})^{-1}P_{{\mathcal{N}}_{i}}\big{|}_{{\mathcal{N}}_{i}}\,,\quad
z\in{\mathbb{C}}\backslash{\mathbb{R}},\end{split}$ (1.3)

with $I_{{\mathcal{N}}_{i}}$ the identity operator in ${\mathcal{N}}_{i}$, and
$P_{{\mathcal{N}}_{i}}$ the orthogonal projection in ${\mathcal{H}}$ onto
${\mathcal{N}}_{i}$. The special case $k=1$, was discussed in detail by
Donoghue [17]; for the case $k\in{\mathbb{N}}$ we refer to [31].

More generally, given a self-adjoint extension $A$ of ${\hskip
2.27626pt\overset{\textbf{\Large.}}{A}}$ in ${\mathcal{H}}$ and a closed,
linear subspace ${\mathcal{N}}$ of ${\mathcal{N}}_{i}$, the Donoghue
$m$-operator
$M_{A,{\mathcal{N}}}^{Do}(\,\cdot\,)\in{\mathcal{B}}({\mathcal{N}})$
associated with the pair $(A,{\mathcal{N}})$ is defined by

$\displaystyle\begin{split}M_{A,{\mathcal{N}}}^{Do}(z)&=P_{\mathcal{N}}(zA+I_{\mathcal{H}})(A-zI_{{\mathcal{H}}})^{-1}P_{\mathcal{N}}\big{|}_{\mathcal{N}}\\\
&=zI_{\mathcal{N}}+(z^{2}+1)P_{\mathcal{N}}(A-zI_{{\mathcal{H}}})^{-1}P_{\mathcal{N}}\big{|}_{\mathcal{N}}\,,\quad
z\in{\mathbb{C}}\backslash{\mathbb{R}},\end{split}$ (1.4)

with $I_{\mathcal{N}}$ the identity operator in ${\mathcal{N}}$ and
$P_{\mathcal{N}}$ the orthogonal projection in ${\mathcal{H}}$ onto
${\mathcal{N}}$.

Since $M_{A,{\mathcal{N}}}^{Do}(z)$ is analytic for
$z\in{\mathbb{C}}\backslash{\mathbb{R}}$ and satisfies (see [29, Theorem 5.3])

$\displaystyle[\text{\rm Im}(z)]^{-1}\text{\rm
Im}\big{(}M_{A,{\mathcal{N}}}^{Do}(z)\big{)}\geqslant
2\Big{[}\big{(}|z|^{2}+1\big{)}+\big{[}\big{(}|z|^{2}-1\big{)}^{2}+4(\text{\rm
Re}(z))^{2}\big{]}^{1/2}\Big{]}^{-1}I_{{\mathcal{N}}},$ $\displaystyle\hskip
270.30118ptz\in{\mathbb{C}}\backslash{\mathbb{R}},$ (1.5)

$M_{A,{\mathcal{N}}}^{Do}(\,\cdot\,)$ is a
${\mathcal{B}}({\mathcal{N}})$-valued Nevanlinna–Herglotz function. Thus,
$M_{A,{\mathcal{N}}}^{Do}(\,\cdot\,)$ admits the representation

$M_{A,{\mathcal{N}}}^{Do}(z)=\int_{\mathbb{R}}d\Omega_{A,{\mathcal{N}}}^{Do}(\lambda)\bigg{[}\frac{1}{\lambda-z}-\frac{\lambda}{\lambda^{2}+1}\bigg{]},\quad
z\in{\mathbb{C}}\backslash{\mathbb{R}},$ (1.6)

where the ${\mathcal{B}}({\mathcal{N}})$-valued measure
$\Omega_{A,{\mathcal{N}}}^{Do}(\,\cdot\,)$ satisfies

$\displaystyle\Omega_{A,{\mathcal{N}}}^{Do}(\lambda)=(\lambda^{2}+1)(P_{{\mathcal{N}}}E_{A}(\lambda)P_{{\mathcal{N}}}\big{|}_{{\mathcal{N}}}),$
(1.7)
$\displaystyle\int_{\mathbb{R}}d\Omega_{A,{\mathcal{N}}}^{Do}(\lambda)\,(1+\lambda^{2})^{-1}=I_{{\mathcal{N}}},$
(1.8)
$\displaystyle\int_{\mathbb{R}}d(\xi,\Omega_{A,{\mathcal{N}}}^{Do}(\lambda)\xi)_{{\mathcal{N}}}=\infty\,\text{
for all }\,\xi\in{\mathcal{N}}\backslash\\{0\\},$ (1.9)

with $E_{A}(\,\cdot\,)$ the family of strongly right-continuous spectral
projections of $A$ in ${\mathcal{H}}$ (see [25] for details). Operators of the
type $M_{A,{\mathcal{N}}}^{Do}(\,\cdot\,)$ and some of its variants have
attracted considerable attention in the literature. They appear to go back to
Krein [45] (see also [46]), Saakjan [70], and independently, Donoghue [17].
The interested reader can find a wealth of additional information in the
context of (1.3)–(1.9) in [2], [3], [6]–[7], [10], [11]–[12], [15]–[16],
[25]–[31], [32], [37], [47], [49], [51], [52], [53], [54], [56], [57]–[58],
[63], [65], [69], and the references therein.

Without going into further details (see [29, Corollary 5.8] for details) we
note that the prime reason for the interest in
$M_{A,{\mathcal{N}}_{i}}^{Do}(\,\cdot\,)$ lies in the fundamental fact that
the entire spectral information of $A$ contained in its family of spectral
projections $E_{A}(\,\cdot\,)$, is already encoded in the
${\mathcal{B}}({\mathcal{N}}_{i})$-valued measure
$\Omega_{A,{\mathcal{N}}_{i}}^{Do}(\,\cdot\,)$ (including multiplicity
properties of the spectrum of $A$) if and only if ${\hskip
2.27626pt\overset{\textbf{\Large.}}{A}}$ is completely non-self-adjoint in
${\mathcal{H}}$ (that is, if and only if ${\hskip
2.27626pt\overset{\textbf{\Large.}}{A}}$ has no invariant subspace on which it
is self-adjoint, see [29, Lemma 5.4]).

We also note that a particularly attractive feature of the Donoghue
$m$-operator, that distinguishes it from the Weyl–Titchmarsh–Kodaira
$m$-operator, consists of the explicit appearance of the resolvent
$(A-zI_{{\mathcal{H}}})^{-1}$, $z\in{\mathbb{C}}\backslash{\mathbb{R}}$, in
it’s definition (1.3) (resp., (1.4)).

In the remainder of this paper, we will exclusively focus on the particular
case ${\mathcal{N}}={\mathcal{N}}_{i}=\ker\big{(}\big{(}{\hskip
2.27626pt\overset{\textbf{\Large.}}{A}}\big{)}{\raisebox{2.84526pt}{${}^{*}$}}-iI_{{\mathcal{H}}}\big{)}$,
with ${\hskip 2.27626pt\overset{\textbf{\Large.}}{A}}$ being a singular
Sturm–Liouville operator.

Turning to the content of each section, we discuss the necessary background in
connection to singular Sturm–Liouville operators in Section 2. In Sections 3
and 4 we recall the Donoghue $m$-functions in the two limit circle and one
limit circle endpoint cases, respectively, following [27]. The Jacobi operator
and its Donoghue $m$-functions are the topic of Section 5, with Appendices A–C
providing a detailed treatment of solutions of the Jacobi differential
equation and the associated hypergeometric differential equations.

Finally, some comments 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. Moreover, we denote the scalar product and
norm in $L^{2}((a,b);rdx)$ by $(\,\cdot\,,\,\cdot\,)_{L^{2}((a,b);rdx)}$
(linear in the second argument) and $\|\,\cdot\,\|_{L^{2}((a,b);rdx)}$.

##  2\. Some Background

In this section we briefly recall the basics of singular Sturm–Liouville
operators. The material is standard and can be found, for instance, in [5, Ch.
6], [14, Chs. 8, 9], [18, Sects. 13.6, 13.9, 13.10], [19], [33, Ch. 4], [39,
Ch. III], [59, Ch. V], [60], [64, Ch. 6], [72, Ch. 9], [73, Sect. 8.3], [74,
Ch. 13], [76, Chs. 4, 6–8].

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 $\tau$ of the form

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

If $f\in AC_{loc}((a,b))$, then the quasi-derivative of $f$ is defined to be
$f^{[1]}:=pf^{\prime}$. Moreover, the Wronskian of two functions $f,g\in
AC_{loc}((a,b))$ is defined by

$W(f,g)(x)=f(x)g^{[1]}(x)-f^{[1]}(x)g(x)\,\text{ for a.e.~{}$x\in(a,b)$}.$
(2.2)

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

$\displaystyle T_{max}f=\tau f,$ $\displaystyle
f\in\operatorname{dom}(T_{max})=\big{\\{}g\in
L^{2}((a,b);rdx)\,\big{|}\,g,g^{[1]}\in{AC_{loc}((a,b))};$ (2.3)
$\displaystyle\hskip 179.25244pt\tau g\in L^{2}((a,b);rdx)\big{\\}}.$

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

$\displaystyle\dot{T}_{min}f=\tau f,$ $\displaystyle
f\in\operatorname{dom}\big{(}\dot{T}_{min}\big{)}=\big{\\{}g\in
L^{2}((a,b);rdx)\,\big{|}\,g,g^{[1]}\in{AC_{loc}((a,b))};$ (2.4)
$\displaystyle\hskip 88.2037pt\operatorname{supp}\,(g)\subset(a,b)\text{ is
compact; }\tau g\in L^{2}((a,b);rdx)\big{\\}}.$

One can prove that $\dot{T}_{min}$ is closable, and one then defines the
minimal operator $T_{min}$ by $T_{min}=\overline{\dot{T}_{min}}$.

Still assuming Hypothesis 2.1, one can prove the following basic fact,

$\big{(}\dot{T}_{min}\big{)}^{*}=T_{max},$ (2.5)

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

The celebrated Weyl alternative can be stated as follows:

######  Theorem 2.2 (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 as follows.

######  Definition 2.3.

Assume Hypothesis 2.1.
In case $(i)$ in Theorem 2.2, $\tau$ is said to be in the limit circle case at
$b$ $($resp., $a$$)$. $($Frequently, $\tau$ is then called quasi-regular at
$b$ $($resp., $a$$)$.$)$
In case $(ii)$ in Theorem 2.2, $\tau$ is said to be in the limit point case at
$b$ $($resp., $a$$)$.
If $\tau$ is in the limit circle case at $a$ and $b$ then $\tau$ is also
called quasi-regular on $(a,b)$.

The next result links self-adjointness of $T_{min}$ (resp., $T_{max}$) and the
limit point property of $\tau$ at both endpoints. Here, and throughout, we
shall employ the notation

${\mathcal{N}}_{z}=\ker\big{(}T_{max}-zI_{L^{2}((a,b);rdx)}\big{)},\quad
z\in{\mathbb{C}}.$ (2.6)

######  Theorem 2.4.

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 $T_{min}=\overline{\dot{T}_{min}}$. Then

$\displaystyle\begin{split}n_{\pm}(T_{min})&=\dim\big{(}{\mathcal{N}}_{\pm
i}\big{)}\\\ &=\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 if and only if $\tau$ is in
the limit point case at $a$ and $b$.

All self-adjoint extensions of $T_{min}$ are then described as follows:

######  Theorem 2.5.

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

$\displaystyle T_{\gamma,\delta}f=\tau f,\quad\gamma,\delta\in[0,\pi),$ (2.11)
$\displaystyle
f\in\operatorname{dom}(T_{\gamma,\delta})=\bigg{\\{}g\in\operatorname{dom}(T_{max})\,\bigg{|}\,\begin{aligned}
\cos(\gamma)\widetilde{g}_{1}(a)+\sin(\gamma)\widetilde{g}_{2}(a)&=0,\\\
\cos(\delta)\widetilde{g}_{1}(b)+\sin(\delta)\widetilde{g}_{2}(b)&=0\end{aligned}\bigg{\\}}.$

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

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

where $\varphi\in[0,\pi)$, 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 $)$.

One can now detail the characterization of $\operatorname{dom}(T_{min})$ by

$\displaystyle\begin{split}&T_{min}f=\tau f,\\\
&f\in\operatorname{dom}(T_{min})=\big{\\{}g\in\operatorname{dom}(T_{max})\,\big{|}\,\widetilde{g}_{1}(a)=\widetilde{g}_{2}(a)=\widetilde{g}_{1}(b)=\widetilde{g}_{2}(b)=0\big{\\}}.\end{split}$
(2.13)

Finally, we turn to the characterization of generalized boundary values in the
case where $T_{min}$ is bounded from below following [26] and [60].

We briefly recall the basics of oscillation theory with particular emphasis on
principal and nonprincipal solutions, a notion originally due to Leighton and
Morse [50] (see also Rellich [66], [67] and Hartman and Wintner [36,
Appendix]). Our outline below follows [13], [18, Sects. 13.6, 13.9, 13.10],
[33, Ch. 7], [35, Ch. XI], [60], [76, Chs. 4, 6–8].

######  Definition 2.6.

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
$T_{min}\geqslant\lambda_{0}I_{L^{2}((a,b);rdx)}$, if

$\big{(}u,[T_{min}-\lambda_{0}I_{L^{2}((a,b);rdx)}]u\big{)}_{L^{2}((a,b);rdx)}\geqslant
0,\quad u\in\operatorname{dom}(T_{min}).$ (2.14)

The following is a key result.

