$\displaystyle\begin{split}&m(z)=c+dz+\int_{{\mathbb{R}}}d\omega(\lambda)\bigg{[}\frac{1}{\lambda-z}-\frac{\lambda}{1+\lambda^{2}}\bigg{]},\quad
z\in{\mathbb{C}}_{+},\\\\[5.69054pt] &\,c=\text{\rm Re}[m(i)],\quad
d=\lim_{\eta\uparrow\infty}m(i\eta)/(i\eta)\geqslant 0.\end{split}$ (8.6)

Conversely, any function $m$ of the type (8.6) $($with the measure $\omega$
satisfying (8.5) and some $c\in{\mathbb{R}}$, $d\geqslant 0$$)$ is N–H.
Moreover,

$\displaystyle\begin{split}&{\mathbb{C}}_{+}\ni
z\mapsto-e^{-ir\pi}z^{r}\,\text{ is N--H if and only if }\,r\in[0,1],\\\
&{\mathbb{C}}_{+}\ni z\mapsto e^{-ir\pi}z^{r}\,\text{ is N--H if and only if
}\,r\in[-1,0].\end{split}$ (8.7)

If the measure $d\omega$ in the representation (8.6) is pure point, then $m$
is meromorphic and real-valued on the real line (with all residues strictly
negative). In this case $m$ monotonically increases from $-\infty$ to
$+\infty$ between any consecutive poles, hence there is a unique zero between
any two poles, an interlacing property. Conversely, any meromorphic function
real-valued on ${\mathbb{R}}$ with all its residues strictly negative, is a
meromorphic N-H function corresponding to a pure point measure $d\omega$ in
(8.6). (We refer to [31] and the extensive literature cited therein for
various properties of N–H functions.)

Using [53, eq. 5.11.13], the behavior of $\widehat{m}_{\alpha,\beta}$ at
infinity reads

$\displaystyle\begin{split}\widehat{m}_{\alpha,\beta}(z)\underset{z\rightarrow\infty}{=}\frac{-\Gamma(1-\beta)e^{\mp
i\beta\pi}}{2^{1+\alpha+\beta}\beta\Gamma(1+\beta)}z^{\beta}\big{[}1+O\big{(}z^{-1}\big{)}\big{]},&\\\
z\in{\mathbb{C}}_{\pm},\;|\arg
z|\geqslant\varepsilon,\;\varepsilon>0,\;\alpha\in{\mathbb{R}},\;\beta\in{\mathbb{R}}\backslash{\mathbb{Z}},&\end{split}$
(8.8)

and hence for $\widehat{m}_{\alpha,\beta}$ to be N–H, (LABEL:9.8) necessarily
limits $\beta$ to the interval $[-1,1]$ and due to the $\beta$ and Gamma
factors $\Gamma(1\pm\beta)$ present in (8.8) one necessarily assumes from the
outset that

$\beta\in(-1,1)\backslash\\{0\\}.$ (8.9)

Before continuing, we recall that upon combining (3.24) and (4.8),

$m_{\gamma,\delta,\alpha,\beta}(\,\cdot\,),\;m_{\gamma,\alpha,\beta}(\,\cdot\,)\,\text{
are N--H if $\gamma,\delta\in[0,\pi)$, $\alpha\in{\mathbb{R}}$,
$\beta\in(-1,1)$.}$ (8.10)

In the remainder of this section we prove the N–H property of
$\widehat{m}_{\alpha,\beta}$ from scratch for
$\alpha\in(-{\mathbb{N}})\cup(-1,\infty)$ and
$\beta\in(-1,1)\backslash\\{0\\}$ and relate the situation
$\alpha\in(-{\mathbb{N}})$ to Jacobi polynomials.

######  Theorem 8.1.

Let $\alpha\in{\mathbb{R}}$, $\beta\in{\mathbb{R}}\backslash{\mathbb{Z}}$.
Then $\widehat{m}_{\alpha,\beta}$ as introduced in (8) has the
Nevanlinna–Herglotz property if and only if
$\alpha\in(-{\mathbb{N}})\cup(-1,\infty)$ and $\beta\in(-1,0)\cup(0,1)$.

###### Proof.

One recalls from (8.9) that $\beta\in(-1,1)\backslash\\{0\\}$ is a necessary
condition for $\widehat{m}_{\alpha,\beta}$ to be N–H.

Next, we recall the formula111Formula (8.11) is also recorded in [49, p. 5],
however, the minus sign in the last term in the first line of the formula must
be replaced by a plus sign. in [15, eq. 1.4.(3), p. 8],

$\displaystyle\frac{\Gamma(1+a-b)\Gamma(1-b)}{\Gamma(a)}\frac{\Gamma((a/2)+\zeta)\Gamma((a/2)-\zeta)}{\Gamma(1+(a/2)-b+\zeta)\Gamma(1+(a/2)-b-\zeta)}$
$\displaystyle\quad=\sum_{n\in{\mathbb{N}}_{0}}\frac{(a)_{n}(b)_{n}}{(1+a-b)_{n}(n!)}\bigg{[}\frac{1}{n+(a/2)+\zeta}+\frac{1}{n+(a/2)-\zeta}\bigg{]},$
(8.11) $\displaystyle\hskip 71.13188pta,b\in{\mathbb{C}},\;\text{\rm
Re}(b)<1,\;\zeta\in{\mathbb{C}}\backslash\\{\pm({\mathbb{N}}_{0}+(a/2))\\},$

proven in [15, Sect. 2.4] (see (A.11) for the introduction of Pochhammer’s
symbol $(c)_{n}$, $c\in{\mathbb{C}}$, $n\in{\mathbb{N}}_{0}$). Identifying

$\zeta=\sigma_{\alpha,\beta}(z)/2,\quad a=1+\alpha+\beta,\quad b=1+\beta,$
(8.12)

in (8.11), (8) for $\beta\in(-1,0)$ can be rewritten in the form

$\displaystyle\widehat{m}_{\alpha,\beta}(z)$
$\displaystyle=\frac{\Gamma(1+\alpha+\beta)}{2^{1+\alpha+\beta}\Gamma(1+\beta)\Gamma(1+\alpha)}\sum_{n\in{\mathbb{N}}_{0}}\frac{(1+\alpha+\beta)_{n}(1+\beta)_{n}}{(1+\alpha)_{n}(n!)}$
$\displaystyle\hskip
143.6865pt\times\frac{2n+1+\alpha+\beta}{n(n+1+\alpha+\beta)-z}$
$\displaystyle=\frac{1}{2^{1+\alpha+\beta}\Gamma(1+\beta)^{2}}\sum_{n\in{\mathbb{N}}_{0}}\frac{\Gamma(n+1+\alpha+\beta)\Gamma(n+1+\beta)}{\Gamma(n+1+\alpha)(n!)}$
$\displaystyle\hskip
110.96556pt\times\frac{2n+1+\alpha+\beta}{n(n+1+\alpha+\beta)-z},\quad\beta\in(-1,0),$
(8.13)

where we employed

$(\gamma)_{n}=\Gamma(n+\gamma)/\Gamma(\gamma),\quad\gamma\in{\mathbb{C}},\;n\in{\mathbb{N}}_{0},$
(8.14)

in the second equality in (8.13).

Since $\widehat{m}_{\alpha,\beta}$ is meromorphic and real-valued on
${\mathbb{R}}$, characterizing its N–H property is now equivalent to
characterizing when all residues in (8.13) are strictly negative. By
inspection, this is clear from elementary Gamma function properties (see, e.g.
[1, p. 255–256]) for $\alpha>-1$ and $\beta\in(-1,0)$. To prove that
$\widehat{m}_{\alpha,\beta}$ is not N–H for
$\alpha\in(-\infty,-1]\backslash(-{\mathbb{N}})$ it suffices to establish some
strictly positive residues in (8.13). For this purpose suppose that
$\alpha\in(-k-1,-k)$ for some $k\in{\mathbb{N}}$ and $\beta\in(-1,0)$: Then
one verifies that $\Gamma(n+1+\beta)>0$, $n\in{\mathbb{N}}_{0}$, and choosing
$n=0$ one confirms that

$\frac{\Gamma(1+\alpha+\beta)[1+\alpha+\beta]}{\Gamma(1+\alpha)}<0,\quad
1+\alpha+\beta\in(-k,-k+1).$ (8.15)

Similarly, if $n=k$,

$\frac{\Gamma(k+1+\alpha+\beta)[2k+1+\alpha+\beta]}{\Gamma(k+1+\alpha)}<0,\quad
1+\alpha+\beta\in(-k-1,-k).$ (8.16)

Since $k\in{\mathbb{N}}$ is arbitrary, (8.15) and (8.16) prove the non N–H
property of $\widehat{m}_{\alpha,\beta}$ for
$\alpha\in(-\infty,-1]\backslash(-{\mathbb{N}})$ and $\beta\in(-1,0)$.

To treat the case $\beta\in(0,1)$ one replaces $\widehat{m}_{\alpha,\beta}$ by
$-1/\widehat{m}_{\alpha,\beta}$ to obtain

$\displaystyle-1/\widehat{m}_{\alpha,\beta}(z)$
$\displaystyle\quad=\frac{2^{1+\alpha+\beta}\beta\Gamma(1+\beta)}{\Gamma(1-\beta)}\frac{\Gamma([1+\alpha-\beta+\sigma_{\alpha,\beta}(z)]/2)\Gamma([1+\alpha-\beta-\sigma_{\alpha,\beta}(z)]/2)}{\Gamma([1+\alpha+\beta+\sigma_{\alpha,\beta}(z)]/2)\Gamma([1+\alpha+\beta-\sigma_{\alpha,\beta}(z)]/2)},$
$\displaystyle\hskip
81.09052pt\alpha\in{\mathbb{R}},\;\beta\in{\mathbb{R}}\backslash{\mathbb{Z}},\;z\in{\mathbb{C}}\backslash\\{(n-\beta)(n+1+\alpha)\\}_{n\in{\mathbb{N}}_{0}}.$
(8.17)

Identifying

$\zeta=\sigma_{\alpha,\beta}(z)/2,\quad a=1+\alpha-\beta,\quad b=1-\beta,$
(8.18)

in (8.11), (8.17) for $\beta\in(0,1)$ can be rewritten as

$\displaystyle-1/\widehat{m}_{\alpha,\beta}(z)$
$\displaystyle=\frac{-2^{1+\alpha+\beta}\beta\Gamma(1+\alpha-\beta)}{\Gamma(-\beta)\Gamma(1+\alpha)}\sum_{n\in{\mathbb{N}}_{0}}\frac{(1+\alpha-\beta)_{n}(1-\beta)_{n}}{(1+\alpha)_{n}(n!)}$
$\displaystyle\hskip
139.41832pt\times\frac{2n+1+\alpha-\beta}{(n-\beta)(n+1+\alpha)-z}$
$\displaystyle=\frac{2^{1+\alpha+\beta}\beta^{2}}{\Gamma(1-\beta)^{2}}\sum_{n\in{\mathbb{N}}_{0}}\frac{\Gamma(n+1+\alpha-\beta)\Gamma(n+1-\beta)}{\Gamma(n+1+\alpha)(n!)}$
$\displaystyle\hskip
79.6678pt\times\frac{2n+1+\alpha-\beta}{(n-\beta)(n+1+\alpha)-z},\quad\beta\in(0,1).$
(8.19)

The analysis for $\alpha\in{\mathbb{R}}\backslash(-{\mathbb{N}})$,
$\beta\in(-1,0)$ now parallels the one for $\beta\in(-1,0)$.

The remaining assertion for $\alpha\in(-{\mathbb{N}})$ is obtained as follows.
Suppose $\alpha=-k$ for some $k\in{\mathbb{N}}$. Then the fact

$1/\Gamma(n+1-k)=0,\quad 0\leqslant n\leqslant k-1,$ (8.20)

yields that the first $k$ terms in the sums on the right-hand sides of (8.13)
and (8.19) (i.e., the terms with $n=0,\dots,k-1$) are simply absent. The
remaining N–H analysis is not affected by the missing of the first $k$ poles
in (8.13) and (8.19). ∎

######  Remark 8.2.

An alternative direct proof of the N–H property of
$\widehat{m}_{\alpha,\beta}$ can be obtained by employing its Hadamard
factorization as follows. Using the Weierstrass factorization formula for the
$\Gamma$-function (also known as Euler’s product formula, cf. [1, eq. 6.1.3]),

$\displaystyle\frac{1}{\Gamma(z)}=ze^{\gamma_{E}}\prod_{n=1}^{\infty}\Big{[}\Big{(}1+\frac{z}{n}\Big{)}e^{-\frac{z}{n}}\Big{]},\quad
z\in{\mathbb{C}}$ (8.21)

(with $\gamma_{E}=0.57721\dots$ Euler’s constant), one verifies that

$\widehat{m}_{\alpha,\beta}(z)=-\frac{\Gamma(2+\alpha+\beta)}{2^{1+\alpha+\beta}\beta\Gamma(1+\beta)\Gamma(2+\alpha)}\frac{z-z_{-\beta}}{z}\prod_{n=1}^{\infty}\bigg{[}1-\frac{z}{z_{n-\beta}}\bigg{]}\bigg{/}\bigg{[}1-\frac{z}{z_{n}}\bigg{]},$
(8.22)

employing the notation

${\mathbb{R}}\ni t\mapsto z_{t}=t(t+1+\alpha+\beta).$ (8.23)

In particular, the possible poles and zeros of $\widehat{m}_{\alpha,\beta}$
are given by

$\displaystyle z_{n}=n(n+1+\alpha+\beta),\quad n\in{\mathbb{N}}_{0},$ (8.24)

and

$\displaystyle z_{n-\beta}=(n-\beta)(n+1+\alpha),\quad n\in{\mathbb{N}}_{0},$
(8.25)

respectively. (Cancellations between poles and zeros are possible, see the
case $\alpha\in(-{\mathbb{N}})$ in the proof of Theorem 8.1.) To characterize
the N–H property of $\widehat{m}_{\alpha,\beta}$ one then employs [48, Theorem
1 on p. 308] which amounts to establishing the interlacing property of zeros
and poles of $\widehat{m}_{\alpha,\beta}$. $\diamond$

We conclude with some additional comments regarding the Nevanlinna–Herglotz
property of $\widehat{m}_{\alpha,\beta}$ for $\alpha\in-{\mathbb{N}}$. For
this purpose we take a closer look at the Jacobi polynomials for $\alpha=-k$,
$k\in{\mathbb{N}}$. We start with the general definition given in [58, eq.
4.12.2],

$\displaystyle\begin{split}P_{n}^{\alpha,\beta}(x)=\frac{(1+\alpha)_{n}}{n!}\mathstrut_{2}F_{1}(-n,n+\alpha+\beta+1;1+\alpha;(1-x)/2),&\\\
n\in{\mathbb{N}}_{0},\;\alpha,\beta\in{\mathbb{R}},\;x\in(-1,1)&\end{split}$
(8.26)

(see (A.11) for the introduction of Pochhammer’s symbol $(c)_{n}$,
$c\in{\mathbb{C}}$, $n\in{\mathbb{N}}_{0}$), where for $\alpha=-k$,
$k\in{\mathbb{N}}$, $n\geqslant k$, an appropriate limit has to be taken. For
this we can use a special case of the formula [1, eq. 15.1.2] to compute

$\displaystyle P_{n}^{-k,\beta}(x)$
$\displaystyle=\lim_{\alpha\rightarrow-k}P_{n}^{\alpha,\beta}(x)=\frac{(-1)^{k}(n+1-k+\beta)_{k}}{(n-k)!k!(n+1-k)_{k}}\bigg{(}\frac{1-x}{2}\bigg{)}^{k}$
$\displaystyle\quad\,\times\mathstrut_{2}F_{1}(-n+k,n+1+\beta;k+1;(1-x)/2),$
(8.27) $\displaystyle\hskip 74.83089ptn\geqslant
k,\;n,k\in{\mathbb{N}},\;x\in(-1,1),$

where for $n<k$, the prefactor in (8.26) is zero whenever $\alpha=-k$, while
the hypergeometric function remains well-defined and hence

$\displaystyle P_{n}^{-k,\beta}(x)=0,\quad 0\leqslant n\leqslant k-1,\
n\in{\mathbb{N}}_{0},\ k\in{\mathbb{N}},\;x\in(-1,1).$ (8.28)

The Jacobi polynomials $P_{n}(\,\cdot\,)$ satisfy (in the sense of
distributions)

$\displaystyle\tau_{\alpha,\beta}P_{n}^{\alpha,\beta}(x)=z_{n}P_{n}^{\alpha,\beta}(x),\quad
n\in{\mathbb{N}}_{0},\;\alpha,\beta\in{\mathbb{R}},\;x\in(-1,1),$ (8.29)

with $\tau_{\alpha,\beta}$ given by (4.1) and $z_{n}$ introduced in (8.24). In
particular,
$\big{\\{}P_{n}^{\alpha,\beta}(\,\cdot\,)\big{\\}}_{n\in{\mathbb{N}}_{0}}$
constitutes an orthogonal basis in the Hilbert space
$L^{2}\big{(}(-1,1);r_{\alpha,\beta}dx\big{)}$ for $\alpha>-1$,
$\beta\in(-1,1)$.

For $\alpha\leqslant-1$, the Jacobi differential expression
$\tau_{\alpha,\beta}$ is in the limit-point case at $x=1$ (cf. Section 3).
Hence up to a multiplicative constant, there can be at most one distributional
solution $y$ of $\tau_{\alpha,\beta}y=zy$, that lies in
$L^{2}\big{(}(c,1);r_{\alpha,\beta}dx\big{)}$ for some $c\in(-1,1)$, and from
(8) one infers that $P_{n}^{-k,\beta}(x)$ with $n\geqslant k$ are indeed
square integrable locally near $x=1$ with respect to the weight function
$r_{-k,\beta}$. As $P_{n}^{-k,\beta}(\,\cdot\,)$, $n\geqslant k$, satisfy the
Friedrichs boundary condition at $x=-1$ for $\beta\in(0,1)$, and the Neumann
boundary condition for $\beta\in(-1,0)$, one concludes that they are
distributional eigensolutions of $\tau_{-k,\beta}$ corresponding to $z_{n}$
saisfying the corresponding boundary condition at $x=-1$ (no boundary
condition at $x=1$ is needed due to the limit-point case at $x=1$). Moreover,
from [1, eq. 15.1.2] it also follows that the corresponding $m$-function is
again $\widehat{m}_{\alpha,\beta}$ with poles exactly at
$z_{n}=n(n+1-k+\beta)$, $n\geqslant k$. Thus, one can summarize the case
$\alpha\in(-{\mathbb{N}})$ as follows.

######  Theorem 8.3.

Suppose $\alpha=-k\in(-{\mathbb{N}})$, $\beta\in(-1,1)\backslash\\{0\\}$, and
consider the self-adjoint operator

$T_{-k,\beta,Jac}=\begin{cases}T_{0,-k,\beta},&\beta\in(0,1),\\\
T_{\pi/2,-k,\beta},&\beta\in(-1,0),\end{cases}\quad k\in{\mathbb{N}}$ (8.30)

$($cf. Section 6$)$. Then

$\displaystyle M_{-k,\beta}(z)$
$\displaystyle=\begin{cases}m_{0,-k,\beta}(z),&\beta\in(0,1),\\\
m_{\frac{\pi}{2},-k,\beta}(z),&\beta\in(-1,0),\end{cases}$
$\displaystyle=\frac{-\Gamma(1-\beta)}{2^{\beta+1-k}\beta\Gamma(1+\beta)}$
(8.31)
$\displaystyle\quad\times\frac{\Gamma([1-k+\beta+\sigma_{-k,\beta}(z)]/2)\Gamma([1-k+\beta-\sigma_{-k,\beta}(z)]/2)}{\Gamma([1-k-\beta+\sigma_{-k,\beta}(z)]/2)\Gamma((1-k-\beta-\sigma_{-k,\beta}(z)]/2)},$
$\displaystyle\hskip 165.02606ptk\in{\mathbb{N}},\
\beta\in(-1,1)\backslash\\{0\\}.$

Moreover, the spectrum of $T_{-k,\beta,Jac}$ is given by

$\sigma(T_{-k,\beta,Jac})=\\{n(n+1-k+\beta)\\}_{n\in{\mathbb{N}},\,n\geqslant
k},\quad\beta\in(-1,1)\backslash\\{0\\},$ (8.32)

with corresponding eigenfunctions the Jacobi polynomials,
$\big{\\{}P_{n}^{-k,\beta}(\,\cdot\,)\big{\\}}_{n\in{\mathbb{N}},\,n\geqslant
k}$. In particular,
$\big{\\{}P_{n}^{-k,\beta}(\,\cdot\,)\big{\\}}_{n\in{\mathbb{N}},\,n\geqslant
k}$ forms a complete orthogonal basis in the Hilbert space
$L^{2}\big{(}(-1,1);r_{-k,\beta}dx\big{)}$.