######  Theorem 2.7.

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 $x\downarrow a$,}$
(2.15) $\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 $x\uparrow b$,}$
(2.16)
$\displaystyle\int_{a}^{c}dx\,p(x)^{-1}u_{a}(\lambda,x)^{-2}=\int_{d}^{b}dx\,p(x)^{-1}u_{b}(\lambda,x)^{-2}=\infty,$
(2.17)
$\displaystyle\int_{a}^{c}dx\,p(x)^{-1}{\widehat{u}_{a}(\lambda,x)}^{-2}<\infty,\quad\int_{d}^{b}dx\,p(x)^{-1}{\widehat{u}_{b}(\lambda,x)}^{-2}<\infty.$
(2.18)

######  Definition 2.8.

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\,)$$)$ in Theorem 2.7 $(iii)$ is called a principal
$($or minimal $)$ solution of $\tau u=\lambda u$ at $a$ $($resp., $b$$)$. A
real-valued solution ${\mathchoice{\mathop{\vbox{\halign{#\cr\kern
0.80002pt$\hss\leavevmode\resizebox{4.57967pt}{3.8889pt}{\rotatebox[origin={c}]{90.0}{(}}\hss$\crcr\nointerlineskip\cr$\hss\displaystyle
u\hss$\crcr}}}\limits}{\mathop{\vbox{\halign{#\cr\kern
0.80002pt$\hss\leavevmode\resizebox{4.57967pt}{3.8889pt}{\rotatebox[origin={c}]{90.0}{(}}\hss$\crcr\nointerlineskip\cr$\hss\textstyle
u\hss$\crcr}}}\limits}{\mathop{\vbox{\halign{#\cr\kern
0.80002pt$\hss\leavevmode\resizebox{3.20576pt}{2.72223pt}{\rotatebox[origin={c}]{90.0}{(}}\hss$\crcr\nointerlineskip\cr$\hss\scriptstyle
u\hss$\crcr}}}\limits}{\mathop{\vbox{\halign{#\cr\kern
0.80002pt$\hss\leavevmode\resizebox{2.28983pt}{1.94444pt}{\rotatebox[origin={c}]{90.0}{(}}\hss$\crcr\nointerlineskip\cr$\hss\scriptscriptstyle
u\hss$\crcr}}}\limits}}_{a}(\lambda,\,\cdot\,)$ $($resp.,
${\mathchoice{\mathop{\vbox{\halign{#\cr\kern
0.80002pt$\hss\leavevmode\resizebox{4.57967pt}{3.8889pt}{\rotatebox[origin={c}]{90.0}{(}}\hss$\crcr\nointerlineskip\cr$\hss\displaystyle
u\hss$\crcr}}}\limits}{\mathop{\vbox{\halign{#\cr\kern
0.80002pt$\hss\leavevmode\resizebox{4.57967pt}{3.8889pt}{\rotatebox[origin={c}]{90.0}{(}}\hss$\crcr\nointerlineskip\cr$\hss\textstyle
u\hss$\crcr}}}\limits}{\mathop{\vbox{\halign{#\cr\kern
0.80002pt$\hss\leavevmode\resizebox{3.20576pt}{2.72223pt}{\rotatebox[origin={c}]{90.0}{(}}\hss$\crcr\nointerlineskip\cr$\hss\scriptstyle
u\hss$\crcr}}}\limits}{\mathop{\vbox{\halign{#\cr\kern
0.80002pt$\hss\leavevmode\resizebox{2.28983pt}{1.94444pt}{\rotatebox[origin={c}]{90.0}{(}}\hss$\crcr\nointerlineskip\cr$\hss\scriptscriptstyle
u\hss$\crcr}}}\limits}}_{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$$)$.
In particular, $\widehat{u}_{a}(\lambda,\,\cdot\,)$ $($resp.,
$\widehat{u}_{b}(\lambda,\,\cdot\,)$$)$ in (2.15)–(2.18) are nonprincipal
solutions at $a$ $($resp., $b$$)$.

Next, we revisit in Theorem 2.5 how the generalized boundary values are
utilized in the description of all self-adjoint extensions of $T_{min}$ in the
case where $T_{min}$ is bounded from below.

######  Theorem 2.9 ([26, Theorem 4.5]).

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

$W(\widehat{u}_{a}(\lambda_{0},\,\cdot\,),u_{a}(\lambda_{0},\,\cdot\,))=W(\widehat{u}_{b}(\lambda_{0},\,\cdot\,),u_{b}(\lambda_{0},\,\cdot\,))=1.$
(2.19)

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}$ (2.20)

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

$\displaystyle\begin{split}\widetilde{g}(a)&=-W(v_{2},g)(a)=\widetilde{g}_{1}(a)=-W(u_{a}(\lambda_{0},\,\cdot\,),g)(a)=\lim_{x\downarrow
a}\frac{g(x)}{\widehat{u}_{a}(\lambda_{0},x)},\\\
\widetilde{g}(b)&=-W(v_{2},g)(b)=\widetilde{g}_{1}(b)=-W(u_{b}(\lambda_{0},\,\cdot\,),g)(b)=\lim_{x\uparrow
b}\frac{g(x)}{\widehat{u}_{b}(\lambda_{0},x)},\end{split}$ (2.21)
$\displaystyle\begin{split}{\widetilde{g}}^{\,\prime}(a)&=W(v_{1},g)(a)=\widetilde{g}_{2}(a)=W(\widehat{u}_{a}(\lambda_{0},\,\cdot\,),g)(a)=\lim_{x\downarrow
a}\frac{g(x)-\widetilde{g}(a)\widehat{u}_{a}(\lambda_{0},x)}{u_{a}(\lambda_{0},x)},\\\
{\widetilde{g}}^{\,\prime}(b)&=W(v_{1},g)(b)=\widetilde{g}_{2}(b)=W(\widehat{u}_{b}(\lambda_{0},\,\cdot\,),g)(b)=\lim_{x\uparrow
b}\frac{g(x)-\widetilde{g}(b)\widehat{u}_{b}(\lambda_{0},x)}{u_{b}(\lambda_{0},x)}.\end{split}$
(2.22)

In particular, the limits on the right-hand sides in (2.21), (2.22) exist.

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 Niessen and Zettl [60](see
also [30], [40], [41], [42], [55], [67], [68], [75]):

######  Theorem 2.10.

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 $T_{F}=T_{0,0}$ of $T_{min}$ is characterized by

$\displaystyle T_{F}f=\tau f,\quad
f\in\operatorname{dom}(T_{F})=\big{\\{}g\in\operatorname{dom}(T_{max})\,\big{|}\,\widetilde{g}(a)=\widetilde{g}(b)=0\big{\\}}.$
(2.23)

##  3\. Donoghue $m$-functions: Two Limit Circle Endpoints

The Donoghue $m$-functions in the case where $\tau$ is in the limit circle
case at $a$ and $b$ is the primary topic of this section following [27, Sect.
6].

######  Hypothesis 3.1.

In addition to Hypothesis 2.1 assume that $\tau$ is in the limit circle case
at $a$ and $b$. Moreover, for $z\in\rho(T_{0,0})$, let
$\\{u_{j}(z,\,\cdot\,)\\}_{j=1,2}$ denote solutions to $\tau u=zu$ which
satisfy the boundary conditions

$\begin{split}\widetilde{u}_{1}(z,a)=0,&\quad\widetilde{u}_{1}(z,b)=1,\\\
\widetilde{u}_{2}(z,a)=1,&\quad\widetilde{u}_{2}(z,b)=0.\end{split}$ (3.1)

Assume Hypotheses 3.1. By Theorem 2.5 or Theorem 2.9, the following statements
$(i)$–$(iii)$ hold.
$(i)$ If $\gamma,\delta\in[0,\pi)$, then the operator $T_{\gamma,\delta}$
defined by

$\displaystyle T_{\gamma,\delta}f=T_{max}f,$ (3.2) $\displaystyle
f\in\operatorname{dom}(T_{\gamma,\delta})=\bigg{\\{}g\in\operatorname{dom}(T_{max})\,\bigg{|}\,\begin{aligned}
\cos(\gamma)\widetilde{g}(a)+\sin(\gamma)\widetilde{g}^{\,\prime}(a)&=0,\\\
\cos(\delta)\widetilde{g}(b)+\sin(\delta)\widetilde{g}^{\,\prime}(b)&=0\end{aligned}\bigg{\\}},$

is a self-adjoint extension of $T_{min}$.
$(ii)$ If $\varphi\in[0,\pi)$ and $R\in\text{{SL}}(2,{\mathbb{R}})$, then the
operator $T_{\varphi,R}$ defined by

$\displaystyle T_{\varphi,R}f=T_{max}f,$ (3.3) $\displaystyle
f\in\operatorname{dom}(T_{\varphi,R})=\bigg{\\{}g\in\operatorname{dom}(T_{max})\,\bigg{|}\,\begin{pmatrix}\widetilde{g}(b)\\\
\widetilde{g}^{\,\prime}(b)\end{pmatrix}=e^{i\varphi}R\begin{pmatrix}\widetilde{g}(a)\\\
\widetilde{g}^{\,\prime}(a)\end{pmatrix}\bigg{\\}},$

is a self-adjoint extension of $T_{min}$.
$(iii)$ If $T$ is a self-adjoint extension of $T_{min}$, then either
$T=T_{\gamma,\delta}$ for some $\gamma,\delta\in[0,\pi)$, or $T=T_{\varphi,R}$
for some $\varphi\in[0,\pi)$ and some $R\in\text{{SL}}(2,{\mathbb{R}})$.
Notational Convention. To describe all possible self-adjoint boundary
conditions associated with self-adjoint extensions of $T_{min}$ effectively,
we will frequently employ the notation $T_{A,B}$, $M_{A,B}^{Do}(\,\cdot\,)$,
etc., where $A,B$ represents $\gamma,\delta$ in the case of separated boundary
conditions and $\varphi,R$ in the context of coupled boundary conditions.

Choosing $\gamma=\delta=0$ in (3.2) yields the self-adjoint extension with
Dirichlet-type boundary conditions at $a$ and $b$, equivalently, the
Friedrichs extension $T_{F}$ of $T_{min}$:

$\operatorname{dom}(T_{0,0})=\operatorname{dom}(T_{F})=\\{g\in\operatorname{dom}(T_{max})\,|\,\widetilde{g}(a)=\widetilde{g}(b)=0\\}.$
(3.4)

Since the coefficients of the Sturm–Liouville differential expression are
real, the following conjugation property holds:

$\overline{u_{j}(z,\,\cdot\,)}=u_{j}(\overline{z},\,\cdot\,),\quad
z\in\rho(T_{0,0}),\,j\in\\{1,2\\}.$ (3.5)

Applying (3.1), one computes

$\begin{split}W(u_{1}(z,\,\cdot\,),u_{2}(z,\,\cdot\,)(a)&=-\widetilde{u}_{1}^{\,\prime}(z,a),\\\
W(u_{1}(z,\,\cdot\,),u_{2}(z,\,\cdot\,)(b)&=\widetilde{u}_{2}^{\,\prime}(z,b),\quad
z\in\rho(T_{0,0}).\end{split}$ (3.6)

In particular, since the Wronskian of two solutions is constant,

$\widetilde{u}_{2}^{\,\prime}(z,b)=-\widetilde{u}_{1}^{\,\prime}(z,a),\quad
z\in\rho(T_{0,0}).$ (3.7)

We begin by recalling the orthonormal basis for ${\mathcal{N}}_{\pm i}$ given
by $\\{v_{j}(\pm i,\,\cdot\,)\\}_{j=1,2}$,

$\displaystyle v_{1}(\pm i,\,\cdot\,)$ $\displaystyle=c_{1}(\pm i)u_{1}(\pm
i,\,\cdot\,),$ (3.8) $\displaystyle v_{2}(\pm i,\,\cdot\,)$
$\displaystyle=c_{2}(\pm i)\bigg{[}u_{2}(\pm i,\,\cdot\,)-\frac{(u_{1}(\pm
i,\,\cdot\,),u_{2}(\pm i,\,\cdot\,))_{L^{2}((a,b);rdx)}}{\|u_{1}(\pm
i,\,\cdot\,)\|_{L^{2}((a,b);rdx)}^{2}}u_{1}(\pm i,\,\cdot\,)\bigg{]}$ (3.9)
$\displaystyle\;=c_{2}(\pm i)\bigg{[}u_{2}(\pm i,\,\cdot\,)-\frac{\text{\rm
Im}\big{(}\widetilde{u}_{2}^{\,\prime}(i,b)\big{)}}{\text{\rm
Im}\big{(}\widetilde{u}_{1}^{\,\prime}(i,b)\big{)}}u_{1}(\pm
i,\,\cdot\,)\bigg{]},$

with

$\displaystyle c_{1}(\pm i)$ $\displaystyle=\|u_{1}(\pm
i,\,\cdot\,)\|_{L^{2}((a,b);rdx)}^{-1}=\big{[}\mp\text{\rm
Im}\big{(}\widetilde{u}_{1}^{\,\prime}(\pm i,b)\big{)}\big{]}^{-1/2},$ (3.10)
$\displaystyle c_{2}(\pm i)$ $\displaystyle=\bigg{\|}u_{2}(\pm
i,\,\cdot\,)-\frac{\text{\rm
Im}\big{(}\widetilde{u}_{2}^{\,\prime}(i,b)\big{)}}{\text{\rm
Im}\big{(}\widetilde{u}_{1}^{\,\prime}(i,b)\big{)}}u_{1}(\pm
i,\,\cdot\,)\bigg{\|}_{L^{2}((a,b);rdx)}^{-1}$ (3.11)
$\displaystyle\;=\bigg{[}\pm\text{\rm
Im}\big{(}\widetilde{u}_{2}^{\,\prime}(\pm
i,a)\big{)}\pm\frac{\big{[}\text{\rm
Im}\big{(}\widetilde{u}_{2}^{\,\prime}(\pm i,b)\big{)}\big{]}^{2}}{\text{\rm
Im}\big{(}\widetilde{u}_{1}^{\,\prime}(\pm i,b)\big{)}}\bigg{]}^{-1/2}.$

The Donoghue $m$-function $M_{T_{A,B},\,{\mathcal{N}}_{i}}^{Do}(\,\cdot\,)$
with $T_{A,B}$ any self-adjoint extension of $T_{min}$ is provided next (cf.
Theorems 6.1–6.3 in [27]).