######  Remark 8.4.

The statement concerning completeness in Theorem 8.3 can also be derived
directly from the formula (cf. [58, eq. 4.22.2])

$\displaystyle\binom{n}{k}P^{-k,\beta}_{n}(x)=\binom{n+\beta}{k}\Big{(}\frac{x-1}{2}\Big{)}^{k}P^{k,\beta}_{n-k}(x),\quad
n\in{\mathbb{N}},\;n\geqslant k,\;\beta\in(-1,1)\backslash\\{0\\},$ (8.33)

and the known completeness of the set of Jacobi polynomials
$\big{\\{}P^{k,\beta}_{n-k}(\,\cdot\,)\big{\\}}_{n\in{\mathbb{N}},\,n\geqslant
k}$ in $L^{2}\big{(}(-1,1);r_{\alpha,\beta}dx\big{)}$,
$\beta\in(-1,1)\backslash\\{0\\}$ (see, e.g., [4, Corollary 5.7.5], [6,
Theorem 4.1.5], [58, Theorem 3.1.5]). $\diamond$

######  Remark 8.5.

$(i)$ One notes that the case $\beta\in(0,1),\ \alpha\in(-{\mathbb{N}})$ in
Theorem 8.3 was already explicitly covered in (6.26), but at first glance the
two results seem to differ. This apparent discrepancy can be addressed through
a simple change of variables. From (6.26), one has for
$\alpha=-k\in(-{\mathbb{N}})$ and using the substitution $m=n+k\geqslant k,\
n\in{\mathbb{N}}_{0}$,

$\displaystyle\begin{split}\sigma(T_{F,-k,\beta})=\\{(n+k)(n+1+\beta)\\}_{n\in{\mathbb{N}}_{0}}=\\{m(m-k+1+\beta)\\}_{m\in{\mathbb{N}},\,m\geqslant
k},&\\\ \beta\in(0,1),&\end{split}$ (8.34)

in agreement with (8.31).

The remaining case when $\beta\in(-1,0)$ in Theorem 8.3 can similarly be
verified by using Theorem 6.1 with $\gamma=\pi/2$ along with (6.18). In
particular, the $m$-function for $\gamma=\pi/2$ for this case is simply the
negative reciprocal of (6.18), which is formally (6.26) with only the $\beta$
range changed.
$(ii)$ To directly compare the $m$-functions instead, by $k$ applications of
the functional equation $\Gamma(z+1)=z\Gamma(z)$ to each of the Gamma
functions appearing in (8.31), one can verify this $m$-function equals
$m_{0,-k,\beta}(z)$ for $\beta\in(0,1)$ given in (6.26). From the previous
discussion, $m_{\frac{\pi}{2},-k,\beta}(z)$ for $\beta\in(-1,0)$ is formally
the same as $m_{0,-k,\beta}(z)$ with only the $\beta$ range changed; hence
this case follows from the previous one. $\diamond$

##  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])

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

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)

Making the substitution $\xi=(1+x)/2$ in (A.2) yields

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

Similarly, the substitution $\xi=(1-x)/2$ in (A.2) implies

$\displaystyle\begin{split}\xi(1-\xi)y^{\prime\prime}(z,\xi)+[\alpha+1-(\alpha+\beta+2)\xi]y^{\prime}(z,\xi)+zy(z,\xi)=0,\\\
\alpha,\beta\in{\mathbb{R}},\;\xi\in(0,1),\end{split}$ (A.4)

which has effectively only interchanged the roles of the parameters $\alpha$
and $\beta$.

At the endpoints of the Jacobi equation, $x=-1$ and $x=1$ respectively, the
substitutions used to arrive at (A.3) and (A.4) both give $\xi=0$, hence we
next consider solutions of (A.1) near $\xi=0$ (cf. [1, Eqs. 15.5.3, 15.5.4])

$\displaystyle
w_{1,0}(\xi)=\mathstrut_{2}F_{1}(a,b;c;\xi)=\sum_{n\in{\mathbb{N}}_{0}}\frac{(a)_{n}(b)_{n}}{(c)_{n}}\frac{\xi^{n}}{n!},\quad
a,b\in{\mathbb{C}},\;c\in{\mathbb{C}}\backslash(-{\mathbb{N}}_{0}),\;\xi\in(0,1),$
$\displaystyle
w_{2,0}(\xi)=\xi^{1-c}\mathstrut_{2}F_{1}(a-c+1,b-c+1;2-c;\xi),\quad
a,b\in{\mathbb{C}},\;(c-1)\in{\mathbb{C}}\backslash{\mathbb{N}},$ (A.5)
$\displaystyle\hskip 280.25981pt\xi\in(0,1).$

In addition, see [1, p. 563],

$\text{$w_{1,0}$ and $w_{2,0}$ are linearly independent if
}\,(a-b),(c-a-b),c\in{\mathbb{C}}\backslash{\mathbb{Z}}.$ (A.6)

For $c=k+1\in{\mathbb{N}}$ we will employ (cf. [1, eq. 15.5.16–19])

$\displaystyle w_{1,0}(\xi)$
$\displaystyle=\mathstrut_{2}F_{1}(a,b;k+1;\xi),\quad
a,b\in{\mathbb{C}},\;\xi\in(0,1),$ $\displaystyle w_{2,0}^{\text{\rm
ln},k}(\xi)$ $\displaystyle=\mathstrut_{2}F_{1}(a,b;k+1;\xi)\text{\rm
ln}(\xi)+\sum_{n=1}^{\infty}\frac{(a)_{n}(b)_{n}}{(k+1)_{n}n!}\xi^{n}\big{[}\psi(a+n)-\psi(a)$
(A.7)
$\displaystyle\quad+\psi(b+n)-\psi(b)-\psi(k+1+n)+\psi(k+1)-\psi(n+1)-\gamma_{E}\big{]}$
$\displaystyle\quad-\sum_{n=1}^{k}\frac{(n-1)!(-k)_{n}}{(1-a)_{k}(1-b)_{k}}\xi^{-n},\quad
a,b\in{\mathbb{C}}\backslash(-{\mathbb{N}}_{0}),\;\xi\in(0,1),$

where the superscipt “ln” indicates the presence of a logarithmic term
(familiar from Frobenius theory). For the special case $c=1$ $(k=0)$, the last
sum disappears resulting in

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

In the case $c=1-k\in-{\mathbb{N}}_{0}$ or $k\in{\mathbb{N}}$ we can use [1,
Eq. 15.5.20,21]

$\displaystyle w_{1,0}(\xi)$
$\displaystyle=\xi^{k}\mathstrut_{2}F_{1}(a+k,b+k;1+k;\xi),\quad
a,b\in{\mathbb{C}},\;\xi\in(0,1),$ $\displaystyle w_{2,0}^{\text{\rm
ln},-k}(\xi)$
$\displaystyle=\xi^{k}\mathstrut_{2}F_{1}(a+k,b+k;1+k;\xi)\text{\rm ln}(\xi)$
$\displaystyle+\xi^{k}\sum_{n\in{\mathbb{N}}}$
$\displaystyle\xi^{n}\frac{(a+k)_{n}(b+k)_{n}}{(1+k)_{n}n!}[\psi(a+k+n)-\psi(a+n)+\psi(b+k+n)$
(A.9) $\displaystyle-\psi(b+n)-\psi(k+1+n)+\psi(k+1)-\psi(n+1)-\gamma_{E}]$
$\displaystyle-\sum_{n=1}^{k}\frac{(n-1)!(-k)_{n}}{(1-a-k)_{n}(1-b-k)_{n}}\xi^{k-n},\quad
a,b\in{\mathbb{C}}\backslash(-{\mathbb{N}}_{0}),\;\xi\in(0,1).$

In addition,

$\text{$w_{1,0}$ and $w_{2,0}^{\text{\rm ln},k}$ are linearly independent if
}\,a,b\in{\mathbb{C}}\backslash(-{\mathbb{N}}_{0}),\,k\in{\mathbb{N}}_{0}.$
(A.10)

Here $\mathstrut{}_{2}F_{1}(\,\cdot\,,\,\cdot\,;\,\cdot\,;\,\cdot\,)$ 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.11)

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

Comparing (A.1) and (A.3), and identifying

$\displaystyle\begin{split}&a=[1+\alpha+\beta+\sigma_{\alpha,\beta}(z)]/2,\quad
b=[1+\alpha+\beta-\sigma_{\alpha,\beta}(z)]/2,\\\ &c=1+\beta,\\\
&\sigma_{\alpha,\beta}(z)=\big{[}(1+\alpha+\beta)^{2}+4z\big{]}^{1/2},\\\
&\xi=(1+x)/2,\end{split}$ (A.12)

in (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
$x=-1$,

$\displaystyle
y_{1,\alpha,\beta,-1}(z,x)=\mathstrut_{2}F_{1}(a_{\alpha,\beta,\sigma_{\alpha,\beta}(z)},a_{\alpha,\beta,-\sigma_{\alpha,\beta}(z)};1+\beta;(1+x)/2),$
(A.13) $\displaystyle\hskip
213.39566pt\beta\in{\mathbb{R}}\backslash(-{\mathbb{N}}),$ $\displaystyle
y_{2,\alpha,\beta,-1}(z,x)=(1+x)^{-\beta}\mathstrut_{2}F_{1}(a_{\alpha,-\beta,\sigma_{\alpha,\beta}(z)},a_{\alpha,-\beta,-\sigma_{\alpha,\beta}(z)};1-\beta;(1+x)/2),$
$\displaystyle\hskip
261.76526pt\beta\in{\mathbb{R}}\backslash{\mathbb{N}}_{0},$ (A.14)
$\displaystyle
y_{2,\alpha,0,-1}(z,x)=\mathstrut_{2}F_{1}(a_{\alpha,0,\sigma_{\alpha,0}(z)},a_{\alpha,0,-\sigma_{\alpha,0}(z)};1;(1+x)/2)\,\text{\rm
ln}((1+x)/2)$
$\displaystyle\quad+\sum_{n\in{\mathbb{N}}}\frac{(a_{\alpha,0,\sigma_{\alpha,0}(z)})_{n}(a_{\alpha,0,-\sigma_{\alpha,0}(z)})_{n}}{2^{n}(n!)^{2}}(1+x)^{n}$
(A.15) $\displaystyle\hskip
39.83368pt\times[\psi(a_{\alpha,0,\sigma_{\alpha,0}(z)}+n)-\psi(a_{\alpha,0,\sigma_{\alpha,0}(z)})+\psi(a_{\alpha,0,-\sigma_{\alpha,0}(z)}+n)$
$\displaystyle\hskip
54.06006pt-\psi(a_{\alpha,0,-\sigma_{\alpha,0}(z)})-2\psi(n+1)-2\gamma_{E}],\quad\beta=0;$
$\displaystyle\hskip
145.10922pt\alpha\in{\mathbb{R}},\;z\in{\mathbb{C}},\;x\in(-1,1),$

where we once again used the abbreviation

$\displaystyle
a_{\mu,\nu,\pm\sigma}=[1+\mu+\nu\pm\sigma]/2,\quad\mu,\nu,\sigma\in{\mathbb{C}}.$
(A.16)