######  Theorem 3.2.

Assume Hypothesis 3.1 and let $\\{v_{j}(i,\,\cdot\,)\\}_{j=1,2}$ be the
orthonormal basis for ${\mathcal{N}}_{i}$ defined in (3.8)–(3.11). The
Donoghue $m$-function
$M_{T_{0,0},\,{\mathcal{N}}_{i}}^{Do}(\,\cdot\,):{\mathbb{C}}\backslash{\mathbb{R}}\to{\mathcal{B}}({\mathcal{N}}_{i})$
for $T_{0,0}$ satisfies

$\displaystyle M_{T_{0,0},\,{\mathcal{N}}_{i}}^{Do}(\pm i)$ $\displaystyle=\pm
iI_{{\mathcal{N}}_{i}},$ $\displaystyle
M_{T_{0,0},\,{\mathcal{N}}_{i}}^{Do}(z)$
$\displaystyle=-\sum_{j,k=1}^{2}[i\delta_{j,k}+W_{j,k}(z)](v_{k}(i,\,\cdot\,),\,\cdot\,)_{L^{2}((a,b);rdx)}v_{j}(i,\,\cdot\,)\big{|}_{{\mathcal{N}}_{i}},$
(3.12)
$\displaystyle=-iI_{{\mathcal{N}}_{i}}-\sum_{j,k=1}^{2}W_{j,k}(z)\big{(}v_{k}(i,\,\cdot\,),\,\cdot\,\big{)}_{L^{2}((a,b);rdx)}v_{j}(i,\,\cdot\,)\big{|}_{{\mathcal{N}}_{i}},$
$\displaystyle\hskip
176.407ptz\in{\mathbb{C}}\backslash{\mathbb{R}},\,z\neq\pm i,$

where the matrix $\big{(}W_{j,k}(\,\cdot\,)\big{)}_{j,k=1}^{2}$,
$z\in{\mathbb{C}}\backslash{\mathbb{R}}$, $z\neq\pm i$, is given by

$\displaystyle
W_{1,1}(z)=[c_{1}(i)]^{2}\big{[}\widetilde{u}_{1}^{\,\prime}(z,b)-\widetilde{u}_{1}^{\,\prime}(-i,b)\big{]},$
(3.13) $\displaystyle W_{1,2}(z)=c_{1}(i)c_{2}(i)\Bigg{\\{}\frac{\text{\rm
Im}\big{(}\widetilde{u}_{2}^{\,\prime}(i,b)\big{)}}{\text{\rm
Im}\big{(}\widetilde{u}_{1}^{\,\prime}(i,b)\big{)}}\big{[}\widetilde{u}_{1}^{\,\prime}(-i,b)-\widetilde{u}_{1}^{\,\prime}(z,b)\big{]}$
(3.14) $\displaystyle\hskip
93.89418pt+\widetilde{u}_{2}^{\,\prime}(z,b)+\widetilde{u}_{1}^{\,\prime}(-i,a)\Bigg{\\}},$
$\displaystyle W_{2,1}(z)=-c_{1}(i)c_{2}(i)\Bigg{\\{}\frac{\text{\rm
Im}\big{(}\widetilde{u}_{2}^{\,\prime}(i,b)\big{)}}{\text{\rm
Im}\big{(}\widetilde{u}_{1}^{\,\prime}(i,b)\big{)}}\big{[}\widetilde{u}_{1}^{\,\prime}(z,b)-\widetilde{u}_{1}^{\,\prime}(-i,b)\big{]}$
(3.15) $\displaystyle\hskip
99.58464pt+\widetilde{u}_{2}^{\,\prime}(-i,b)+\widetilde{u}_{1}^{\,\prime}(z,a)\Bigg{\\}},$
$\displaystyle
W_{2,2}(z)=[c_{2}(i)]^{2}\Bigg{\\{}\Bigg{[}\widetilde{u}_{2}^{\,\prime}(-i,b)-\widetilde{u}_{2}^{\,\prime}(z,b)$
(3.16) $\displaystyle\hskip 88.2037pt+\frac{\text{\rm
Im}\big{(}\widetilde{u}_{2}^{\,\prime}(i,b)\big{)}}{\text{\rm
Im}\big{(}\widetilde{u}_{1}^{\,\prime}(i,b)\big{)}}\big{[}\widetilde{u}_{1}^{\,\prime}(z,b)-\widetilde{u}_{1}^{\,\prime}(-i,b)\big{]}\Bigg{]}\frac{\text{\rm
Im}\big{(}\widetilde{u}_{2}^{\,\prime}(i,b)\big{)}}{\text{\rm
Im}\big{(}\widetilde{u}_{1}^{\,\prime}(i,b)\big{)}}$ $\displaystyle\hskip
85.35826pt+\widetilde{u}_{2}^{\,\prime}(-i,a)-\widetilde{u}_{2}^{\,\prime}(z,a)+\frac{\text{\rm
Im}\big{(}\widetilde{u}_{2}^{\,\prime}(i,b)\big{)}}{\text{\rm
Im}\big{(}\widetilde{u}_{1}^{\,\prime}(i,b)\big{)}}\big{[}\widetilde{u}_{1}^{\,\prime}(z,a)-\widetilde{u}_{1}^{\,\prime}(-i,a)\big{]}\Bigg{\\}}.$

Furthermore, the following items $(i)$–$(v)$ hold.
$(i)$ If $\gamma,\delta\in(0,\pi)$, then the Donoghue $m$-function
$M_{T_{\gamma,\delta},\,{\mathcal{N}}_{i}}^{Do}(\,\cdot\,):{\mathbb{C}}\backslash{\mathbb{R}}\to{\mathcal{B}}({\mathcal{N}}_{i})$
for $T_{\gamma,\delta}$ satisfies

$\displaystyle M_{T_{\gamma,\delta},\,{\mathcal{N}}_{i}}^{Do}(\pm i)=\pm
iI_{{\mathcal{N}}_{i}},$ $\displaystyle
M_{T_{\gamma,\delta},\,{\mathcal{N}}_{i}}^{Do}(z)=M_{T_{0,0},\,{\mathcal{N}}_{i}}^{Do}(z)$
(3.17)
$\displaystyle\quad+(i-z)\sum_{j,k,\ell=1}^{2}\big{[}K_{\gamma,\delta}(z)^{-1}\big{]}_{j,k}W_{\ell,k}^{Kr}(z)(u_{j}(\overline{z},\,\cdot\,),\,\cdot\,)_{L^{2}((a,b);rdx)}v_{\ell}(i,\,\cdot\,)\big{|}_{{\mathcal{N}}_{i}},$
$\displaystyle\hskip
248.96161ptz\in{\mathbb{C}}\backslash{\mathbb{R}},\,z\neq\pm i,$

where the invertible matrix $K_{\gamma,\delta}(\,\cdot\,)$ and
$\big{(}W_{\ell,k}^{Kr}(\,\cdot\,)\big{)}_{\ell,k=1}^{2}$ are given by

$\displaystyle K_{\gamma,\delta}(z)$
$\displaystyle=\begin{pmatrix}\cot(\delta)+\widetilde{u}_{1}^{\,\prime}(z,b)&-\widetilde{u}_{1}^{\,\prime}(z,a)\\\\[5.69054pt]
\widetilde{u}_{2}^{\,\prime}(z,b)&-\cot(\gamma)-\widetilde{u}_{2}^{\,\prime}(z,a)\end{pmatrix},$
(3.18) $\displaystyle W_{1,1}^{Kr}(z)$
$\displaystyle=c_{1}(i)\big{[}\widetilde{u}_{1}^{\,\prime}(z,b)-\widetilde{u}_{1}^{\,\prime}(-i,b)\big{]},$
(3.19) $\displaystyle W_{1,2}^{Kr}(z)$
$\displaystyle=c_{1}(i)\big{[}\widetilde{u}_{2}^{\,\prime}(z,b)+\widetilde{u}_{1}^{\,\prime}(-i,a)\big{]},$
(3.20) $\displaystyle W_{2,1}^{Kr}(z)$
$\displaystyle=\widetilde{v}_{2}(-i,b)\widetilde{u}_{1}^{\,\prime}(z,b)-\widetilde{v}_{2}^{\,\prime}(-i,b)-\widetilde{v}_{2}(-i,a)\widetilde{u}_{1}^{\,\prime}(z,a)$
(3.21) $\displaystyle=-c_{2}(i)\bigg{\\{}\frac{\text{\rm
Im}\big{(}\widetilde{u}_{2}^{\,\prime}(i,b)\big{)}}{\text{\rm
Im}\big{(}\widetilde{u}_{1}^{\,\prime}(i,b)\big{)}}\big{[}\widetilde{u}_{1}^{\,\prime}(z,b)-\widetilde{u}_{1}^{\,\prime}(-i,b)\big{]}+\widetilde{u}_{2}^{\,\prime}(-i,b)+\widetilde{u}_{1}^{\,\prime}(z,a)\bigg{\\}},$
$\displaystyle W_{2,2}^{Kr}(z)$
$\displaystyle=\widetilde{v}_{2}(-i,b)\widetilde{u}_{2}^{\,\prime}(z,b)-\widetilde{v}_{2}(-i,a)\widetilde{u}_{2}^{\,\prime}(z,a)+\widetilde{v}_{2}^{\,\prime}(-i,a)$
(3.22) $\displaystyle=-c_{2}(i)\bigg{\\{}\frac{\text{\rm
Im}\big{(}\widetilde{u}_{2}^{\,\prime}(i,b)\big{)}}{\text{\rm
Im}\big{(}\widetilde{u}_{1}^{\,\prime}(i,b)\big{)}}\big{[}\widetilde{u}_{2}^{\,\prime}(z,b)+\widetilde{u}_{1}^{\,\prime}(-i,a)\big{]}+\widetilde{u}_{2}^{\,\prime}(z,a)-\widetilde{u}_{2}^{\,\prime}(-i,a)\bigg{\\}}.$

$(ii)$ If $\varphi\in[0,\pi)$ and $R\in\text{{SL}}(2,{\mathbb{R}})$ with
$R_{1,2}\neq 0$, then the Donoghue $m$-function
$M_{T_{\varphi,R},\,{\mathcal{N}}_{i}}^{Do}(\,\cdot\,):{\mathbb{C}}\backslash{\mathbb{R}}\to{\mathcal{B}}({\mathcal{N}}_{i})$
for $T_{\varphi,R}$ satisfies

$\displaystyle M_{T_{\varphi,R},\,{\mathcal{N}}_{i}}^{Do}(\pm i)=\pm
iI_{{\mathcal{N}}_{i}},$ $\displaystyle
M_{T_{\varphi,R},\,{\mathcal{N}}_{i}}^{Do}(z)=M_{T_{0,0},\,{\mathcal{N}}_{i}}^{Do}(z)$
(3.23)
$\displaystyle\quad+(i-z)\sum_{j,k,\ell=1}^{2}\big{[}K_{\varphi,R}(z)^{-1}\big{]}_{j,k}W_{\ell,k}^{Kr}(z)(u_{j}(\overline{z},\,\cdot\,),\,\cdot\,)_{L^{2}((a,b);rdx)}v_{\ell}(i,\,\cdot\,)\big{|}_{{\mathcal{N}}_{i}},$
$\displaystyle\hskip
251.80705ptz\in{\mathbb{C}}\backslash{\mathbb{R}},\,z\neq\pm i,$

where $\big{(}W_{\ell,k}^{Kr}(\,\cdot\,)\big{)}_{\ell,k=1}^{2}$ is once again
given in (3.19)–(3.22) and the invertible matrix $K_{\varphi,R}(\,\cdot\,)$ is
given by

$K_{\varphi,R}(z)=\begin{pmatrix}-\dfrac{R_{2,2}}{R_{1,2}}+\widetilde{u}_{1}^{\,\prime}(z,b)&\dfrac{e^{-i\varphi}}{R_{1,2}}-\widetilde{u}_{1}^{\,\prime}(z,a)\\\\[11.38109pt]
\dfrac{e^{i\varphi}}{R_{1,2}}+\widetilde{u}_{2}^{\,\prime}(z,b)&-\dfrac{R_{1,1}}{R_{1,2}}-\widetilde{u}_{2}^{\,\prime}(z,a)\end{pmatrix}.$
(3.24)

$(iii)$ If $\gamma\in(0,\pi)$, then the Donoghue $m$-function
$M_{T_{\gamma,0},\,{\mathcal{N}}_{i}}^{Do}(\,\cdot\,):{\mathbb{C}}\backslash{\mathbb{R}}\to{\mathcal{B}}({\mathcal{N}}_{i})$
for $T_{\gamma,0}$ satisfies

$\displaystyle M_{T_{\gamma,0},\,{\mathcal{N}}_{i}}^{Do}(\pm i)$
$\displaystyle=\pm iI_{{\mathcal{N}}_{i}},$ $\displaystyle
M_{T_{\gamma,0},\,{\mathcal{N}}_{i}}^{Do}(z)$
$\displaystyle=M_{T_{0,0},\,{\mathcal{N}}_{i}}^{Do}(z)$ (3.25)
$\displaystyle\quad+\frac{z-i}{\cot(\gamma)+\widetilde{u}_{2}^{\,\prime}(z,a)}(u_{2}(\overline{z},\,\cdot\,),\,\cdot\,)_{L^{2}((a,b);rdx)}\sum_{\ell=1}^{2}W_{\ell,2}^{Kr}(z)v_{\ell}(i,\,\cdot\,)\big{|}_{{\mathcal{N}}_{i}},$
$\displaystyle\hskip
220.50885ptz\in{\mathbb{C}}\backslash{\mathbb{R}},\,z\neq\pm i,$

where $\cot(\gamma)+\widetilde{u}_{2}^{\,\prime}(z,a)\neq 0$ and the scalars
$\big{\\{}W_{\ell,2}^{Kr}(\,\cdot\,)\big{\\}}_{\ell=1,2}$ are given by (3.20)
and (3.22).
$(iv)$ If $\delta\in(0,\pi)$, then the Donoghue $m$-function
$M_{T_{0,\delta},\,{\mathcal{N}}_{i}}^{Do}(\,\cdot\,):{\mathbb{C}}\backslash{\mathbb{R}}\to{\mathcal{B}}({\mathcal{N}}_{i})$
for $T_{0,\delta}$ satisfies

$\displaystyle M_{T_{0,\delta},\,{\mathcal{N}}_{i}}^{Do}(\pm i)$
$\displaystyle=\pm iI_{{\mathcal{N}}_{i}},$ $\displaystyle
M_{T_{0,\delta},\,{\mathcal{N}}_{i}}^{Do}(z)$
$\displaystyle=M_{T_{0,0},\,{\mathcal{N}}_{i}}^{Do}(z)$ (3.26)
$\displaystyle\quad-\frac{z-i}{\cot(\delta)+\widetilde{u}_{1}^{\,\prime}(z,b)}(u_{1}(\overline{z},\,\cdot\,),\,\cdot\,)_{L^{2}((a,b);rdx)}\sum_{\ell=1}^{2}W_{\ell,1}^{Kr}(z)v_{\ell}(i,\,\cdot\,)\big{|}_{{\mathcal{N}}_{i}},$
$\displaystyle\hskip
217.6634ptz\in{\mathbb{C}}\backslash{\mathbb{R}},\,z\neq\pm i,$

where $\cot(\delta)+\widetilde{u}_{1}^{\,\prime}(z,b)\neq 0$ and the scalars
$\big{\\{}W_{\ell,1}^{Kr}(\,\cdot\,)\big{\\}}_{\ell=1,2}$ are given by (3.19)
and (3.21).
$(v)$ If $\varphi\in[0,\pi)$ and $R\in\text{{SL}}(2,{\mathbb{R}})$ with
$R_{1,2}=0$, then the Donoghue $m$-function
$M_{T_{\varphi,R},\,{\mathcal{N}}_{i}}^{Do}(\,\cdot\,):{\mathbb{C}}\backslash{\mathbb{R}}\to{\mathcal{B}}({\mathcal{N}}_{i})$
for $T_{\varphi,R}$ satisfies

$\displaystyle M_{T_{\varphi,R},\,{\mathcal{N}}_{i}}^{Do}(\pm i)=\pm
iI_{{\mathcal{N}}_{i}},$ $\displaystyle
M_{T_{\varphi,R},\,{\mathcal{N}}_{i}}^{Do}(z)=M_{T_{0,0},\,{\mathcal{N}}_{i}}^{Do}(z)$
(3.27)
$\displaystyle\quad-\frac{z-i}{k_{\varphi,R}(z)}(u_{\varphi,R}(\overline{z},\,\cdot\,),\,\cdot\,)_{L^{2}((a,b);rdx)}\sum_{\ell=1}^{2}\big{[}e^{-i\varphi}R_{2,2}W_{\ell,2}^{Kr}(z)+W_{\ell,1}^{Kr}(z)\big{]}v_{\ell}(i,\,\cdot\,)\big{|}_{{\mathcal{N}}_{i}},$
$\displaystyle\hskip
284.52756ptz\in{\mathbb{C}}\backslash{\mathbb{R}},\,z\neq\pm i,$

where the matrix $\big{(}W_{\ell,k}^{Kr}(\,\cdot\,)\big{)}_{\ell,k=1}^{2}$ is
once again given in (3.19)–(3.22) and the nonzero scalar
$k_{\varphi,R}(\,\cdot\,)$ is given by

$\displaystyle
k_{\varphi,R}(z)=-R_{2,1}R_{2,2}-e^{i\varphi}R_{2,2}\widetilde{u}_{\varphi,R}^{\,\prime}(z,a)+\widetilde{u}_{\varphi,R}^{\,\prime}(z,b),$
(3.28)

where

$\displaystyle
u_{\varphi,R}(\zeta,\,\cdot\,)=e^{-i\varphi}R_{2,2}u_{2}(\zeta,\,\cdot\,)+u_{1}(\zeta,\,\cdot\,),\quad\zeta\in\rho(T_{0,0}).$
(3.29)

######  Remark 3.3.

For the Krein extension, $T_{0,R_{K}}$, under the additional assumption that
$T_{min}\geqslant\varepsilon I_{{}_{L^{2}((a,b);rdx)}}$ for some
$\varepsilon>0$, applying [24, Theorem 3.5 $(ii)$], one computes for the
matrix $K_{0,R_{K}}$,

$K_{0,R_{K}}(z)=\begin{pmatrix}\widetilde{u}_{1}^{\,\prime}(z,b)-\widetilde{u}_{1}^{\,\prime}(0,b)&\widetilde{u}_{1}^{\,\prime}(0,a)-\widetilde{u}_{1}^{\,\prime}(z,a)\\\\[11.38109pt]
\widetilde{u}_{2}^{\,\prime}(z,b)-\widetilde{u}_{2}^{\,\prime}(0,b)&\widetilde{u}_{2}^{\,\prime}(0,a)-\widetilde{u}_{2}^{\,\prime}(z,a)\end{pmatrix},\quad
z\in\rho(T_{0,0})\cap\rho(T_{0,R_{K}}),$ (3.30)

where we note that $0\in\sigma(T_{0,R_{K}})$. $\diamond$

##  4\. Donoghue $m$-functions: One Limit Circle Endpoint

In this section we recall the Donoghue $m$-functions in the case where $\tau$
is in the limit circle case at precisely one endpoint (which we choose to be
$a$ without loss of generality) following [27, Sect. 5].

######  Hypothesis 4.1.

In addition to Hypothesis 2.1 assume that $\tau$ is in the limit circle case
at $a$ and in the limit point case at $b$. Moreover, for $z\in\rho(T_{0})$,
let $\psi(z,\,\cdot\,)$ denote the unique solution to $(\tau-z)y=0$ that
satisfies $\psi(z,\,\cdot\,)\in L^{2}((a,b);rdx)$ and
$\widetilde{\psi}(z,a)=1$.

Assume Hypothesis 4.1. By Theorem 2.5 or Theorem 2.9, the following statements
$(i)$ and $(ii)$ hold.
$(i)$ If $\gamma\in[0,\pi)$, then the operator $T_{\gamma}$ defined by

$\begin{split}&T_{\gamma}f=T_{max}f,\\\
&f\in\operatorname{dom}(T_{\gamma})=\\{g\in\operatorname{dom}(T_{max})\,|\,\cos(\gamma)\widetilde{g}(a)+\sin(\gamma)\widetilde{g}^{\,\prime}(a)=0\\},\end{split}$
(4.1)

is a self-adjoint extension of $T_{min}$.
$(ii)$ If $T$ is a self-adjoint extension of $T_{min}$, then $T=T_{\gamma}$
for some $\gamma\in[0,\pi)$.
Statements analogous to $(i)$ and $(ii)$ hold if $\tau$ is in the limit point
case at $a$ and in the limit circle case at $b$; for brevity we omit the
details.

Choosing $\gamma=0$ in (4.1) yields the self-adjoint extension $T_{0}$ with a
Dirichlet-type boundary condition at $a$:

$\operatorname{dom}(T_{0})=\\{g\in\operatorname{dom}(T_{max})\,|\,\widetilde{g}(a)=0\\}.$
(4.2)

Since the coefficients $p$, $q$, and $r$ are real-valued, the solution
$\psi(z,\,\cdot\,)$ has the following conjugation property:

$\overline{\psi(z,\,\cdot\,)}=\psi(\overline{z},\,\cdot\,),\quad
z\in\rho(T_{0}).$ (4.3)

We now turn to the Donoghue $m$-function
$M_{T_{\gamma},\,{\mathcal{N}}_{i}}^{Do}(\,\cdot\,)$ with $T_{\gamma}$ any
self-adjoint extension of $T_{min}$ (cf. Theorems 5.1 and 5.2 in [27]).

######  Theorem 4.2.

Assume Hypothesis 4.1 and let $\gamma\in[0,\pi)$. The Donoghue $m$-function
$M_{T_{\gamma},\,{\mathcal{N}}_{i}}^{Do}(\,\cdot\,):{\mathbb{C}}\backslash{\mathbb{R}}\to{\mathcal{B}}({\mathcal{N}}_{i})$
for $T_{\gamma}$ satisfies

$\displaystyle M_{T_{\gamma},\,{\mathcal{N}}_{i}}^{Do}(\pm i)$
$\displaystyle=\pm iI_{{\mathcal{N}}_{i}},\quad\gamma\in[0,\pi),$
$\displaystyle M_{T_{0},\,{\mathcal{N}}_{i}}^{Do}(z)$
$\displaystyle=\Bigg{[}-i+\frac{\widetilde{\psi}^{\,\prime}(z,a)-\widetilde{\psi}^{\,\prime}(-i,a)}{\text{\rm
Im}\big{(}\widetilde{\psi}^{\,\prime}(i,a)\big{)}}\Bigg{]}I_{{\mathcal{N}}_{i}},\quad
z\in{\mathbb{C}}\backslash{\mathbb{R}},\,z\neq\pm i,$ $\displaystyle
M_{T_{\gamma},\,{\mathcal{N}}_{i}}^{Do}(z)$
$\displaystyle=M_{T_{0},\,{\mathcal{N}}_{i}}^{Do}(z)$ (4.4)
$\displaystyle\quad+(i-z)\frac{\widetilde{\psi}^{\,\prime}(z,a)-\widetilde{\psi}^{\,\prime}(-i,a)}{\cot(\gamma)+\widetilde{\psi}^{\,\prime}(z,a)}(\psi(\overline{z},\,\cdot\,),\,\cdot\,)_{L^{2}((a,b);rdx)}\psi(i,\,\cdot\,)\bigg{|}_{{\mathcal{N}}_{i}},$
$\displaystyle\hskip
157.91287pt\gamma\in(0,\pi),\,z\in{\mathbb{C}}\backslash{\mathbb{R}},\,z\neq\pm
i.$

##  5\. The Jacobi Operator and its Donoghue $m$-functions

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

$\displaystyle\begin{split}\tau_{\alpha,\beta}=-(1-x)^{-\alpha}(1+x)^{-\beta}(d/dx)\big{(}(1-x)^{\alpha+1}(1+x)^{\beta+1}\big{)}(d/dx),&\\\
x\in(-1,1),\;\alpha,\beta\in{\mathbb{R}},\end{split}$ (5.1)

that is, in connection with Sections 2 one now has

$\displaystyle a=-1,\quad b=1,$ $\displaystyle
p(x)=p_{\alpha,\beta}(x)=(1-x)^{\alpha+1}(1+x)^{\beta+1},\quad
q(x)=q_{\alpha,\beta}(x)=0,$ (5.2) $\displaystyle
r(x)=r_{\alpha,\beta}(x)=(1-x)^{\alpha}(1+x)^{\beta},\quad
x\in(-1,1),\quad\alpha,\beta\in{\mathbb{R}}$

(see, e.g. [1, Ch. 22], [4], [9], [20, Sect. 23], [38, Ch. 4], [44, Sects.
VII.6.1, XIV.2], [62, Ch. 18], [71, 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 Hilbert space leading to
an equivalent differential expression in the Hilbert space $L^{2}((-1,1);dx)$
(cf. [18, p. 1510–1520], [20, Sect. 37], [34]). For more recent developments
see, for instance, [21], [22], [23], [43], [48].

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

$y_{2}(x)=y_{1}(x)\int_{c}^{x}dx^{\prime}\,p(x^{\prime})^{-1}y_{1}(x^{\prime})^{-2},\quad
c,x\in(a,b).$ (5.3)

Returning to the concrete Jacobi case at hand, one can choose

$\displaystyle y_{1}(x)=1,\quad x\in(-1,1),$ $\displaystyle
y_{2}(x)=\int_{0}^{x}dx^{\prime}\,(1-x^{\prime})^{-1-\alpha}(1+x^{\prime})^{-1-\beta}$
(5.4)
$\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, one has 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 at $-1$ if and only if $\alpha\in{\mathbb{R}}$,
$\beta\in[0,1)$,}\\\ \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 at $+1$ if and only
if $\alpha\in[0,1)$, $\beta\in{\mathbb{R}}$,}\\\ \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}$ (5.5)

The maximal and preminimal operators, $T_{max,\alpha,\beta}$ and
$T_{min,0,\alpha,\beta}$, associated to $\tau_{\alpha,\beta}$ in
$L^{2}((-1,1);r_{\alpha,\beta}dx)$ are then given by

$\displaystyle T_{max,\alpha,\beta}f=\tau_{\alpha,\beta}f,$ $\displaystyle
f\in\operatorname{dom}(T_{max,\alpha,\beta})=\big{\\{}g\in
L^{2}((-1,1);r_{\alpha,\beta}dx)\,\big{|}\,g,g^{[1]}\in AC_{loc}((-1,1));$
$\displaystyle\hskip 193.47882pt\tau_{\alpha,\beta}g\in
L^{2}((-1,1);r_{\alpha,\beta}dx)\big{\\}},$ (5.6)

and

$\displaystyle T_{min,0,\alpha,\beta}f=\tau_{\alpha,\beta}f,$ $\displaystyle
f\in\operatorname{dom}(T_{min,0,\alpha,\beta})=\big{\\{}g\in
L^{2}((-1,1);r_{\alpha,\beta}dx)\,\big{|}\,g,g^{[1]}\in AC_{loc}((-1,1));$
(5.7) $\displaystyle\hskip
62.59596pt\operatorname{supp}\,(g)\subset(-1,1)\text{ is compact;
}\tau_{\alpha,\beta}g\in L^{2}((-1,1);r_{\alpha,\beta}dx)\big{\\}}.$

The fact (5.4) 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:

$\displaystyle\begin{split}u_{-1,\alpha,\beta}(0,x)&=\begin{cases}-2^{-\alpha-1}\beta^{-1}(1+x)^{-\beta}[1+O(1+x)],&\beta\in(-\infty,0),\\\
1,&\beta\in[0,\infty),\end{cases}\\\
\widehat{u}_{-1,\alpha,\beta}(0,x)&=\begin{cases}1,&\beta\in(-\infty,0),\\\
-2^{-\alpha-1}\text{\rm ln}((1+x)/2),&\beta=0,\\\
2^{-\alpha-1}\beta^{-1}(1+x)^{-\beta}[1+O(1+x)],&\beta\in(0,\infty),\end{cases}\end{split}\quad\alpha\in{\mathbb{R}},$
(5.8)

and

$\displaystyle\begin{split}u_{+1,\alpha,\beta}(0,x)&=\begin{cases}2^{-\beta-1}\alpha^{-1}(1-x)^{-\alpha}[1+O(1-x)],&\alpha\in(-\infty,0),\\\
1,&\alpha\in[0,\infty),\end{cases}\\\
\widehat{u}_{+1,\alpha,\beta}(0,x)&=\begin{cases}1,&\alpha\in(-\infty,0),\\\
2^{-\beta-1}\text{\rm ln}((1-x)/2),&\alpha=0,\\\
-2^{-\beta-1}\alpha^{-1}(1-x)^{-\alpha}[1+O(1-x)],&\alpha\in(0,\infty),\end{cases}\end{split}\quad\beta\in{\mathbb{R}}.$
(5.9)

Combining the fact (5.5) with Theorem 2.2, $T_{min,0,\alpha,\beta}$ 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 (5.8), (5.9) at $\pm 1$,
according to (2.22), (2.23), generalized boundary values for
$g\in\operatorname{dom}(T_{max,\alpha,\beta})$ are of the form

$\displaystyle\begin{split}\widetilde{g}(-1)&=\begin{cases}g(-1),&\beta\in(-1,0),\\\
-2^{\alpha+1}\lim_{x\downarrow-1}g(x)/\text{\rm ln}((1+x)/2),&\beta=0,\\\
\beta
2^{\alpha+1}\lim_{x\downarrow-1}(1+x)^{\beta}g(x),&\beta\in(0,1),\end{cases}\\\
{\widetilde{g}}^{\,\prime}(-1)&=\begin{cases}g^{[1]}(-1),&\beta\in(-1,0),\\\
\lim_{x\downarrow-1}\big{[}g(x)+\widetilde{g}(-1)2^{-\alpha-1}\text{\rm
ln}((1+x)/2)\big{]},&\beta=0,\\\
\lim_{x\downarrow-1}\big{[}g(x)-\widetilde{g}(-1)2^{-\alpha-1}\beta^{-1}(1+x)^{-\beta}\big{]},&\beta\in(0,1),\end{cases}\end{split}\quad\alpha\in{\mathbb{R}},$
(5.10)
$\displaystyle\begin{split}\widetilde{g}(1)&=\begin{cases}g(1),&\alpha\in(-1,0),\\\
2^{\beta+1}\lim_{x\uparrow 1}g(x)/\text{\rm ln}((1-x)/2),&\alpha=0,\\\ -\alpha
2^{\beta+1}\lim_{x\uparrow 1}(1-x)^{\alpha}g(x),&\alpha\in(0,1),\end{cases}\\\
{\widetilde{g}}^{\,\prime}(1)&=\begin{cases}g^{[1]}(1),&\alpha\in(-1,0),\\\
\lim_{x\uparrow 1}\big{[}g(x)-\widetilde{g}(1)2^{-\beta-1}\text{\rm
ln}((1-x)/2)\big{]},&\alpha=0,\\\ \lim_{x\uparrow
1}\big{[}g(x)+\widetilde{g}(1)2^{-\beta-1}\alpha^{-1}(1-x)^{-\alpha}\big{]},&\alpha\in(0,1),\end{cases}\end{split}\quad\beta\in{\mathbb{R}}.$
(5.11)