In the case $\beta=k\in{\mathbb{N}}_{0}$ and $\alpha\in[0,\infty)$ we define
(cf. (A.7))

$\displaystyle
y_{1,\alpha,k,-1}(z,x)=\mathstrut_{2}F_{1}(a_{\alpha,k,\sigma_{\alpha,k}(z)},a_{\alpha,k,-\sigma_{\alpha,k}(z)};1+k;(1+x)/2),$
(A.17) $\displaystyle
y_{2,\alpha,k,-1}(z,x)=\mathstrut_{2}F_{1}(a_{\alpha,k,\sigma_{\alpha,k}(z)},a_{\alpha,k,-\sigma_{\alpha,k}(z)};1+k;(1+x)/2)\text{\rm
ln}((1+x)/2)$
$\displaystyle\quad+\sum_{n=1}^{\infty}\frac{(a_{\alpha,k,\sigma_{\alpha,k}(z)})_{n}(a_{\alpha,k,-\sigma_{\alpha,k}(z)})_{n}}{2^{n}(1+k)_{n}n!}(1+x)^{n}$
(A.18) $\displaystyle\hskip
39.83368pt\times\,\big{[}\psi(a_{\alpha,k,\sigma_{\alpha,k}(z)}+n)-\psi(a_{\alpha,k,\sigma_{\alpha,k}(z)})+\psi(a_{\alpha,k,-\sigma_{\alpha,k}(z)}+n)$
$\displaystyle\hskip
54.06006pt-\,\psi(a_{\alpha,k,-\sigma_{\alpha,k}(z)})-\psi(1+k+n)+\psi(1+k)-\psi(n+1)-\gamma_{E}\big{]}$
$\displaystyle\hskip
68.28644pt-\sum_{n=1}^{k}\frac{2^{n}(n-1)!(-k)_{n}}{(1-a_{\alpha,k,\sigma_{\alpha,k}(z)})_{k}(1-a_{\alpha,k,-\sigma_{\alpha,k}(z)})_{k}}(1+x)^{-n};$
$\displaystyle\hskip 189.21065pt\alpha\geqslant
0,\;z\in{\mathbb{C}},\;x\in(-1,1),$

while for $\beta=-k\in-{\mathbb{N}}$ and $\alpha\in[0,\infty)$ one obtains
from (A.9),

$\displaystyle y_{2,\alpha,-k,-1}(z,x)$
$\displaystyle=(1+x)^{k}\mathstrut_{2}F_{1}(a_{\alpha,k,\sigma_{\alpha,k}(z)}+k,a_{\alpha,k,-\sigma_{\alpha,k}(z)}+k;1+k;(1+x)/2),$
(A.19) $\displaystyle y_{1,\alpha,-k,-1}(z,x)$
$\displaystyle=(1+x)^{k}\mathstrut_{2}F_{1}(a_{\alpha,k,\sigma_{\alpha,k}(z)}+k,a_{\alpha,k,-\sigma_{\alpha,k}(z)}+k;1+k;(1+x)/2)$
$\displaystyle\times\,\text{\rm ln}((1+$ $\displaystyle
x)/2)+\sum_{n=1}^{\infty}\frac{(a_{\alpha,k,\sigma_{\alpha,k}(z)}+k)_{n}(a_{\alpha,k,-\sigma_{\alpha,k}(z)}+k)_{n}}{2^{n}(1+k)_{n}n!}(1+x)^{n}$
(A.20) $\displaystyle\times\,\big{[}\psi($ $\displaystyle
a_{\alpha,k,\sigma_{\alpha,k}(z)}+k+n)-\psi(a_{\alpha,k,\sigma_{\alpha,k}(z)}+n)+\psi(a_{\alpha,k,-\sigma_{\alpha,k}(z)}+k+n)$
$\displaystyle-\,\psi($ $\displaystyle
a_{\alpha,k,-\sigma_{\alpha,k}(z)}+n)-\psi(k+1+n)+\psi(1+k)-\psi(n+1)-\gamma_{E}\big{]}$
$\displaystyle-\sum_{n=1}^{k}\frac{2^{n}(n-1)!(-k)_{n}}{(1-a_{\alpha,k,\sigma_{\alpha,k}(z)}-k)_{n}(1-a_{\alpha,k,-\sigma_{\alpha,k}(z)}-k)_{n}}(1+x)^{-n};$
$\displaystyle\hskip 157.91287pt\alpha\geqslant
0,\;z\in{\mathbb{C}},\;x\in(-1,1).$

Again, $y_{1,\alpha,\beta,-1}(z,\,\cdot\,)$ and
$y_{2,\alpha,\beta,-1}(z,\,\cdot\,)$ are linearly independent for
$\alpha\in{\mathbb{R}}$, $\beta\in{\mathbb{R}}\backslash{\mathbb{Z}}$.
Similarly, $y_{1,\alpha,0,-1}(z,\,\cdot\,)$ and
$y_{2,\alpha,0,-1}(z,\,\cdot\,)$ are linearly independent for
$\alpha\in{\mathbb{R}}$. We note that (A.17)–(A.20) make sense for
$\alpha\in{\mathbb{R}}$, but we will choose different solutions for
$\alpha\in(-\infty,0)$ which greatly simplify the computation of the
$m$-functions in Section 7:

$\displaystyle y_{1,\alpha,\pm k,-1}(z,x)$
$\displaystyle=(1-x)^{-\alpha}y_{1,-\alpha,\pm k,-1}(z+(1\pm k)\alpha),x),$
$\displaystyle y_{2,\alpha,\pm k,-1}(z,x)$
$\displaystyle=(1-x)^{-\alpha}y_{2,-\alpha,\pm k,-1}(z+(1\pm k)\alpha,x);$
(A.21) $\displaystyle\hskip
71.98543pt\alpha\in(-\infty,0),\;k\in{\mathbb{Z}}\backslash\\{0\\}.$

That (A.21) are indeed solutions to the Jacobi differential equation
$\tau_{\alpha,\pm k}y=zy(z,\,\cdot\,)$ follows from (4.5).
In precisely the same manner one obtains for solutions of
$\tau_{\alpha,\beta}y(z,\,\cdot\,)=0$ near $x=1$,

$\displaystyle
y_{1,\alpha,\beta,1}(z,x)=\mathstrut_{2}F_{1}(a_{\alpha,\beta,\sigma_{\alpha,\beta}(z)},a_{\alpha,\beta,-\sigma_{\alpha,\beta}(z)};1+\alpha;(1-x)/2),$
(A.22) $\displaystyle\hskip
213.39566pt\alpha\in{\mathbb{R}}\backslash(-{\mathbb{N}}),$ $\displaystyle
y_{2,\alpha,\beta,1}(z,x)=(1-x)^{-\alpha}\mathstrut_{2}F_{1}(a_{-\alpha,\beta,\sigma_{\alpha,\beta}(z)},a_{-\alpha,\beta,-\sigma_{\alpha,\beta}(z)};1-\alpha;(1-x)/2),$
$\displaystyle\hskip 256.0748pt\alpha\in{\mathbb{R}}\backslash{\mathbb{N}},$
(A.23) $\displaystyle
y_{2,0,\beta,1}(z,x)=\mathstrut_{2}F_{1}(a_{0,\beta,\sigma_{0,\beta}(z)},a_{0,\beta,-\sigma_{0,\beta}(z)};1;(1-x)/2)\,\text{\rm
ln}((1-x)/2)$
$\displaystyle\quad+\sum_{n\in{\mathbb{N}}}\frac{(a_{0,\beta,\sigma_{0,\beta}(z)})_{n}(a_{0,\beta,-\sigma_{0,\beta}(z)})_{n}}{2^{n}(n!)^{2}}(1-x)^{n}$
(A.24) $\displaystyle\hskip
39.83368pt\times[\psi(a_{0,\beta,\sigma_{0,\beta}(z)}+n)-\psi(a_{0,\beta,\sigma_{0,\beta}(z)})+\psi(a_{0,\beta,-\sigma_{0,\beta}(z)}+n)$
$\displaystyle\hskip
54.06006pt-\psi(a_{0,\beta,-\sigma_{0,\beta}(z)})-2\psi(n+1)-2\gamma_{E}],\quad\alpha=0;$
$\displaystyle\hskip
146.53151pt\beta\in{\mathbb{R}},\;z\in{\mathbb{C}},\;x\in(-1,1).$

Again, $y_{1,\alpha,\beta,1}(z,\,\cdot\,)$ and
$y_{2,\alpha,\beta,1}(z,\,\cdot\,)$ are linearly independent for
$\alpha\in{\mathbb{R}}\backslash{\mathbb{Z}}$, $\beta\in{\mathbb{R}}$.
Similarly, $y_{1,0,\beta,1}(z,\,\cdot\,)$ and $y_{2,0,\beta,1}(z,\,\cdot\,)$
are linearly independent for $\beta\in{\mathbb{R}}$. In the limit-point case
at $x=1$, that is, for $\alpha\in(-\infty,-1]\cup[1,\infty)$, we only need the
principal solutions, which are $y_{1,\alpha,\beta,1}(z,\,\cdot\,)$ for
$\alpha\in[1,\infty)$ and $y_{2,\alpha,\beta,1}(z,\,\cdot\,)$ for
$\alpha\in(-\infty,-1]$. Thus, we see from (A.22) and (A.23) that these cases
are already covered, and we do not need to define an additional solution for
$\alpha\in{\mathbb{Z}}\backslash\\{0\\}$.

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.25)

Moreover $y_{j,\alpha,\beta,\pm 1}(z,x)$ satisfy the relations (cf. (4.5) and
(5.16))

$\displaystyle y_{1,\alpha,\beta,-1}(z,x)$
$\displaystyle=(1+x)^{-\beta}y_{2,\alpha,-\beta,-1}(z+(1+\alpha)\beta,x),$
(A.26) $\displaystyle\hskip
92.47145pt\alpha\in{\mathbb{R}},\;\beta\in{\mathbb{R}}\backslash\\{0\\},$
$\displaystyle y_{2,\alpha,\beta,-1}(z,x)$
$\displaystyle=(1+x)^{-\beta}y_{1,\alpha,-\beta,-1}(z+(1+\alpha)\beta,x),$
(A.27) $\displaystyle\hskip
92.47145pt\alpha\in{\mathbb{R}},\;\beta\in{\mathbb{R}}\backslash\\{0\\},$
$\displaystyle y_{1,\alpha,\beta,1}(z,x)$
$\displaystyle=(1-x)^{-\alpha}y_{2,-\alpha,\beta,1}(z+(1+\beta)\alpha,x),$
(A.28) $\displaystyle\hskip
86.78099pt\alpha\in{\mathbb{R}}\backslash\\{0\\},\;\beta\in{\mathbb{R}},$
$\displaystyle y_{2,\alpha,\beta,1}(z,x)$
$\displaystyle=(1-x)^{-\alpha}y_{1,-\alpha,\beta,1}(z+(1+\beta)\alpha,x),$
(A.29) $\displaystyle\hskip
86.78099pt\alpha\in{\mathbb{R}}\backslash\\{0\\},\;\beta\in{\mathbb{R}},$

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.30)

######  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 [53,
eq. 18.5.7])

$\displaystyle\begin{split}P_{n}^{\alpha,\beta}(x)=\dfrac{(\alpha+1)_{n}}{n!}\mathstrut_{2}F_{1}(-n,\,n+\alpha+\beta+1;\,\alpha+1;\,(1-x)/2),&\\\
n\in{\mathbb{N}}_{0},\;-\alpha\notin{\mathbb{N}},\;-n-\alpha-\beta-1\notin{\mathbb{N}},\;x\in(-1,1),&\end{split}$
(A.31)

and can be defined by continuity for all parameters
$\alpha,\beta\in{\mathbb{R}}$. One notes that
$P_{n}^{\alpha,\beta}(\,\cdot\,)$ 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. [58, p. 64]). It satisfies the equation

$\displaystyle\tau_{\alpha,\beta}P_{n}^{\alpha,\beta}(x)=\lambda^{\alpha,\beta}_{n}P_{n}^{\alpha,\beta}(x),\quad
x\in(-1,1),$ (A.32)

with

$\displaystyle\lambda^{\alpha,\beta}_{n}=n(n+1+\alpha+\beta).$ (A.33)

In particular, one obtains (cf. [36, p. 87], [58, p. 62])

$P_{0}^{\alpha,\beta}(x)=1,\quad
P_{1}^{\alpha,\beta}(x)=[(2+\alpha+\beta)x+\alpha-\beta]/2,\,\text{
etc.,}\;x\in(-1,1).$ (A.34)

In addition, 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 at $x=1$ (resp., $x=-1$) if $\alpha\in[0,\infty)$ (resp.,
$\beta\in[0,\infty)$).

More generally, all quasi-rational solutions, meaning the logarithmic
derivative being rational, can be derived from the Jacobi polynomials together
with (4.5) and are summarized in Table 1, which is taken from [9]. Here
$(1-x)^{-\alpha}P_{n}^{-\alpha,\beta}(x)$ satisfy at $x=1$ the Friedrichs
boundary condition (b.c.) for $\alpha\in(-\infty,0]$ and the Neumann b.c. for
$\alpha\in(0,1)$, while at $x=-1$ they satisfy the Friedrichs b.c. for
$\beta\in[0,\infty)$ and the Neumann b.c. 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, implying the Friedrichs b.c. at $x=1$
for $\alpha\in[0,\infty)$, and the Neumann b.c. for $\alpha\in(-1,0)$, and at
$x=-1$, the Friedrichs b.c. for $\beta\in(-\infty,0]$, the Neumann b.c. 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 b.c. for $\alpha\in(-\infty,0]$ (resp.,
$\beta\in(-\infty,0]$) and the Neumann b.c. for $\alpha\in(0,1)$ (resp.,
$\beta\in(0,1)$). $\diamond$

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 $\tau_{\alpha,\beta}$

##  Appendix B Connection Formulas

In this appendix we provide the connection formulas utilized to find the
solution behaviors in Appendix C, (C.1)–(C.12). We express them using
$w_{1,0}(\xi)$ and $w_{2,0}(\xi)$ $\big{(}w_{2,0}^{\text{\rm
ln},k}(\xi)\big{)}$ and their analogs $w_{1,1}(\xi)$ and $w_{2,1}(\xi)$
$\big{(}w_{2,1}^{\text{\rm ln},k}(\xi)\big{)}$ at the endpoint $\xi=1$.

We recall the relations (A.12) connecting the parameters $a,b,c$ and
$\alpha,\beta$ and note that $\xi\in(0,1)$ for the remainder of this appendix.
$\mathbf{(I)}$ 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 solutions $w_{1,1}(\xi)$ and $w_{2,1}(\xi)$ can be obtained from (A.5)
by the change of parameters

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

or, explicitly,

$\displaystyle w_{1,1}(\xi)$
$\displaystyle=\mathstrut_{2}F_{1}(a,b;a+b-c+1;1-\xi),$ (B.2) $\displaystyle
w_{2,1}(\xi)$
$\displaystyle=(1-\xi)^{c-a-b}\mathstrut_{2}F_{1}(c-a,c-b;c-a-b+1;1-\xi).$
(B.3)

The two connection formulas are given by (cf. [53, 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.4) $\displaystyle w_{2,0}(\xi)$
$\displaystyle=\frac{\Gamma(2-c)\Gamma(c-a-b)}{\Gamma(1-a)\Gamma(1-b)}w_{1,1}(\xi)+\frac{\Gamma(2-c)\Gamma(a+b-c)}{\Gamma(a-c+1)\Gamma(b-c+1)}w_{2,1}(\xi).$
(B.5)

One notes that poles occur on the right-hand sides of (B.4), (B.5) whenever
$(a+b-c)\in{\mathbb{Z}}$. Using (B.1) we 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.6) $\displaystyle w_{2,1}(\xi)$
$\displaystyle=\frac{\Gamma(1+c-a-b)\Gamma(1-c)}{\Gamma(1-a)\Gamma(1-b)}w_{1,0}(\xi)+\frac{\Gamma(1+c-a-b)\Gamma(c-1)}{\Gamma(c-a)\Gamma(c-b)}w_{2,0}(\xi).$
(B.7)

$\mathbf{(II)}$ 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 $w_{1,0}(\xi)=\mathstrut_{2}F_{1}(a,b;a+b;\xi)$ can be expanded
at $\xi=1$ (cf. [1, eq. 15.3.10]):

$\displaystyle\begin{split}\mathstrut_{2}F_{1}(a,b;a+b;\xi)&=\frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)}\sum_{n\in{\mathbb{N}}_{0}}\frac{(a)_{n}(b)_{n}}{(n!)^{2}}[2\psi(n+1)-\psi(a+n)-\psi(b+n)\\\
&\hskip 115.23373pt-\text{\rm ln}(1-\xi)](1-\xi)^{n}.\end{split}$ (B.8)

Meanwhile, two linearly independent solutions at $\xi=1$ are given by (cf.
(A.8)):

$\displaystyle w_{1,1}(\xi)$ $\displaystyle=\mathstrut_{2}F_{1}(a,b;1;1-\xi),$
(B.9) $\displaystyle w_{2,1}^{\text{\rm ln},0}(\xi)$
$\displaystyle=\mathstrut_{2}F_{1}(a,b;1;1-\xi)\text{\rm
ln}(1-\xi)+\sum_{n\in{\mathbb{N}}}\frac{(a)_{n}(b)_{n}}{(n!)^{2}}(1-\xi)^{n}$
(B.10)
$\displaystyle\quad\,\times[\psi(a+n)-\psi(a)+\psi(b+n)-\psi(b)-2\psi(n+1)-2\gamma_{E}].$

The connection formula for $w_{1,1}(\xi)$ is given by (B.6) with $a+b=c$. To
obtain the second connection formula we need to compare the expansion of
$w_{2,1}^{\text{\rm ln}}(\xi)$ at $\xi=1$, with the expansion of
$\mathstrut{}_{2}F_{1}(a,b;a+b;\xi)$ at $\xi=1$ using (B.8). We obtain

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

The reflection formula for the $\psi$-function [1, eq. 6.3.7],

$\displaystyle\psi(1-z)-\psi(z)=\pi\cot(\pi z),$ (B.12)

also implies

$\displaystyle w_{1,0}(\xi)$
$\displaystyle=-\frac{\Gamma(1-a)\Gamma(1-b)}{\Gamma(1-a-b)\pi[\cot(\pi
a)+\cot(\pi b)]}$
$\displaystyle\qquad\times\big{[}[\psi(a)+\psi(b)+2\gamma_{E}]w_{1,1}(\xi)+w_{2,1}^{\text{\rm
ln},0}(\xi)\big{]},$ (B.13) $\displaystyle w_{2,0}(\xi)$
$\displaystyle=\frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b-1)\pi[\cot(\pi a)+\cot(\pi
b)]}$
$\displaystyle\quad\,\times\big{[}[\psi(1-a)+\psi(1-b)+2\gamma_{E}]w_{1,1}(\xi)+w_{2,1}^{\text{\rm
ln},0}(\xi)\big{]}.$ (B.14)

$\mathbf{(III)}$ 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)–(B.14) must be changed through the renaming of (B.1) with $c\rightarrow
a+b-c+1=a+b$ as $c=1$. Since $c$ does not appear in (B)–(B.14) (we eliminated
it through $c=a+b$), we can adopt the aforementioned formulas directly only
changing the second index in the $w$’s222Formula (B.15) could have been
obtained directly from (B.4) 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.15) $\displaystyle w_{2,0}^{\text{\rm ln},0}(\xi)$
$\displaystyle=-[\psi(1-a)+\psi(1-b)+2\gamma_{E}]\frac{\Gamma(1-a-b)}{\Gamma(1-a)\Gamma(1-b)}w_{1,1}(\xi)$
$\displaystyle\quad\,-[\psi(a)+\psi(b)+2\gamma_{E}]\frac{\Gamma(a+b-1)}{\Gamma(a)\Gamma(b)}w_{2,1}(\xi),$
(B.16) $\displaystyle w_{1,1}(\xi)$
$\displaystyle=-\frac{\Gamma(1-a)\Gamma(1-b)}{\Gamma(1-a-b)\pi[\cot(\pi
a)+\cot(\pi b)]}$
$\displaystyle\qquad\,\times\big{[}[\psi(a)+\psi(b)+2\gamma_{E}]w_{1,0}(\xi)+w_{2,0}^{\text{\rm
ln},0}(\xi)\big{]},$ (B.17) $\displaystyle w_{2,1}(\xi)$
$\displaystyle=\frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b-1)\pi[\cot(\pi a)+\cot(\pi
b)]}$
$\displaystyle\quad\,\times\big{[}[\psi(1-a)+\psi(1-b)+2\gamma_{E}]w_{1,0}(\xi)+w_{2,0}^{\text{\rm
ln},0}(\xi)\big{]}.$ (B.18)

$\mathbf{(IV)}$ The case $\alpha=\beta=0$, that is, $a+b=c=1$:
For $\alpha=0$ and $\beta=0$ the Jacobi differential expression (4.1) becomes
the Legendre differential expression. This case was treated in detail in [25],
we shall only write down the connection formulas for completeness.