As a result, the minimal operator $T_{min}$ associated to
$\tau_{\alpha,\beta}$, that is, $T_{min}=\overline{T_{min,0}}$, is thus given
by

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

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

######  Remark 5.1.

We now mention a few special cases of interest. The Legendre equation
($\alpha=\beta=0$) has frequently been discussed in the literature, see, for
instance, [26] and the extensive list of references cited therein. The
Gegenbauer, or ultraspherical, equation (see, e.g., [1, Ch. 22], [62, Ch. 18],
[71, 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
$\mu\in{\mathbb{R}}\backslash(-1/2,3/2)$. The Chebyshev equations of the first
and second kinds are two more important special cases, with the first kind
realized by choosing $\mu=0$ in the Gegenbauer equation, or
$\alpha=\beta=-1/2$ in the Jacobi equation (see, e.g., [1, Ch. 22], [62, Ch.
18], [71, Ch. IV]), whereas 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], [62, Ch. 18], [71, Ch. IV]). $\diamond$

We now determine the solutions $\phi_{0,\alpha,\beta}(z,\,\cdot\,)$ and
$\theta_{0,\alpha,\beta}(z,\,\cdot\,)$ of
$\tau_{\alpha,\beta}u=zu,z\in{\mathbb{C}}$, that are subject to the conditions

$\displaystyle\begin{split}\widetilde{\phi}_{0,\alpha,\beta}(z,-1)&=0,\quad\widetilde{\phi}_{0,\alpha,\beta}^{\,\prime}(z,-1)=1,\\\
\widetilde{\theta}_{0,\alpha,\beta}(z,-1)&=1,\quad\widetilde{\theta}_{0,\alpha,\beta}^{\,\prime}(z,-1)=0.\end{split}$
(5.13)

In particular, one obtains from $\eqref{C.13}$,

$\displaystyle\begin{split}\phi_{0,\alpha,\beta}(z,x)&=\begin{cases}-2^{-\alpha-1}\beta^{-1}y_{2,\alpha,\beta,-1}(z,x),&\beta\in(-1,0),\\\
y_{1,\alpha,\beta,-1}(z,x),&\beta\in[0,1),\\\ \end{cases}\\\
\theta_{0,\alpha,\beta}(z,x)&=\begin{cases}y_{1,\alpha,\beta,-1}(z,x),&\beta\in(-1,0),\\\
-2^{-\alpha-1}y_{2,\alpha,0,-1}(z,x),&\beta=0,\\\
2^{-\alpha-1}\beta^{-1}y_{2,\alpha,\beta,-1}(z,x),&\beta\in(0,1),\end{cases}\\\
&\hskip
68.28644pt\alpha\in{\mathbb{R}},\;z\in{\mathbb{C}},\;x\in(-1,1).\end{split}$
(5.14)

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

In this section we compute the Donoghue $m$-function when the Jacobi problem
considered is either in the regular or limit circle case at $\pm 1$.

Using (5.13), the solutions in (3.1) for this example are given by

$\displaystyle
u_{1,\alpha,\beta}(z,x)=\phi_{0,\alpha,\beta}(z,x)/\widetilde{\phi}_{0,\alpha,\beta}(z,1)$
$\displaystyle\hskip
48.65439pt=\begin{cases}y_{2,\alpha,\beta,-1}(z,x)/\widetilde{y}_{2,\alpha,\beta,-1}(z,1),&\beta\in(-1,0),\\\
y_{1,\alpha,\beta,-1}(z,x)/\widetilde{y}_{1,\alpha,\beta,-1}(z,1),&\beta\in[0,1),\end{cases}$
(5.15) $\displaystyle
u_{2,\alpha,\beta}(z,x)=\theta_{0,\alpha,\beta}(z,x)-[\widetilde{\theta}_{0,\alpha,\beta}(z,1)/\widetilde{\phi}_{0,\alpha,\beta}(z,1)]\phi_{0,\alpha,\beta}(z,x)$
$\displaystyle=\begin{cases}y_{1,\alpha,\beta,-1}(z,x)-[\widetilde{y}_{1,\alpha,\beta,-1}(z,1)/\widetilde{y}_{2,\alpha,\beta,-1}(z,1)]y_{2,\alpha,\beta,-1}(z,x),\\\
\hskip 213.39566pt\beta\in(-1,0),\\\
-2^{-\alpha-1}\\{y_{2,\alpha,0,-1}(z,x)-[\widetilde{y}_{2,\alpha,0,-1}(z,1)/\widetilde{y}_{1,\alpha,0,-1}(z,1)]y_{1,\alpha,0,-1}(z,x)\\},\\\
\hskip 277.41437pt\beta=0,\\\
2^{-\alpha-1}\beta^{-1}\\{y_{2,\alpha,\beta,-1}(z,x)-[\widetilde{y}_{2,\alpha,\beta,-1}(z,1)/\widetilde{y}_{1,\alpha,\beta,-1}(z,1)]y_{1,\alpha,\beta,-1}(z,x)\\},\\\
\hfill\beta\in(0,1),\end{cases}$ $\displaystyle\hskip
202.01474pt\alpha\in(-1,1),\;z\in{\mathbb{C}},\;x\in(-1,1),$

where the generalized boundary values are given in (C.2)–(C.4). Hence
substituting (5.15) into (3.8)–(3.11) and applying Theorem 3.2 yields the
(Nevanlinna–Herglotz) Donoghue $m$-function
$M_{T_{A,B,\alpha,\beta},\,{\mathcal{N}}_{i}}^{Do}(\,\cdot\,)$ for any self-
adjoint extension $T_{A,B,\alpha,\beta}$ of $T_{min}$ with
$\alpha,\beta\in(-1,1)$.

As an example of coupled boundary conditions, we consider the Krein–von
Neumann extension following Example 4.3 found in [24]. 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 choices of $R_{K,\alpha,\beta}$ for the Krein–von Neumann
extension $T_{0,R_{K},\alpha,\beta}$ of $T_{min,\alpha,\beta}$:

$\displaystyle T_{0,R_{K},\alpha,\beta}f=\tau_{\alpha,\beta}f,$ (5.16)
$\displaystyle
f\in\operatorname{dom}(T_{0,R_{K},\alpha,\beta})=\bigg{\\{}g\in\operatorname{dom}(T_{max,\alpha,\beta})\,\bigg{|}\begin{pmatrix}\widetilde{g}(1)\\\
{\widetilde{g}}^{\,\prime}(1)\end{pmatrix}=R_{K,\alpha,\beta}\begin{pmatrix}\widetilde{g}(-1)\\\
{\widetilde{g}}^{\,\prime}(-1)\end{pmatrix}\bigg{\\}},$ $\displaystyle
R_{K,\alpha,\beta}=\begin{cases}\begin{pmatrix}1&2^{-\alpha-\beta-1}\dfrac{\Gamma(-\alpha)\Gamma(-\beta)}{\Gamma(-\alpha-\beta)}\\\
0&1\end{pmatrix},&\alpha,\beta\in(-1,0),\\\\[19.91692pt]
\begin{pmatrix}-2^{-\alpha-\beta-1}\dfrac{\Gamma(-\alpha)\Gamma(-\beta)}{\Gamma(-\alpha-\beta)}&1\\\
-1&0\end{pmatrix},&\alpha\in(-1,0),\;\beta\in(0,1),\\\\[19.91692pt]
\begin{pmatrix}0&-1\\\
1&2^{-\alpha-\beta-1}\dfrac{\Gamma(-\alpha)\Gamma(-\beta)}{\Gamma(-\alpha-\beta)}\end{pmatrix},&\alpha\in(0,1),\;\beta\in(-1,0),\\\\[19.91692pt]
\begin{pmatrix}0&-1\\\
1&-2^{-\beta-1}[\gamma_{E}+\psi(-\beta)]\end{pmatrix},&\alpha=0,\;\beta\in(-1,0)\\\\[19.91692pt]
\begin{pmatrix}2^{-\alpha-1}[\gamma_{E}+\psi(-\alpha)]&1\\\
-1&0\end{pmatrix},&\alpha\in(-1,0),\;\beta=0,\end{cases}$ (5.17)

where we interpret $1/\Gamma(0)=0$,
$\psi(\,\cdot\,)=\Gamma^{\prime}(\,\cdot\,)/\Gamma(\,\cdot\,)$ denotes the
Digamma function, and $\gamma_{E}=-\psi(1)=0.57721\dots$ represents Euler’s
constant. Obviously, $\det(R_{K,\alpha,\beta})=1$ in all five cases.
Furthermore, as $R_{1,2}\neq 0$ for each case, Theorem 3.2 $(ii)$ applies and
one obtains the Donoghue $m$-function
$M_{T_{0,R_{K},\alpha,\beta},\,{\mathcal{N}}_{i}}^{Do}(\,\cdot\,)$ for the
Krein–von Neumann extension $T_{0,R_{K},\alpha,\beta}$ by utilizing (5.15) and
(5.17) as well as the explicit form of $K_{0,R_{K}}(\,\cdot\,)$ in (3.30). We
note once again that
$M_{T_{0,R_{K},\alpha,\beta},\,{\mathcal{N}}_{i}}^{Do}(\,\cdot\,)$ is a
Nevanlinna–Herglotz function.

In the remaining four cases not covered by (5.17), given by all combinations
of $\alpha=0,\ \beta=0,\ \alpha\in(0,1)$, and $\beta\in(0,1)$, one observes
that [24, Theorem 3.5] is not applicable as the underlying minimal operator,
$T_{min,\alpha,\beta}$, is nonnegative but not strictly positive. In
particular, the Jacobi polynomials satisfy Friedrichs boundary conditions for
$\alpha,\beta\in[0,1)$, hence $0\in\sigma(T_{F,\alpha,\beta}),\
\alpha,\beta\in[0,1)$ and $T_{min,\alpha,\beta}\geqslant 0$ is nonnegative,
but not strictly positive when $\alpha,\beta\in[0,1)$.

###  5.2. Precisely One Interval Endpoint in the Limit Point Case

In this section we determine the Donoghue $m$-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 recall from [26, Sect. 6] that the Weyl–Titchmarsh–Kodaira solution and
$m$-function corresponding to the Friedrichs (resp., Dirichlet) boundary
condition at $x=-1$ is determined via the requirement

$\displaystyle\begin{split}\psi_{0,\alpha,\beta}(z,\,\cdot\,)=\theta_{0,\alpha,\beta}(z,\,\cdot\,)+m_{0,\alpha,\beta}(z)\phi_{0,\alpha,\beta}(z,\,\cdot\,)\in
L^{2}((c,1);r_{\alpha,\beta}dx),&\\\
z\in{\mathbb{C}}\backslash\sigma(T_{F,\alpha,\beta}),\;\alpha\in(-\infty,-1]\cup[1,\infty),\;\beta\in(-1,1),\;c\in(-1,1).&\end{split}$
(5.18)

In particular, since
$\widetilde{\psi}_{0,\alpha,\beta}^{\,\prime}(z,-1)=m_{0,\alpha,\beta}(z)$ one
finds from Theorem 4.2,

$\displaystyle\begin{split}M_{T_{0,\alpha,\beta},\,{\mathcal{N}}_{i}}^{Do}(z)&=\Bigg{[}-i+\dfrac{m_{0,\alpha,\beta}(z)-m_{0,\alpha,\beta}(-i)}{\text{\rm
Im}(m_{0,\alpha,\beta}(i))}\Bigg{]}I_{{\mathcal{N}}_{i}},\\\
M_{T_{\gamma,\alpha,\beta},\,{\mathcal{N}}_{i}}^{Do}(z)&=M_{T_{0,\alpha,\beta},\,{\mathcal{N}}_{i}}^{Do}(z)+(i-z)\frac{m_{0,\alpha,\beta}(z)-m_{0,\alpha,\beta}(-i)}{\cot(\gamma)+m_{0,\alpha,\beta}(z)}\\\
&\quad\;\times(\psi_{0,\alpha,\beta}(\overline{z},\,\cdot\,),\,\cdot\,)_{L^{2}((a,b);rdx)}\psi_{0,\alpha,\beta}(i,\,\cdot\,)\big{|}_{{\mathcal{N}}_{i}},\quad\gamma\in(0,\pi),\\\
&\hskip
59.75095pt\alpha\in(-\infty,-1]\cup[1,\infty),\;\beta\in(-1,1),\;z\in{\mathbb{C}}\backslash{\mathbb{R}},\end{split}$
(5.19)

where $\psi_{0,\alpha,\beta}(z,\,\cdot\,)$ and
$m_{0,\alpha,\beta}(z,\,\cdot\,)$ are given by the following:
$\mathbf{(I)}$ The Case $\alpha\in[1,\infty)$ and $\beta\in(-1,0)$:

$\displaystyle\begin{split}&\psi_{0,\alpha,\beta}(z,x)=y_{1,\alpha,\beta,-1}(z,x)-2^{-\alpha-1}\beta^{-1}y_{2,\alpha,\beta,-1}(z,x)m_{0,\alpha,\beta}(z),\\\
&m_{0,\alpha\,\beta}(z)=2^{1+\alpha+\beta}\beta\frac{\Gamma(1+\beta)}{\Gamma(1-\beta)}\\\
&\hskip
51.21504pt\times\frac{\Gamma([1+\alpha-\beta+\sigma_{\alpha,\beta}(z)]/2)\Gamma([1+\alpha-\beta-\sigma_{\alpha,\beta}(z)]/2)}{\Gamma([1+\alpha+\beta+\sigma_{\alpha,\beta}(z)]/2)\Gamma([1+\alpha+\beta-\sigma_{\alpha,\beta}(z)]/2)},\\\
&\hskip
133.72786ptz\in\rho(T_{F,\alpha,\beta}),\;\alpha\in[1,\infty),\;\beta\in(-1,0),\\\
&\sigma(T_{F,\alpha,\beta})=\\{(n-\beta)(n+1+\alpha)\\}_{n\in{\mathbb{N}}_{0}},\quad\alpha\in[1,\infty),\;\beta\in(-1,0),\end{split}$
(5.20)

with

$\sigma_{\alpha,\beta}(z)=\big{[}(1+\alpha+\beta)^{2}+4z\big{]}^{1/2}.$ (5.21)

$\mathbf{(II)}$ The Case $\alpha\in[1,\infty)$ and $\beta=0$:

$\displaystyle\psi_{0,\alpha,0}(z,x)=-2^{-\alpha-1}y_{2,\alpha,0,-1}(z,x)+y_{1,\alpha,0,-1}(z,x)m_{0,\alpha,0}(z),$
$\displaystyle
m_{0,\alpha,0}(z)=-2^{-\alpha-1}\\{2\gamma_{E}+\psi([1+\alpha+\sigma_{\alpha,0}(z)]/2)+\psi([1+\alpha-\sigma_{\alpha,0}(z)]/2)\\},$
$\displaystyle\hskip
193.47882ptz\in\rho(T_{F,\alpha,0}),\;\alpha\in[1,\infty),\;\beta=0,$
$\displaystyle\sigma(T_{F,\alpha,0})=\\{n(n+1+\alpha)\\}_{n\in{\mathbb{N}}_{0}},\quad\alpha\in[1,\infty),\;\beta=0.$
(5.22)

$\mathbf{(III)}$ The Case $\alpha\in[1,\infty)$ and $\beta\in(0,1)$:

$\displaystyle\begin{split}\psi_{0,\alpha,\beta}(z,x)&=2^{-\alpha-1}\beta^{-1}y_{2,\alpha,\beta,-1}(z,x)+y_{1,\alpha,\beta,-1}(z,x)m_{0,\alpha,\beta}(z),\\\
m_{0,\alpha\,\beta}(z)&=\beta^{-1}2^{-1-\alpha-\beta}\frac{-\Gamma(1-\beta)}{\Gamma(1+\beta)}\\\
&\quad\times\frac{\Gamma([1+\alpha+\beta+\sigma_{\alpha,\beta}(z)]/2)\Gamma([1+\alpha+\beta-\sigma_{\alpha,\beta}(z)]/2)}{\Gamma([1+\alpha-\beta+\sigma_{\alpha,\beta}(z)]/2)\Gamma((1+\alpha-\beta-\sigma_{\alpha,\beta}(z))/2)},\\\
&\hskip
101.00737ptz\in\rho(T_{F,\alpha,\beta}),\;\alpha\in[1,\infty),\;\beta\in(0,1),\\\
\sigma(T_{F,\alpha,\beta})&=\\{n(n+1+\alpha+\beta)\\}_{n\in{\mathbb{N}}_{0}},\quad\alpha\in[1,\infty),\;\beta\in(0,1).\end{split}$
(5.23)

$\mathbf{(IV)}$ The Case $\alpha\in(-\infty,-1]$ and $\beta\in(-1,0)$:

$\displaystyle\begin{split}\psi_{0,\alpha,\beta}(z,x)&=y_{1,\alpha,\beta,-1}(z,x)-2^{-\alpha-1}\beta^{-1}y_{2,\alpha,\beta,-1}(z,x)m_{0,\alpha,\beta}(z),\\\
m_{0,\alpha\,\beta}(z)&=2^{1+\alpha+\beta}\beta\frac{\Gamma(1+\beta)}{\Gamma(1-\beta)}\\\
&\quad\times\frac{\Gamma([1-\alpha-\beta+\sigma_{\alpha,\beta}(z)]/2)\Gamma([1-\alpha-\beta-\sigma_{\alpha,\beta}(z)]/2)}{\Gamma([1+\beta-\alpha+\sigma_{\alpha,\beta}(z)]/2)\Gamma([1+\beta-\alpha-\sigma_{\alpha,\beta}(z)]/2)},\\\
&\hskip 72.55461pt\
z\in\rho(T_{F,\alpha,\beta}),\;\alpha\in(-\infty,-1],\;\beta\in(-1,0),\\\
\sigma(T_{F,\alpha,\beta})&=\\{(n-\alpha-\beta)(n+1)\\}_{n\in{\mathbb{N}}_{0}},\quad\alpha\in(-\infty,-1],\;\beta\in(-1,0).\end{split}$
(5.24)

$\mathbf{(V)}$ The Case $\alpha\in(-\infty,-1]$ and $\beta=0$:

$\displaystyle\psi_{0,\alpha,0}(z,x)=-2^{-\alpha-1}y_{2,\alpha,0,-1}(z,x)+y_{1,\alpha,0,-1}(z,x)m_{0,\alpha,0}(z),$
$\displaystyle
m_{0,\alpha,0}(z)=-2^{-\alpha-1}\\{2\gamma_{E}+\psi([1-\alpha+\sigma_{\alpha,0}(z)]/2)+\psi([1-\alpha-\sigma_{\alpha,0}(z)]/2)\\},$
$\displaystyle\hskip
177.82971ptz\in\rho(T_{F,\alpha,0}),\;\alpha\in(-\infty,-1],\;\beta=0,$
$\displaystyle\sigma(T_{F,\alpha,0})=\\{(n-\alpha)(n+1)\\}_{n\in{\mathbb{N}}_{0}},\quad\alpha\in(-\infty,-1],\;\beta=0.$
(5.25)

$\mathbf{(VI)}$ The Case $\alpha\in(-\infty,-1]$ and $\beta\in(0,1)$:

$\displaystyle\begin{split}\psi_{0,\alpha,\beta}(z,x)&=2^{-\alpha-1}\beta^{-1}y_{2,\alpha,\beta,-1}(z,x)+y_{1,\alpha,\beta,-1}(z,x)m_{0,\alpha,\beta}(z),\\\
m_{0,\alpha\,\beta}(z)&=-\beta^{-1}2^{-1-\alpha-\beta}\frac{\Gamma(1-\beta)}{\Gamma(1+\beta)}\\\
&\quad\times\frac{\Gamma([1+\beta-\alpha+\sigma_{\alpha,\beta}(z)]/2)\Gamma([1+\beta-\alpha-\sigma_{\alpha,\beta}(z)]/2)}{\Gamma([1-\alpha-\beta+\sigma_{\alpha,\beta}(z)]/2)\Gamma([1-\alpha-\beta-\sigma_{\alpha,\beta}(z)]/2)},\\\
&\hskip
83.93553ptz\in\rho(T_{F,\alpha,\beta}),\;\alpha\in(-\infty,-1],\;\beta\in(0,1),\\\
\sigma(T_{F,\alpha,\beta})&=\\{(n-\alpha)(n+1+\beta)\\}_{n\in{\mathbb{N}}_{0}},\quad\alpha\in(-\infty,-1],\;\beta\in(0,1).\end{split}$
(5.26)

##  Appendix A The Hypergeometric and Jacobi Differential Equations

In this appendix we provide the connection between the hypergeometric
differential equation (cf. [1, Sect. 15.5])

$\displaystyle\xi(1-\xi)\ddot{w}(\xi)+[c-(a+b+1)\xi]\dot{w}(\xi)-abw(\xi)=0,\quad\xi\in(0,1),$
(A.1)

(where $\dot{\ }=d/d\xi$) and the Jacobi differential equation

$\displaystyle\begin{split}\tau_{\alpha,\beta}y(z,x)=-(1-x^{2})y^{\prime\prime}(z,x)+[\alpha-\beta+(\alpha+\beta+2)x]y^{\prime}(z,x)=zy(z,x),&\\\
\alpha,\beta\in{\mathbb{R}},\ x\in(-1,1),&\end{split}$ (A.2)

(where ${}^{\prime}=d/dx$). Making the substitution $\xi=(1+x)/2$ in (A.2)
yields

$\displaystyle\begin{split}\xi(1-\xi)\ddot{y}(z,\xi)+[\beta+1-(\alpha+\beta+2)\xi]\dot{y}(z,\xi)+zy(z,\xi)=0,&\\\
\alpha,\beta\in{\mathbb{R}},\ \xi\in(0,1).&\end{split}$ (A.3)

which is equal to (A.1) once one identifies,

$\displaystyle a=[1+\alpha+\beta+\sigma_{\alpha,\beta}(z)]/2,\quad
b=[1+\alpha+\beta-\sigma_{\alpha,\beta}(z)]/2,\quad c=1+\beta,$
$\displaystyle\sigma_{\alpha,\beta}(z)=\big{[}(1+\alpha+\beta)^{2}+4z\big{]}^{1/2}.$
(A.4)

At the endpoint $x=-1$ of the Jacobi equation the substitution used to arrive
at (A.3) yields $\xi=0$, hence we next consider solutions of (A.1) near
$\xi=0$ (cf. [1, Eqs. 15.5.3, 15.5.4]) (analogous solutions near $\xi=1$ are
found in (A))

$\displaystyle
w_{1,0}(\xi)=F(a,b;c;\xi)=\sum_{n\in{\mathbb{N}}_{0}}\frac{(a)_{n}(b)_{n}}{(c)_{n}}\frac{\xi^{n}}{n!},\quad
a,b\in{\mathbb{C}},\;c\in{\mathbb{C}}\backslash(-{\mathbb{N}}_{0}),$
$\displaystyle w_{2,0}(\xi)=\xi^{1-c}F(a-c+1,b-c+1;2-c;\xi),\quad
a,b\in{\mathbb{C}},\;(c-1)\in{\mathbb{C}}\backslash{\mathbb{N}},$ (A.5)
$\displaystyle\hskip 272.57745pt\xi\in(0,1).$

Here $F(\,\cdot\,,\,\cdot\,;\,\cdot\,;\,\cdot\,)$ (frequently written as
$\mathstrut{}_{2}F_{1}(\,\cdot\,,\,\cdot\,;\,\cdot\,;\,\cdot\,)$) denotes the
hypergeometric function $($see, e.g., [1, Ch. 15]$)$,
$\psi(\,\cdot\,)=\Gamma^{\prime}(\,\cdot\,)/\Gamma(\,\cdot\,)$ the Digamma
function, $\gamma_{E}=-\psi(1)=0.57721\dots$ represents Euler’s constant, and

$(\zeta)_{0}=1,\quad(\zeta)_{n}=\Gamma(\zeta+n)/\Gamma(\zeta),\;n\in{\mathbb{N}},\quad\zeta\in{\mathbb{C}}\backslash(-{\mathbb{N}}_{0}),$
(A.6)

abbreviates Pochhammer’s symbol (see, e.g., [1, Ch. 6]).

In addition,

$\text{$w_{1,0}$ and $w_{2,0}$ are linearly independent if
}\,c\in{\mathbb{C}}\backslash{\mathbb{Z}},$ (A.7)

which can be seen by noticing the different behaviors of $w_{1,0}(\xi)$,
$w_{2,0}(\xi)$ around $\xi=0$. One notes that only the case
$c=1+\beta\in(0,2)$ is needed. Thus, for $c=1$ we will use instead

$\displaystyle w_{1,0}(\xi)=F(a,b;1;\xi),\quad a,b\in{\mathbb{C}},$
$\displaystyle w_{2,0}^{\text{\rm ln}}(\xi)=F(a,b;1;\xi)\,\text{\rm
ln}(\xi)+\sum_{n\in{\mathbb{N}}}\frac{(a)_{n}(b)_{n}}{(n!)^{2}}\xi^{n}$ (A.8)
$\displaystyle\quad\times[\psi(a+n)-\psi(a)+\psi(b+n)-\psi(b)-2\psi(n+1)-2\gamma_{E}],\quad
a,b\in{\mathbb{C}}\backslash(-{\mathbb{N}}_{0}),$ $\displaystyle\hskip
303.59085pt\xi\in(0,1),$

where the superscipt “ln” indicates the presence of a logarithmic term
(familiar from Frobenius theory).

Using (A.4) in formulas (A.5) and (A.8), one obtains for the solutions of the
Jacobi differential equation
$\tau_{\alpha,\beta}y(z,\,\cdot\,)=zy(z,\,\cdot\,)$ (cf. (A.2)) near $x=-1$,

$\displaystyle
y_{1,\alpha,\beta,-1}(z,x)=F(a_{\alpha,\beta,\sigma_{\alpha,\beta}(z)},a_{\alpha,\beta,-\sigma_{\alpha,\beta}(z)};1+\beta;(1+x)/2),$
(A.9) $\displaystyle\hskip
213.39566pt\beta\in{\mathbb{R}}\backslash(-{\mathbb{N}}),$ $\displaystyle
y_{2,\alpha,\beta,-1}(z,x)=(1+x)^{-\beta}F(a_{\alpha,-\beta,\sigma_{\alpha,\beta}(z)},a_{\alpha,-\beta,-\sigma_{\alpha,\beta}(z)};1-\beta;(1+x)/2),$
$\displaystyle\hskip
261.76526pt\beta\in{\mathbb{R}}\backslash{\mathbb{N}}_{0},$ (A.10)
$\displaystyle
y_{2,\alpha,0,-1}(z,x)=F(a_{\alpha,0,\sigma_{\alpha,0}(z)},a_{\alpha,0,-\sigma_{\alpha,0}(z)};1;(1+x)/2)\,\text{\rm
ln}((1+x)/2)$
$\displaystyle\quad+\sum_{n\in{\mathbb{N}}}\frac{(a_{\alpha,0,\sigma_{\alpha,0}(z)})_{n}(a_{\alpha,0,-\sigma_{\alpha,0}(z)})_{n}}{2^{n}(n!)^{2}}(1+x)^{n}$
(A.11) $\displaystyle\hskip
39.83368pt\times[\psi(a_{\alpha,0,\sigma_{\alpha,0}(z)}+n)-\psi(a_{\alpha,0,\sigma_{\alpha,0}(z)})+\psi(a_{\alpha,0,-\sigma_{\alpha,0}(z)}+n)$
$\displaystyle\hskip
54.06006pt-\psi(a_{\alpha,0,-\sigma_{\alpha,0}(z)})-2\psi(n+1)-2\gamma_{E}],\quad\beta=0,$
$\displaystyle\hskip 145.10922pt\alpha\in{\mathbb{R}},\ z\in{\mathbb{C}},\
x\in(-1,1),$

where we abbreviated

$\displaystyle
a_{\mu,\nu,\pm\sigma}=[1+\mu+\nu\pm\sigma]/2,\quad\mu,\nu,\sigma\in{\mathbb{C}}.$
(A.12)

Again one observes that for $z\in{\mathbb{C}}$,
$y_{1,\alpha,\beta,-1}(z,\,\cdot\,)$ and $y_{2,\alpha,\beta,-1}(z,\,\cdot\,)$
are linearly independent for $\alpha\in{\mathbb{R}}$,
$\beta\in{\mathbb{R}}\backslash{\mathbb{Z}}$. Similarly, for
$z\in{\mathbb{C}}$, $y_{1,\alpha,0,-1}(z,\,\cdot\,)$ and
$y_{2,\alpha,0,-1}(z,\,\cdot\,)$ are linearly independent for
$\alpha\in{\mathbb{R}}$.

In precisely the same manner solutions of (A.1) are given by

$\displaystyle w_{1,1}(\xi)$ $\displaystyle=F(a,b;a+b-c+1;1-\xi),\quad
a,b\in{\mathbb{C}},\;c-a-b\in{\mathbb{C}}\backslash{\mathbb{N}},$
$\displaystyle w_{2,1}(\xi)$
$\displaystyle=(1-\xi)^{c-a-b}F(c-a,c-b;c-a-b+1;1-\xi),$ (A.13)
$\displaystyle\hskip
106.69783pta,b\in{\mathbb{C}},\;a+b-c\in{\mathbb{C}}\backslash{\mathbb{N}},$

and for $a+b=c$,

$\displaystyle w_{1,1}(\xi)$ $\displaystyle=F(a,b;1;1-\xi),\quad
a,b\in{\mathbb{C}},$ $\displaystyle w_{2,1}^{\text{\rm ln}}(\xi)$
$\displaystyle=F(a,b;1;1-\xi)\text{\rm
ln}(1-\xi)+\sum_{n\in{\mathbb{N}}}\frac{(a)_{n}(b)_{n}}{(n!)^{2}}(1-\xi)^{n}$
(A.14)
$\displaystyle\quad\,\times[\psi(a+n)-\psi(a)+\psi(b+n)-\psi(b)-2\psi(n+1)-2\gamma_{E}],$
$\displaystyle\hskip 184.9429pta,b\in{\mathbb{C}},\;\xi\in(0,1).$

which are obtained from (A.5) and (A.8) by the change of variables

$(a,\,b,\,c,\,\xi)\rightarrow(a,\,b,\,a+b-c+1,\,1-\xi).$ (A.15)

Together with the identification $x=(1+\xi)/2$ and (A.4) one obtains the
following solutions of $\tau_{\alpha,\beta}y(z,\,\cdot\,)=zy(z,\,\cdot\,)$
near $x=+1$,

$\displaystyle
y_{1,\alpha,\beta,+1}(z,x)=F(a_{\alpha,\beta,\sigma_{\alpha,\beta}(z)},a_{\alpha,\beta,-\sigma_{\alpha,\beta}(z)};1+\alpha;(1-x)/2),$
(A.16) $\displaystyle\hskip
213.39566pt\alpha\in{\mathbb{R}}\backslash(-{\mathbb{N}}),$ $\displaystyle
y_{2,\alpha,\beta,+1}(z,x)=(1-x)^{-\alpha}F(a_{-\alpha,\beta,\sigma_{\alpha,\beta}(z)},a_{-\alpha,\beta,-\sigma_{\alpha,\beta}(z)};1-\alpha;(1-x)/2)$
$\displaystyle\hskip 256.0748pt\alpha\in{\mathbb{R}}\backslash{\mathbb{N}},$
(A.17) $\displaystyle
y_{2,0,\beta,+1}(z,x)=F(a_{0,\beta,\sigma_{0,\beta}(z)},a_{0,\beta,-\sigma_{0,\beta}(z)};1;(1-x)/2)\,\text{\rm
ln}((1-x)/2)$
$\displaystyle\quad+\sum_{n\in{\mathbb{N}}}\frac{(a_{0,\beta,\sigma_{0,\beta}(z)})_{n}(a_{0,\beta,-\sigma_{0,\beta}(z)})_{n}}{2^{n}(n!)^{2}}(1-x)^{n}$
(A.18) $\displaystyle\hskip
39.83368pt\times[\psi(a_{0,\beta,\sigma_{0,\beta}(z)}+n)-\psi(a_{0,\beta,\sigma_{0,\beta}(z)})+\psi(a_{0,\beta,-\sigma_{0,\beta}(z)}+n)$
$\displaystyle\hskip
54.06006pt-\psi(a_{0,\beta,-\sigma_{0,\beta}(z)})-2\psi(n+1)-2\gamma_{E}],\quad\alpha=0,$
$\displaystyle\hskip 145.10922pt\beta\in{\mathbb{R}},\ z\in{\mathbb{C}},\
x\in(-1,1).$

Again, for $z\in{\mathbb{C}}$, $y_{1,\alpha,\beta,+1}(z,\,\cdot\,)$ and
$y_{2,\alpha,\beta,+1}(z,\,\cdot\,)$ are linearly independent for
$\alpha\in{\mathbb{R}}\backslash{\mathbb{Z}}$, $\beta\in{\mathbb{R}}$.
Similarly, for $z\in{\mathbb{C}}$, $y_{1,0,\beta,+1}(z,\,\cdot\,)$ and
$y_{2,0,\beta,+1}(z,\,\cdot\,)$ are linearly independent for
$\beta\in{\mathbb{R}}$.

In the limit-point case at $x=1$, where $\alpha\in(-\infty,-1]\cup[1,\infty)$,
one only needs the principal solutions, which are
$y_{1,\alpha,\beta,+1}(z,\,\cdot\,)$ for $\alpha\geqslant 1$ and
$y_{2,\alpha,\beta,+1}(z,\,\cdot\,)$ for $\alpha\leqslant-1$. Thus, one
concludes from (A.16) and (A.17) that these case are already covered, and one
does not have to define an additional solution for
$\alpha\in{\mathbb{Z}}\setminus\\{0\\}$.

Since
$(a_{\alpha,\beta,\sigma_{\alpha,\beta}(z)})_{n}(a_{\alpha,\beta,-\sigma_{\alpha,\beta}(z)})_{n}$,
$n\in{\mathbb{N}}_{0}$, depends polynomially on $z\in{\mathbb{C}}$, one infers
that

for fixed $x\in(0,1)$, $y_{j,\alpha,\beta,\pm 1}(z,x)$, $j=1,2$, are entire
with respect to $z\in{\mathbb{C}}$. (A.19)

Moreover $y_{j,\alpha,\beta,\pm 1}(z,x)$ satisfy the relations (cf. (A.1))

$\displaystyle y_{1,\alpha,\beta,-1}(z,x)$
$\displaystyle=(1+x)^{-\beta}y_{2,\alpha,-\beta,-1}(z+(1+\alpha)\beta,x),$
(A.20) $\displaystyle\hskip
89.626pt\alpha\in{\mathbb{R}},\,\beta\in{\mathbb{R}}\setminus\\{0\\},$
$\displaystyle y_{2,\alpha,\beta,-1}(z,x)$
$\displaystyle=(1+x)^{-\beta}y_{1,\alpha,-\beta,-1}(z+(1+\alpha)\beta,x),$
(A.21) $\displaystyle\hskip
89.626pt\alpha\in{\mathbb{R}},\,\beta\in{\mathbb{R}}\setminus\\{0\\},$
$\displaystyle y_{1,\alpha,\beta,+1}(z,x)$
$\displaystyle=(1-x)^{-\alpha}y_{2,-\alpha,\beta,+1}(z+(1+\beta)\alpha,x),$
(A.22) $\displaystyle\hskip
89.626pt\alpha\in{\mathbb{R}}\setminus\\{0\\},\,\beta\in{\mathbb{R}},$
$\displaystyle y_{2,\alpha,\beta,+1}(z,x)$
$\displaystyle=(1-x)^{-\alpha}y_{1,-\alpha,\beta,+1}(z+(1+\beta)\alpha,x),$
(A.23) $\displaystyle\hskip
90.19518pt\alpha\in{\mathbb{R}}\setminus\\{0\\},\,\beta\in{\mathbb{R}},$

where we used the fact

$\displaystyle\sigma_{\alpha,\beta}(z)=\begin{cases}\sigma_{\alpha,-\beta}(z+(1+\alpha)\beta),\\\
\sigma_{-\alpha,\beta}(z+(1+\beta)\alpha),\\\
\sigma_{-\alpha,-\beta}(z+\alpha+\beta).\end{cases}$ (A.24)

######  Remark A.1.

We conclude this appendix by briefly discussing Jacobi polynomials and quasi-
rational eigenfunctions. The $n$th Jacobi polynomial is defined as (see [61,
Eq. 18.5.7])

$\displaystyle\begin{split}P_{n}^{\alpha,\beta}(x):=\dfrac{(\alpha+1)_{n}}{n!}F(-n,\,n+\alpha+\beta+1;\,\alpha+1;\,(1-x)/2),\\\
n\in{\mathbb{N}}_{0},\ -\alpha\notin{\mathbb{N}},\
-n-\alpha-\beta-1\notin{\mathbb{N}},\end{split}$ (A.25)

and can be defined by continuity for all parameters
$\alpha,\beta\in{\mathbb{R}}$. Note that $P_{n}^{\alpha,\beta}(x)$ is a
polynomial of degree at most $n$, and has strictly smaller degree if and only
if $-n-\alpha-\beta\in\\{1,\dots,n\\}$ (cf. [71, p. 64]). It satisfies the
equation

$\displaystyle\tau_{\alpha,\beta}P_{n}^{\alpha,\beta}(x)=\lambda^{\alpha,\beta}_{n}P_{n}^{\alpha,\beta}(x),$
(A.26)

with

$\displaystyle\lambda^{\alpha,\beta}_{n}:=n(n+1+\alpha+\beta).$ (A.27)

In particular, one can verify that the Jacobi polynomials are solutions of the
Jacobi differential equation (A.2) with Neumann boundary conditions at $x=+1$
(resp. $x=-1$) if $\alpha\in(-1,0)$ (resp. $\beta\in(-1,0)$) and Friedrichs
boundary conditions if $\alpha\geqslant 0$ (resp. $\beta\geqslant 0$).

More generally, all quasi-rational solutions, meaning the logarithmic
derivative being rational, can be derived from the the Jacobi polynomials
together with

$\displaystyle(1+x)^{-\beta}\circ\tau_{\alpha,-\beta}\circ(1+x)^{\beta}=\tau_{\alpha,\beta}+(1+\alpha)\beta,$
$\displaystyle(1-x)^{-\alpha}\circ\tau_{-\alpha,\beta}\circ(1-x)^{\alpha}=\tau_{\alpha,\beta}+(1+\beta)\alpha,$
(A.28)
$\displaystyle(1-x)^{-\alpha}(1+x)^{-\beta}\circ\tau_{-\alpha,-\beta}\circ(1-x)^{\alpha}(1+x)^{\beta}=\tau_{\alpha,\beta}+\alpha+\beta,$

where $(1+x)^{\pm\beta}$ and $(1-x)^{\pm\alpha}$ are regarded as formal
multiplication operators. This is summarized in Table 1, which is taken from
[8].

Eigenfunctions | Eigenvalues
---|---
$P_{n}^{\alpha,\beta}(x)$ | $n(n+1+\alpha+\beta)$
$(1-x)^{-\alpha}P_{n}^{-\alpha,\beta}(x)$ | $n(n+1-\alpha+\beta)-\alpha(1+\beta)$
$(1+x)^{-\beta}P_{n}^{\alpha,-\beta}(x)$ | $n(n+1+\alpha-\beta)-\beta(1+\alpha)$
$(1-x)^{-\alpha}(1+x)^{-\beta}P_{n}^{-\alpha,-\beta}(x)$ | $n(n+1-\alpha-\beta)-(\alpha+\beta)$
Table 1. Formal quasi-rational eigensolutions of $\tau_{\alpha,\beta}$

Here $(1-x)^{-\alpha}P_{n}^{-\alpha,\beta}(x)$ satisfy at $x=+1$ the
Friedrichs boundary condition for $\alpha\leqslant 0$ and Neumann for
$\alpha\in(0,1)$, while at $x=-1$ they satisfy the Friedrichs for
$\beta\geqslant 0$ and Neumann for $\beta\in(-1,0)$. For
$(1+x)^{-\beta}P_{n}^{\alpha,-\beta}(x)$ the roles of $\alpha$ and $\beta$
interchange compared to the last case, meaning Friedrichs at $x=+1$ for
$\alpha\geqslant 0$, Neumann for $\alpha\in(-1,0)$, and at $x=-1$, Friedrichs
for $\beta\leqslant 0$, Neumann for $\beta\in(0,1)$. Finally
$(1-x)^{-\alpha}(1+x)^{-\beta}P_{n}^{-\alpha,-\beta}(x)$ satisfy at $x=+1$
(resp. $x=-1$) the Friedrichs boundary condition for $\alpha\leqslant 0$
(resp. $\beta\leqslant 0$) and Neumann for $\alpha\in(0,1)$ (resp.
$\beta\in(0,1)$). $\diamond$

##  Appendix B Connection Formulas

In this appendix we provide the connection formulas utilized to find the
solution behaviors in Appendix C. We express them using $w_{1,0}(\xi)$ and
$w_{2,0}(\xi)$ $\big{(}w_{2,0}^{\text{\rm ln}}(\xi)\big{)}$ and their analogs
$w_{1,1}(\xi)$ and $w_{2,1}(\xi)$ $\big{(}w_{2,1}^{\text{\rm ln}}(\xi)\big{)}$
at the endpoint $\xi=1$.

We recall the relations (A.4) connecting the parameters $a,b,c$ and
$\alpha,\beta$.
$\mathbf{(I)}$ The case $\alpha\in{\mathbb{R}}\backslash{\mathbb{Z}}$,
$\beta\in(-1,1)\backslash\\{0\\}$, that is, $c\in(0,2)\backslash\\{1\\}$,
$a+b-c\in{\mathbb{R}}\backslash{\mathbb{Z}}$ :
The two connection formulas are given by (cf. [62, Eq. 15.10.21–22])

$\displaystyle w_{1,0}(\xi)$
$\displaystyle=\frac{\Gamma(c)\Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)}w_{1,1}(\xi)+\frac{\Gamma(c)\Gamma(a+b-c)}{\Gamma(a)\Gamma(b)}w_{2,1}(\xi),$
(B.1) $\displaystyle w_{2,0}(\xi)$
$\displaystyle=\frac{\Gamma(2-c)\Gamma(c-a-b)}{\Gamma(1-a)\Gamma(1-b)}w_{1,1}(\xi)+\frac{\Gamma(2-c)\Gamma(a+b-c)}{\Gamma(a-c+1)\Gamma(b-c+1)}w_{2,1}(\xi).$
(B.2)

One notes that poles occur on the right-hand side of (B.1), (B.2) whenever
$(a+b-c)\in{\mathbb{Z}}$. Using (A.15) one can also express $w_{1,1}(\xi)$ or
$w_{2,1}(\xi)$ as a linear combination of $w_{1,0}(\xi)$ and $w_{2,0}(\xi)$:

$\displaystyle w_{1,1}(\xi)$
$\displaystyle=\frac{\Gamma(a+b-c+1)\Gamma(1-c)}{\Gamma(a-c+1)\Gamma(b-c+1)}w_{1,0}(\xi)+\frac{\Gamma(a+b-c+1)\Gamma(c-1)}{\Gamma(a)\Gamma(b)}w_{2,0}(\xi),$
(B.3) $\displaystyle w_{2,1}(\xi)$
$\displaystyle=\frac{\Gamma(1+c-a-b)\Gamma(1-c)}{\Gamma(1-a)\Gamma(1-b)}w_{1,0}(\xi)+\frac{\Gamma(1+c-a-b)\Gamma(c-1)}{\Gamma(c-a)\Gamma(c-b)}w_{2,0}(\xi),$
(B.4)

though, in the following we shall only write down one pair of connection
formulas for brevity. $\mathbf{(II)}$ The case $\alpha=0,\
\beta\in{\mathbb{R}}\backslash{\mathbb{Z}}$, that is,
$c\in{\mathbb{R}}\backslash{\mathbb{Z}}$, $a+b=c$ :
The solution $w_{1,0}(\xi)=F(a,b;a+b;\xi)$ can be expanded at $\xi=1$ (cf. [1,
Eq. 15.3.10]):