The special solutions $w_{1,j}(\xi)$ and $w_{2,j}^{\text{\rm ln},0}(\xi)$ for
$j=1,2$ are given by (A.8) and (B.9), (B.10). One notes that the following
relations hold,

$\displaystyle w_{1,1}(\xi)=w_{1,0}(1-\xi),\quad w_{2,1}^{\text{\rm
ln},0}(\xi)=w_{2,0}^{\text{\rm ln},0}(1-\xi).$ (B.19)

Using [1, eq. 15.3.10] together with
$w_{1,0}=\mathstrut_{2}F_{1}(a,b;a+b,\xi)$ and Euler’s reflection formula for
the $\Gamma$-function [1, eq. 6.1.17],

$\displaystyle\Gamma(z)\Gamma(1-z)=\pi\csc(\pi z),$ (B.20)

one obtains

$\displaystyle w_{1,0}(\xi)=-\pi^{-1}\sin(\pi
a)\big{[}[\psi(a)+\psi(b)+2\gamma_{E}]w_{1,1}(\xi)+w_{2,1}^{\text{\rm
ln},0}(\xi)\big{]}.$ (B.21)

This formula can be also obtained as the limit $\beta\to 0$ in (B). From the
two relations (B.19) we immediately get

$\displaystyle w_{1,1}(\xi)$ $\displaystyle=-\pi^{-1}\sin(\pi
a)\big{[}[\psi(a)+\psi(b)+2\gamma_{E}]w_{1,0}(\xi)+w_{2,0}^{\text{\rm
ln},0}(\xi)\big{]},$ (B.22) $\displaystyle w_{2,0}^{\text{\rm ln},0}(\xi)$
$\displaystyle=\pi^{-1}\sin(\pi
a)\big{\\{}\big{[}[\psi(a)+\psi(b)+2\gamma_{E}]^{2}-\pi^{2}[\sin(\pi
a)]^{-2}\big{]}w_{1,1}(\xi)$
$\displaystyle\quad+[\psi(a)+\psi(b)+2\gamma_{E}]w_{2,1}^{\text{\rm
ln},0}(\xi)\big{\\}},$ (B.23) $\displaystyle w_{2,1}^{\text{\rm ln},0}(\xi)$
$\displaystyle=\pi^{-1}\sin(\pi
a)\big{\\{}\big{[}[\psi(a)+\psi(b)+2\gamma_{E}]^{2}-\pi^{2}[\sin(\pi
a)]^{-2}\big{]}w_{1,0}(\xi)$
$\displaystyle\quad+[\psi(a)+\psi(b)+2\gamma_{E}]w_{2,0}^{\text{\rm
ln},0}(\xi)\big{\\}}.$ (B.24)

$\mathbf{(V)}$ The case $\alpha\in[0,\infty),\,\beta=k\in{\mathbb{N}}$, that
is, $c=1+k$, $a+b\geqslant c$ :
The fundamental system at $\xi=0$ consists of $w_{1,0}(\xi)$ given in (A.5)
and (A.7). At the other endpoint we have the fundamental system (B.2) and
(B.3). In particular, as $a+b-c+1=a+b-k$, we see that $w_{1,1}(\xi)$ has the
form

$\displaystyle w_{1,1}(\xi)=\mathstrut_{2}F_{1}(a,b;a+b-k;1-\xi),\quad
k\in{\mathbb{N}}.$ (B.25)

Thus, one can use [1, eq. 15.3.12] to read off the behavior of $w_{1,1}(\xi)$
at $\xi=0$. Comparing with the expansion of $w_{1,0}(\xi)$ and
$w_{2,0}^{\text{\rm ln}}(\xi)$ at $\xi=0$, one concludes that

$\displaystyle w_{1,1}(\xi)$
$\displaystyle=-\frac{(-1)^{k}\Gamma(a+b-k)}{\Gamma(a-k)\Gamma(b-k)k!}$ (B.26)
$\displaystyle\quad\quad\times\big{[}[\psi(a)+\psi(b)-\psi(k+1)+\gamma_{E}]w_{1,0}(\xi)+w_{2,0}^{\text{\rm
ln},k}(\xi)\big{]}.$

As this is the only connection formula that we need, we just mention that the
other ones can be computed in a similar fashion.

##  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 solutions of the Jacobi differential equation.

We start with the limiting behavior of $y_{j,\alpha,\beta,-1}(z,x)$ as
$x\uparrow 1$ and apply the connection formulas from Appendix B:

$\displaystyle y_{1,\alpha,\beta,-1}(z,x)\underset{x\uparrow
1}{=}\frac{\Gamma(1+\beta)\Gamma(-\alpha)}{\Gamma(a_{-\alpha,\beta,\sigma_{\alpha,\beta}(z)})\Gamma(a_{-\alpha,\beta,-\sigma_{\alpha,\beta}(z)})}[1+O(1-x)]$
$\displaystyle\hskip
76.82234pt+\frac{2^{\alpha}\Gamma(1+\beta)\Gamma(\alpha)}{\Gamma(a_{\alpha,\beta,\sigma_{\alpha,\beta}(z)})\Gamma(a_{\alpha,\beta,-\sigma_{\alpha,\beta}(z)})}(1-x)^{-\alpha}[1+O(1-x)],$
$\displaystyle\hskip
219.08612pt\alpha\in{\mathbb{R}}\backslash{\mathbb{Z}},\;\beta\in(-1,1),$
(C.1) $\displaystyle y_{1,\alpha,\beta,-1}(z,x)\underset{x\uparrow
1}{=}\frac{\Gamma(1+\beta)\Gamma(-\alpha)}{\Gamma(a_{-\alpha,\beta,\sigma_{\alpha,\beta}(z)})\Gamma(a_{-\alpha,\beta,-\sigma_{\alpha,\beta}(z)})}[1+O(1-x)]$
$\displaystyle\hskip
76.82234pt-\frac{(-1)^{-\alpha}2^{\alpha}\Gamma(1+\beta)}{(-\alpha)!\Gamma(a_{\alpha,\beta,\sigma_{\alpha,\beta}(z)})\Gamma(a_{\alpha,\beta,-\sigma_{\alpha,\beta}(z)})}(1-x)^{-\alpha}$
(C.2) $\displaystyle\hskip 76.82234pt\quad\times[\text{\rm
ln}((1-x)/2)+\gamma_{E}-\psi(1-\alpha)+\psi(a_{-\alpha,\beta,\sigma_{\alpha,\beta}(z)})$
$\displaystyle\hskip
76.82234pt+\psi(a_{-\alpha,\beta,-\sigma_{\alpha,\beta}(z)})][1+O(1-x)],\quad\alpha\in(-{\mathbb{N}}),\;\beta\in(-1,1),$
$\displaystyle y_{1,0,\beta,-1}(z,x)\underset{x\uparrow
1}{=}-\frac{\Gamma(1+\beta)}{\Gamma(a_{0,\beta,\sigma_{0,\beta}(z)})\Gamma(a_{0,\beta,-\sigma_{0,\beta}(z)})}[\text{\rm
ln}((1-x)/2)+2\gamma_{E}$ $\displaystyle\hskip
65.44142pt\quad+\psi(a_{0,\beta,\sigma_{0,\beta}(z)})+\psi(a_{0,\beta,-\sigma_{0,\beta}(z)})][1+O(1-x)],$
(C.3) $\displaystyle\hskip 207.7052pt\alpha=0,\;\beta\in(-1,1),$
$\displaystyle y_{1,\alpha,\beta,-1}(z,x)\underset{x\uparrow
1}{=}\frac{2^{\alpha}\Gamma(1+\beta)\Gamma(\alpha)}{\Gamma(a_{\alpha,\beta,\sigma_{\alpha,\beta}(z)})\Gamma(a_{\alpha,\beta,-\sigma_{\alpha,\beta}(z)})}(1-x)^{-\alpha}[1+O(1-x)]$
$\displaystyle\hskip
76.82234pt-\frac{(-1)^{\alpha}\Gamma(1+\beta)}{\alpha!\Gamma(a_{-\alpha,\beta,\sigma_{\alpha,\beta}(z)})\Gamma(a_{-\alpha,\beta,-\sigma_{\alpha,\beta}(z)})}[\text{\rm
ln}((1-x)/2)+\gamma_{E}$ (C.4) $\displaystyle\hskip
76.82234pt-\psi(\alpha+1)+\psi(a_{\alpha,\beta,\sigma_{\alpha,\beta}(z)})+\psi(a_{\alpha,\beta,-\sigma_{\alpha,\beta}(z)})]$
$\displaystyle\hskip
76.82234pt\quad\times[1+O(1-x)],\quad\alpha\in{\mathbb{N}},\;\beta\in(-1,1),$
$\displaystyle y_{2,\alpha,\beta,-1}(z,x)\underset{x\uparrow
1}{=}\frac{2^{-\beta}\Gamma(1-\beta)\Gamma(-\alpha)}{\Gamma(a_{-\alpha,-\beta,\sigma_{\alpha,\beta}(z)})\Gamma(a_{-\alpha,-\beta,-\sigma_{\alpha,\beta}(z)})}[1+O(1-x)]$
$\displaystyle\hskip
76.82234pt+\frac{2^{\alpha-\beta}\Gamma(1-\beta)\Gamma(\alpha)}{\Gamma(a_{\alpha,-\beta,\sigma_{\alpha,\beta}(z)})\Gamma(a_{\alpha,-\beta,-\sigma_{\alpha,\beta}(z)})}(1-x)^{-\alpha}[1+O(1-x)],$
$\displaystyle\hskip
199.16928pt\alpha\in{\mathbb{R}}\backslash{\mathbb{Z}},\;\beta\in(-1,1)\backslash\\{0\\},$
(C.5) $\displaystyle y_{2,\alpha,\beta,-1}(z,x)\underset{x\uparrow
1}{=}\frac{2^{-\beta}\Gamma(1-\beta)\Gamma(-\alpha)}{\Gamma(a_{-\alpha,-\beta,\sigma_{\alpha,\beta}(z)})\Gamma(a_{-\alpha,-\beta,-\sigma_{\alpha,\beta}(z)})}[1+O(1-x)]$
$\displaystyle\hskip
76.82234pt-\frac{(-1)^{-\alpha}2^{\alpha-\beta}\Gamma(1-\beta)}{(-\alpha)!\Gamma(a_{\alpha,-\beta,\sigma_{\alpha,\beta}(z)})\Gamma(a_{\alpha,-\beta,-\sigma_{\alpha,\beta}(z)})}(1-x)^{-\alpha}[\text{\rm
ln}((1-x)/2)$ $\displaystyle\hskip
76.82234pt+\gamma_{E}-\psi(1-\alpha)+\psi(a_{-\alpha,-\beta,\sigma_{\alpha,\beta}(z)})+\psi(a_{-\alpha,-\beta,-\sigma_{\alpha,\beta}(z)})]$
$\displaystyle\hskip
76.82234pt\quad\times[1+O(1-x)],\quad\alpha\in(-{\mathbb{N}}),\;\beta\in(-1,1)\backslash\\{0\\},$
(C.6) $\displaystyle y_{2,0,\beta,-1}(z,x)\underset{x\uparrow
1}{=}-\frac{2^{-\beta}\Gamma(1-\beta)}{\Gamma(a_{0,-\beta,\sigma_{0,\beta}(z)})\Gamma(a_{0,-\beta,-\sigma_{0,\beta}(z)})}[\text{\rm
ln}((1-x)/2)+2\gamma_{E}$ $\displaystyle\hskip
65.44142pt\quad+\psi(a_{0,-\beta,\sigma_{0,\beta}(z)})+\psi(a_{0,-\beta,-\sigma_{0,\beta}(z)})][1+O(1-x)],$
(C.7) $\displaystyle\hskip
202.01474pt\alpha=0,\;\beta\in(-1,1)\backslash\\{0\\},$ $\displaystyle
y_{2,\alpha,\beta,-1}(z,x)\underset{x\uparrow
1}{=}\frac{2^{\alpha-\beta}\Gamma(1-\beta)\Gamma(\alpha)}{\Gamma(a_{\alpha,-\beta,\sigma_{\alpha,\beta}(z)})\Gamma(a_{\alpha,-\beta,-\sigma_{\alpha,\beta}(z)})}(1-x)^{-\alpha}[1+O(1-x)]$
$\displaystyle\hskip
76.82234pt-\frac{(-1)^{\alpha}2^{-\beta}\Gamma(1-\beta)}{\alpha!\Gamma(a_{-\alpha,-\beta,\sigma_{\alpha,\beta}(z)})\Gamma(a_{-\alpha,-\beta,-\sigma_{\alpha,\beta}(z)})}[\text{\rm
ln}((1-x)/2)+\gamma_{E}$ $\displaystyle\hskip
76.82234pt-\psi(\alpha+1)+\psi(a_{\alpha,-\beta,\sigma_{\alpha,\beta}(z)})+\psi(a_{\alpha,-\beta,-\sigma_{\alpha,\beta}(z)})]$
(C.8) $\displaystyle\hskip
76.82234pt\quad\times[1+O(1-x)],\quad\alpha\in{\mathbb{N}},\;\beta\in(-1,1)\backslash\\{0\\},$
$\displaystyle y_{2,\alpha,0,-1}(z,x)\underset{x\uparrow
1}{=}\frac{-[2\gamma_{E}+\psi(a_{-\alpha,0,\sigma_{\alpha,0}(z)})+\psi(a_{-\alpha,0,-\sigma_{\alpha,0}(z)})]\Gamma(-\alpha)}{\Gamma(a_{-\alpha,0,\sigma_{\alpha,0}(z)})\Gamma(a_{-\alpha,0,-\sigma_{\alpha,0}(z)})}$
$\displaystyle\hskip 76.82234pt\quad\times[1+O(1-x)]$ $\displaystyle\hskip
76.82234pt-\frac{[2\gamma_{E}+\psi(a_{\alpha,0,\sigma_{\alpha,0}(z)})+\psi(a_{\alpha,0,-\sigma_{\alpha,0}(z)})]2^{\alpha}\Gamma(\alpha)}{\Gamma(a_{\alpha,0,\sigma_{\alpha,0}(z)})\Gamma(a_{\alpha,0,-\sigma_{\alpha,0}(z)})}(1-x)^{-\alpha}$
$\displaystyle\hskip
76.82234pt\quad\times[1+O(1-x)],\quad\alpha\in{\mathbb{R}}\backslash{\mathbb{Z}},\;\beta=0,$
(C.9) $\displaystyle y_{2,\alpha,0,-1}(z,x)\underset{x\uparrow
1}{=}\frac{-[2\gamma_{E}+\psi(a_{-\alpha,0,\sigma_{\alpha,0}(z)})+\psi(a_{-\alpha,0,-\sigma_{\alpha,0}(z)})]\Gamma(-\alpha)}{\Gamma(a_{-\alpha,0,\sigma_{\alpha,0}(z)})\Gamma(a_{-\alpha,0,-\sigma_{\alpha,0}(z)})}$
$\displaystyle\hskip
76.82234pt\quad\times[1+O(1-x)]-\bigg{(}\frac{\Gamma(a_{-\alpha,0,\sigma_{\alpha,0}(z)})\Gamma(a_{-\alpha,0,-\sigma_{\alpha,0}(z)})}{\Gamma(1-\alpha)}$
$\displaystyle\hskip
76.82234pt\quad\quad-\frac{(-1)^{-\alpha}[2\gamma_{E}+\psi(a_{-\alpha,0,\sigma_{\alpha,0}(z)})+\psi(a_{-\alpha,0,-\sigma_{\alpha,0}(z)})]}{\Gamma(a_{\alpha,0,\sigma_{\alpha,0}(z)})\Gamma(a_{\alpha,0,-\sigma_{\alpha,0}(z)})}$
$\displaystyle\hskip 76.82234pt\quad\times[\text{\rm
ln}((1-x)/2)+\gamma_{E}-\psi(1-\alpha)+\psi(a_{-\alpha,0,\sigma_{\alpha,0}(z)})$
(C.10) $\displaystyle\hskip
76.82234pt\quad\quad+\psi(a_{-\alpha,0,-\sigma_{\alpha,0}(z)})]\bigg{)}2^{\alpha}(1-x)^{-\alpha}[1+O(1-x)],$
$\displaystyle\hskip 233.3125pt\alpha\in(-{\mathbb{N}}),\;\beta=0,$
$\displaystyle y_{2,0,0,-1}(z,x)\underset{x\uparrow
1}{=}-\bigg{(}\Gamma([1+\sigma_{0,0}(z)]/2)\Gamma([1-\sigma_{0,0}(z)]/2)-[\text{\rm
ln}((1-x)/2)+2\gamma_{E}$ $\displaystyle\hskip
65.44142pt\quad+\psi([1+\sigma_{0,0}(z)]/2)+\psi([1-\sigma_{0,0}(z)]/2)]$
(C.11) $\displaystyle\hskip
76.82234pt\quad\times\frac{\psi([1+\sigma_{0,0}(z)]/2)+\psi([1-\sigma_{0,0}(z)]/2)+\gamma_{E}}{\Gamma([1+\sigma_{0,0}(z)]/2)\Gamma([1-\sigma_{0,0}(z)]/2)}\bigg{)}$
$\displaystyle\hskip 76.82234pt\quad\times[1+O(1-x)],\quad\alpha=\beta=0,$
$\displaystyle y_{2,\alpha,0,-1}(z,x)\underset{x\uparrow
1}{=}\frac{-(2\gamma_{E}+\psi(a_{\alpha,0,\sigma_{\alpha,0}(z)})+\psi(a_{\alpha,0,-\sigma_{\alpha,0}(z)}))2^{\alpha}\Gamma(\alpha)}{\Gamma(a_{\alpha,0,\sigma_{\alpha,0}(z)})\Gamma(a_{\alpha,0,-\sigma_{\alpha,0}(z)})}$
$\displaystyle\hskip
76.82234pt\quad\times(1-x)^{-\alpha}[1+O(1-x)]-\bigg{(}\frac{\Gamma(a_{\alpha,0,\sigma_{\alpha,0}(z)})\Gamma(a_{\alpha,0,-\sigma_{\alpha,0}(z)})}{\Gamma(\alpha+1)}$
$\displaystyle\hskip
76.82234pt-\frac{(-1)^{\alpha}[2\gamma_{E}+\psi(a_{\alpha,0,\sigma_{\alpha,0}(z)})+\psi(a_{\alpha,0,-\sigma_{\alpha,0}(z)})]}{\Gamma(a_{-\alpha,0,\sigma_{\alpha,0}(z)})\Gamma(a_{-\alpha,0,-\sigma_{\alpha,0}(z)})}$
(C.12) $\displaystyle\hskip 76.82234pt\quad\times[\text{\rm
ln}((1-x)/2)+\gamma_{E}-\psi(\alpha+1)+\psi(a_{\alpha,0,\sigma_{\alpha,0}(z)})$
$\displaystyle\hskip
76.82234pt\quad+\psi(a_{\alpha,0,-\sigma_{\alpha,0}(z)})]\bigg{)}[1+O(1-x)],\quad\alpha\in{\mathbb{N}},\;\beta=0,$
$\displaystyle\hskip 230.46748ptz\in{\mathbb{C}},\;x\in(-1,1).$

We continue this appendix with a list of generalized boundary values for the
solutions $y_{j,\alpha,\beta,-1}(z,x),\ j=1,2$ at $x=\mp 1$. At the endpoint
$x=-1$ 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.13)

To obtain generalized boundary values at the endpoint $x=1$, one employs the
connection formulas (C.1)–(C.12)), to find for $z\in{\mathbb{C}}$,