$\displaystyle\begin{split}F(a,b;a+b;\xi)&=\frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)}\sum_{n\in{\mathbb{N}}_{0}}\frac{(a)_{n}(b)_{n}}{(n!)^{2}}[2\psi(n+1)-\psi(a+n)-\psi(b+n)\\\
&\hskip 115.23373pt-\text{\rm ln}(1-\xi)](1-\xi)^{n}.\end{split}$ (B.5)

Meanwhile, two linearly independent solutions at $\xi=1$ are taken from
(A.14). The connection formula for $w_{1,1}(\xi)$ is given by (B.3) with
$a+b=c$. To obtain a second connection formula one compares the expansion of
$w_{2,1}^{\text{\rm ln}}(\xi)$ at $\xi=1$ with the expansion of
$F(a,b;a+b;\xi)$ at $\xi=1$, using (B.5), and then obtains

$\displaystyle w_{2,1}^{\text{\rm ln}}(\xi)=$
$\displaystyle-[\psi(1-a)+\psi(1-b)+2\gamma_{E}]\frac{\Gamma(1-a-b)}{\Gamma(1-a)\Gamma(1-b)}w_{1,0}(\xi)$
$\displaystyle-[\psi(a)+\psi(b)+2\gamma_{E}]\frac{\Gamma(a+b-1)}{\Gamma(a)\Gamma(b)}w_{2,0}(\xi).$
(B.6)

$\mathbf{(III)}$ The case
$\alpha\in{\mathbb{R}}\backslash{\mathbb{Z}},\,\beta=0$, that is, $c=1$,
$a+b\in{\mathbb{R}}\backslash{\mathbb{Z}}$ :
This case is analogous to the previous case, with the roles of $\alpha$ and
$\beta$ interchanged. Concretely, this means that the connection formulas
(B.5) and (B) must be changed through the renaming (A.15) with $c\rightarrow
a+b-c+1=a+b$, as $c=1$. As $c$ does not appear in (B.5) and (B) (it was
eliminated via $c=a+b$), one can adopt the aforementioned formulas directly,
only changing the second index in the $w$’s111Formula (B.7) could have been
obtained directly from (B.1) by setting $c=1$.

$\displaystyle w_{1,0}(\xi)$
$\displaystyle=\frac{\Gamma(1-a-b)}{\Gamma(1-a)\Gamma(1-b)}w_{1,1}(\xi)+\frac{\Gamma(a+b-1)}{\Gamma(a)\Gamma(b)}w_{2,1}(\xi),$
(B.7) $\displaystyle w_{2,0}^{\text{\rm ln}}(\xi)$
$\displaystyle=-[\psi(1-a)+\psi(1-b)+2\gamma_{E}]\frac{\Gamma(1-a-b)}{\Gamma(1-a)\Gamma(1-b)}w_{1,1}(\xi)$
$\displaystyle\quad\,-[\psi(a)+\psi(b)+2\gamma_{E}]\frac{\Gamma(a+b-1)}{\Gamma(a)\Gamma(b)}w_{2,1}(\xi).$
(B.8)

$\mathbf{(IV)}$ The case $\alpha=\beta=0$, that is, $a+b=c=1$ :
For $\alpha=0$ and $\beta=0$ the Jacobi differential expression (5.1) becomes
the Legendre differential expression. Since this case was treated in detail in
[26], we shall only present the connection formulas for completeness.

The special solutions $w_{1,i}(\xi)$ and $w_{2,i}^{\text{\rm ln}}(\xi)$ for
$i=1,2$ are given by (A.8) and (A.14), respectively. Note that the following
relations hold

$\displaystyle w_{1,1}(\xi)=w_{1,0}(1-\xi),\qquad w_{2,1}^{\text{\rm
ln}}(\xi)=w_{2,0}^{\text{\rm ln}}(1-\xi).$ (B.9)

Using [1, Eq. 15.3.10] together with $w_{1,0}=F(a,b;a+b,\xi)$ and Euler’s
famous reflection formula, $\Gamma(z)\Gamma(1-z)=\pi\csc(\pi z)$ (cf. [1, Eq.
6.1.17]), one obtains

$\displaystyle w_{1,0}(\xi)=-\pi^{-1}\sin(\pi
a)\big{(}[\psi(a)+\psi(b)+2\gamma_{E}]w_{1,1}(\xi)+w_{2,1}^{\text{\rm
ln}}(\xi)\big{)}.$ (B.10)

The two relations (B.9) immediately imply

$\displaystyle w_{1,1}(\xi)$ $\displaystyle=-\pi^{-1}\sin(\pi
a)\big{(}[\psi(a)+\psi(b)+2\gamma_{E}]w_{1,0}(\xi)+w_{2,0}^{\text{\rm
ln}}(\xi)\big{)},$ (B.11) $\displaystyle w_{2,1}^{\text{\rm ln}}(\xi)$
$\displaystyle=\pi^{-1}\sin(\pi
a)\big{[}\big{(}[\psi(a)+\psi(b)+2\gamma_{E}]^{2}-\pi^{2}[\sin(\pi
a)]^{-2}\big{)}w_{1,0}(\xi)$
$\displaystyle\quad+[\psi(a)+\psi(b)+2\gamma_{E}]w_{2,0}^{\text{\rm
ln}}(\xi)\big{]}.$ (B.12)

##  Appendix C Behavior of $y_{j,\alpha,\beta,\mp 1}(z,x)$, $j=1,2$, near
$x=\pm 1$

In this appendix we focus on the generalized boundary values for the solutions
$y_{j,\alpha,\beta,-1}(z,x),\ j=1,2$ at $x=\mp 1$. One obtains for
$z\in{\mathbb{C}}$,

$\displaystyle\begin{split}\widetilde{y}_{1,\alpha,\beta,-1}(z,-1)&=\begin{cases}1,&\beta\in(-1,0),\\\
0,&\beta=0,\\\ 0,&\beta\in(0,1),\end{cases}\\\
\widetilde{y}_{1,\alpha,\beta,-1}^{\,\prime}(z,-1)&=\begin{cases}0,&\beta\in(-1,0),\\\
1,&\beta=0,\\\ 1,&\beta\in(0,1),\end{cases}\\\
\widetilde{y}_{2,\alpha,\beta,-1}(z,-1)&=\begin{cases}0,&\beta\in(-1,0),\\\
-2^{\alpha+1},&\beta=0,\\\ \beta 2^{\alpha+1},&\beta\in(0,1),\end{cases}\\\
\widetilde{y}_{2,\alpha,\beta,-1}^{\,\prime}(z,-1)&=\begin{cases}-\beta
2^{\alpha+1},&\beta\in(-1,0),\\\ 0,&\beta=0,\\\
0,&\beta\in(0,1),\end{cases}\end{split}\quad\alpha\in{\mathbb{R}},$ (C.1)

and employing connection formulas for the endpoint $x=+1$,

$\displaystyle\begin{split}\widetilde{y}_{1,\alpha,\beta,-1}(z,1)&=\begin{cases}\dfrac{\Gamma(1+\beta)\Gamma(-\alpha)}{\Gamma(a_{-\alpha,\beta,\sigma_{\alpha,\beta}(z)})\Gamma(a_{-\alpha,\beta,-\sigma_{\alpha,\beta}(z)})},&\alpha\in(-1,0),\\\\[8.53581pt]
\dfrac{-2^{1+\alpha+\beta}\Gamma(1+\alpha)\Gamma(1+\beta)}{\Gamma(a_{\alpha,\beta,\sigma_{\alpha,\beta}(z)})\Gamma(a_{\alpha,\beta,-\sigma_{\alpha,\beta}(z)})},&\alpha\in[0,1),\end{cases}\end{split}$
$\displaystyle\begin{split}\widetilde{y}_{1,\alpha,\beta,-1}^{\,\prime}(z,1)&=\begin{cases}\dfrac{2^{1+\alpha+\beta}\Gamma(1+\alpha)\Gamma(1+\beta)}{\Gamma(a_{\alpha,\beta,\sigma_{\alpha,\beta}(z)})\Gamma(a_{\alpha,\beta,-\sigma_{\alpha,\beta}(z)})},&\alpha\in(-1,0),\\\\[8.53581pt]
\dfrac{-\Gamma(1+\beta)}{\Gamma(a_{0,\beta,\sigma_{0,\beta}(z)})\Gamma(a_{0,\beta,-\sigma_{0,\beta}(z)})}[2\gamma_{E}\\\
\quad+\psi(a_{0,\beta,\sigma_{0,\beta}(z)})+\psi(a_{0,\beta,-\sigma_{0,\beta}(z)})],&\alpha=0,\\\\[8.53581pt]
\dfrac{\Gamma(1+\beta)\Gamma(-\alpha)}{\Gamma(a_{-\alpha,\beta,\sigma_{\alpha,\beta}(z)})\Gamma(a_{-\alpha,\beta,-\sigma_{\alpha,\beta}(z)})},&\alpha\in(0,1),\end{cases}\end{split}$
$\displaystyle\hskip 184.9429pt\beta\in(-1,1),$ (C.2)
$\displaystyle\begin{split}\widetilde{y}_{2,\alpha,\beta,-1}(z,1)&=\begin{cases}\dfrac{2^{-\beta}\Gamma(1-\beta)\Gamma(-\alpha)}{\Gamma(a_{-\alpha,-\beta,\sigma_{\alpha,\beta}(z)})\Gamma(a_{-\alpha,-\beta,-\sigma_{\alpha,\beta}(z)})},&\alpha\in(-1,0),\\\\[8.53581pt]
\dfrac{-2^{\alpha+1}\Gamma(1+\alpha)\Gamma(1-\beta)}{\Gamma(a_{\alpha,-\beta,\sigma_{\alpha,\beta}(z)})\Gamma(a_{\alpha,-\beta,-\sigma_{\alpha,\beta}(z)})},&\alpha\in[0,1),\end{cases}\end{split}$
$\displaystyle\begin{split}\widetilde{y}_{2,\alpha,\beta,-1}^{\,\prime}(z,1)&=\begin{cases}\dfrac{2^{\alpha+1}\Gamma(1+\alpha)\Gamma(1-\beta)}{\Gamma(a_{\alpha,-\beta,\sigma_{\alpha,\beta}(z)})\Gamma(a_{\alpha,-\beta,-\sigma_{\alpha,\beta}(z)})},&\alpha\in(-1,0),\\\\[8.53581pt]
\dfrac{-2^{-\beta}\Gamma(1-\beta)}{\Gamma(a_{0,-\beta,\sigma_{0,\beta}(z)})\Gamma(a_{0,-\beta,-\sigma_{0,\beta}(z)})}[2\gamma_{E}\\\
\quad+\psi(a_{0,-\beta,\sigma_{0,\beta}(z)})+\psi(a_{0,-\beta,-\sigma_{0,\beta}(z)})],&\alpha=0,\\\\[8.53581pt]
\dfrac{2^{-\beta}\Gamma(1-\beta)\Gamma(-\alpha)}{\Gamma(a_{-\alpha,-\beta,\sigma_{\alpha,\beta}(z)})\Gamma(a_{-\alpha,-\beta,-\sigma_{\alpha,\beta}(z)})},&\alpha\in(0,1),\end{cases}\end{split}$
$\displaystyle\hskip 177.82971pt\beta\in(-1,1)\backslash\\{0\\},$ (C.3)
$\displaystyle\begin{split}\widetilde{y}_{2,\alpha,0,-1}(z,1)&=\begin{cases}\dfrac{-[2\gamma_{E}+\psi(a_{-\alpha,0,\sigma_{\alpha,0}(z)})+\psi(a_{-\alpha,0,-\sigma_{\alpha,0}(z)})]\Gamma(-\alpha)}{\Gamma(a_{-\alpha,0,\sigma_{\alpha,0}(z)})\Gamma(a_{-\alpha,0,-\sigma_{\alpha,0}(z)})},\\\\[2.84526pt]
\hskip 176.407pt\alpha\in(-1,0),\\\\[8.53581pt]
\dfrac{[2\gamma_{E}+\psi(a_{\alpha,0,\sigma_{\alpha,0}(z)})+\psi(a_{\alpha,0,-\sigma_{\alpha,0}(z)})]\Gamma(1+\alpha)}{2^{-\alpha-1}\Gamma(a_{\alpha,0,\sigma_{\alpha,0}(z)})\Gamma(a_{\alpha,0,-\sigma_{\alpha,0}(z)})},\\\\[2.84526pt]
\hskip 173.56198pt\alpha\in[0,1),\end{cases}\end{split}$
$\displaystyle\begin{split}\widetilde{y}_{2,\alpha,0,-1}^{\,\prime}(z,1)&=\begin{cases}\dfrac{[2\gamma_{E}+\psi(a_{\alpha,0,\sigma_{\alpha,0}(z)})+\psi(a_{\alpha,0,-\sigma_{\alpha,0}(z)})]\Gamma(1+\alpha)}{-2^{-\alpha-1}\Gamma(a_{\alpha,0,\sigma_{\alpha,0}(z)})\Gamma(a_{\alpha,0,-\sigma_{\alpha,0}(z)})},\\\\[2.84526pt]
\hskip 165.02606pt\alpha\in(-1,0),\\\\[2.84526pt]
-\Gamma((1+\sigma_{0,0}(z))/2)\Gamma((1-\sigma_{0,0}(z))/2)\\\
+\dfrac{[2\gamma_{E}+\psi((1+\sigma_{0,0}(z))/2)+\psi((1-\sigma_{0,0}(z))/2)]^{2}}{\Gamma((1+\sigma_{0,0}(z))/2)\Gamma((1-\sigma_{0,0}(z))/2)},\quad\alpha=0,\\\\[8.53581pt]
\dfrac{-[2\gamma_{E}+\psi(a_{-\alpha,0,\sigma_{\alpha,0}(z)})+\psi(a_{-\alpha,0,-\sigma_{\alpha,0}(z)})]\Gamma(-\alpha)}{\Gamma(a_{-\alpha,0,\sigma_{\alpha,0}(z)})\Gamma(a_{-\alpha,0,-\sigma_{\alpha,0}(z)})},\\\\[2.84526pt]
\hskip 183.52019pt\alpha\in(0,1),\end{cases}\end{split}$ $\displaystyle\hskip
223.3543pt\beta=0.$ (C.4)

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