$\displaystyle\begin{split}\widetilde{y}_{1,\alpha,\beta,-1}(z,1)&=\begin{cases}\dfrac{\Gamma(1+\beta)\Gamma(-\alpha)}{\Gamma(a_{-\alpha,\beta,\sigma_{\alpha,\beta}(z)})\Gamma(a_{-\alpha,\beta,-\sigma_{\alpha,\beta}(z)})},&\alpha\in(-1,0),\\\\[8.53581pt]
\dfrac{-2^{1+\alpha+\beta}\Gamma(1+\alpha)\Gamma(1+\beta)}{\Gamma(a_{\alpha,\beta,\sigma_{\alpha,\beta}(z)})\Gamma(a_{\alpha,\beta,-\sigma_{\alpha,\beta}(z)})},&\alpha\in[0,1),\end{cases}\end{split}$
$\displaystyle\begin{split}\widetilde{y}_{1,\alpha,\beta,-1}^{\,\prime}(z,1)&=\begin{cases}\dfrac{2^{1+\alpha+\beta}\Gamma(1+\alpha)\Gamma(1+\beta)}{\Gamma(a_{\alpha,\beta,\sigma_{\alpha,\beta}(z)})\Gamma(a_{\alpha,\beta,-\sigma_{\alpha,\beta}(z)})},&\alpha\in(-1,0),\\\\[8.53581pt]
\dfrac{-\Gamma(1+\beta)}{\Gamma(a_{0,\beta,\sigma_{0,\beta}(z)})\Gamma(a_{0,\beta,-\sigma_{0,\beta}(z)})}[2\gamma_{E}\\\
\quad+\psi(a_{0,\beta,\sigma_{0,\beta}(z)})+\psi(a_{0,\beta,-\sigma_{0,\beta}(z)})],&\alpha=0,\\\\[8.53581pt]
\dfrac{\Gamma(1+\beta)\Gamma(-\alpha)}{\Gamma(a_{-\alpha,\beta,\sigma_{\alpha,\beta}(z)})\Gamma(a_{-\alpha,\beta,-\sigma_{\alpha,\beta}(z)})},&\alpha\in(0,1);\end{cases}\end{split}$
$\displaystyle\hskip 184.9429pt\beta\in(-1,1),$ (C.14)
$\displaystyle\begin{split}\widetilde{y}_{2,\alpha,\beta,-1}(z,1)&=\begin{cases}\dfrac{2^{-\beta}\Gamma(1-\beta)\Gamma(-\alpha)}{\Gamma(a_{-\alpha,-\beta,\sigma_{\alpha,\beta}(z)})\Gamma(a_{-\alpha,-\beta,-\sigma_{\alpha,\beta}(z)})},&\alpha\in(-1,0),\\\\[8.53581pt]
\dfrac{-2^{\alpha+1}\Gamma(1+\alpha)\Gamma(1-\beta)}{\Gamma(a_{\alpha,-\beta,\sigma_{\alpha,\beta}(z)})\Gamma(a_{\alpha,-\beta,-\sigma_{\alpha,\beta}(z)})},&\alpha\in[0,1),\end{cases}\end{split}$
$\displaystyle\begin{split}\widetilde{y}_{2,\alpha,\beta,-1}^{\,\prime}(z,1)&=\begin{cases}\dfrac{2^{\alpha+1}\Gamma(1+\alpha)\Gamma(1-\beta)}{\Gamma(a_{\alpha,-\beta,\sigma_{\alpha,\beta}(z)})\Gamma(a_{\alpha,-\beta,-\sigma_{\alpha,\beta}(z)})},&\alpha\in(-1,0),\\\\[8.53581pt]
\dfrac{-2^{-\beta}\Gamma(1-\beta)}{\Gamma(a_{0,-\beta,\sigma_{0,\beta}(z)})\Gamma(a_{0,-\beta,-\sigma_{0,\beta}(z)})}[2\gamma_{E}\\\
\quad+\psi(a_{0,-\beta,\sigma_{0,\beta}(z)})+\psi(a_{0,-\beta,-\sigma_{0,\beta}(z)})],&\alpha=0,\\\\[8.53581pt]
\dfrac{2^{-\beta}\Gamma(1-\beta)\Gamma(-\alpha)}{\Gamma(a_{-\alpha,-\beta,\sigma_{\alpha,\beta}(z)})\Gamma(a_{-\alpha,-\beta,-\sigma_{\alpha,\beta}(z)})},&\alpha\in(0,1);\end{cases}\end{split}$
$\displaystyle\hskip 177.82971pt\beta\in(-1,1)\backslash\\{0\\},$ (C.15)
$\displaystyle\begin{split}\widetilde{y}_{2,\alpha,0,-1}(z,1)&=\begin{cases}\dfrac{-[2\gamma_{E}+\psi(a_{-\alpha,0,\sigma_{\alpha,0}(z)})+\psi(a_{-\alpha,0,-\sigma_{\alpha,0}(z)})]\Gamma(-\alpha)}{\Gamma(a_{-\alpha,0,\sigma_{\alpha,0}(z)})\Gamma(a_{-\alpha,0,-\sigma_{\alpha,0}(z)})},\\\\[2.84526pt]
\hskip 211.97293pt\alpha\in(-1,0),\\\\[8.53581pt]
\dfrac{[2\gamma_{E}+\psi(a_{\alpha,0,\sigma_{\alpha,0}(z)})+\psi(a_{\alpha,0,-\sigma_{\alpha,0}(z)})]\Gamma(1+\alpha)}{2^{-\alpha-1}\Gamma(a_{\alpha,0,\sigma_{\alpha,0}(z)})\Gamma(a_{\alpha,0,-\sigma_{\alpha,0}(z)})},\\\\[2.84526pt]
\hskip 213.39566pt\alpha\in[0,1),\end{cases}\end{split}$
$\displaystyle\begin{split}\widetilde{y}_{2,\alpha,0,-1}^{\,\prime}(z,1)&=\begin{cases}\dfrac{[2\gamma_{E}+\psi(a_{\alpha,0,\sigma_{\alpha,0}(z)})+\psi(a_{\alpha,0,-\sigma_{\alpha,0}(z)})]\Gamma(1+\alpha)}{-2^{-\alpha-1}\Gamma(a_{\alpha,0,\sigma_{\alpha,0}(z)})\Gamma(a_{\alpha,0,-\sigma_{\alpha,0}(z)})},\\\\[2.84526pt]
\hskip 213.39566pt\alpha\in(-1,0),\\\\[2.84526pt]
-\Gamma([1+\sigma_{0,0}(z)]/2)\Gamma([1-\sigma_{0,0}(z)]/2)\\\
+\dfrac{[2\gamma_{E}+\psi([1+\sigma_{0,0}(z)]/2)+\psi([1-\sigma_{0,0}(z)]/2)]^{2}}{\Gamma([1+\sigma_{0,0}(z)]/2)\Gamma([1-\sigma_{0,0}(z)]/2)},\quad\alpha=0,\\\\[8.53581pt]
\dfrac{-[2\gamma_{E}+\psi(a_{-\alpha,0,\sigma_{\alpha,0}(z)})+\psi(a_{-\alpha,0,-\sigma_{\alpha,0}(z)})]\Gamma(-\alpha)}{\Gamma(a_{-\alpha,0,\sigma_{\alpha,0}(z)})\Gamma(a_{-\alpha,0,-\sigma_{\alpha,0}(z)})},\\\\[2.84526pt]
\hskip 213.39566pt\alpha\in(0,1);\end{cases}\end{split}$ $\displaystyle\hskip
227.62204pt\beta=0.$ (C.16)

Finally, we turn to the limiting behavior of $y_{j,\alpha,\beta,1}(z,x)$ as
$x\downarrow-1$ in some cases, applying the connection formulas (B.6), (B.7)
from Appendix B:

$\displaystyle y_{1,\alpha,\beta,1}(z,x)$
$\displaystyle=\mathstrut_{2}F_{1}(a_{\alpha,\beta,\sigma_{\alpha,\beta}(z)},a_{\alpha,\beta,-\sigma_{\alpha,\beta}(z)};1+\alpha;(1-x)/2)$
$\displaystyle=\frac{\Gamma(1+\alpha)\Gamma(-\beta)}{\Gamma(a_{\alpha,-\beta,\sigma_{\alpha,\beta}(z)})\Gamma(a_{\alpha,-\beta,-\sigma_{\alpha,\beta}(z)})}$
$\displaystyle\qquad\times\mathstrut_{2}F_{1}(a_{\alpha,\beta,\sigma_{\alpha,\beta}(z)},a_{\alpha,\beta,-\sigma_{\alpha,\beta}(z)};1+\beta;(1+x)/2)$
$\displaystyle\quad+\frac{2^{\beta}\Gamma(1+\alpha)\Gamma(\beta)}{\Gamma(a_{\alpha,\beta,\sigma_{\alpha,\beta}(z)})\Gamma(a_{\alpha,\beta,-\sigma_{\alpha,\beta}(z)})}(1+x)^{-\beta}$
$\displaystyle\qquad\times\mathstrut_{2}F_{1}(a_{\alpha,-\beta,\sigma_{\alpha,\beta}(z)},a_{\alpha,-\beta,-\sigma_{\alpha,\beta}(z)};1-\beta;(1+x)/2),$
(C.17) $\displaystyle\hskip
140.84105pt\alpha\in{\mathbb{R}}\backslash(-{\mathbb{N}}),\;\beta\in{\mathbb{R}}\backslash{\mathbb{Z}},$
$\displaystyle y_{2,\alpha,\beta,1}(z,x)$
$\displaystyle=(1-x)^{-\alpha}\mathstrut_{2}F_{1}(a_{-\alpha,\beta,\sigma_{\alpha,\beta}(z)},a_{-\alpha,\beta,-\sigma_{\alpha,\beta}(z)};1-\alpha;(1-x)/2)$
$\displaystyle=\frac{2^{-\alpha}\Gamma(1-\alpha)\Gamma(-\beta)}{\Gamma(a_{-\alpha,-\beta,\sigma_{\alpha,\beta}(z)})\Gamma(a_{-\alpha,-\beta,-\sigma_{\alpha,\beta}(z)})}$
$\displaystyle\qquad\times\mathstrut_{2}F_{1}(a_{\alpha,\beta,\sigma_{\alpha,\beta}(z)},a_{\alpha,\beta,-\sigma_{\alpha,\beta}(z)};1+\beta;(1+x)/2)$
$\displaystyle\quad+\frac{2^{\beta-\alpha}\Gamma(1-\alpha)\Gamma(\beta)}{\Gamma(a_{-\alpha,\beta,\sigma_{\alpha,\beta}(z)})\Gamma(a_{-\alpha,\beta,-\sigma_{\alpha,\beta}(z)})}(1+x)^{-\beta}$
$\displaystyle\qquad\times\mathstrut_{2}F_{1}(a_{\alpha,-\beta,\sigma_{\alpha,\beta}(z)},a_{\alpha,-\beta,-\sigma_{\alpha,\beta}(z)};1-\beta;(1+x)/2),$
(C.18) $\displaystyle\hskip
162.18062pt\alpha\in{\mathbb{R}}\backslash{\mathbb{N}},\;\beta\in{\mathbb{R}}\backslash{\mathbb{Z}};$
$\displaystyle\hskip 165.02606ptz\in{\mathbb{C}},\;x\in(-1,1).$

In particular,

$\displaystyle
y_{1,\alpha,\beta,1}(z,x)\underset{x\downarrow-1}{=}\frac{\Gamma(1+\alpha)\Gamma(-\beta)}{\Gamma(a_{\alpha,-\beta,\sigma_{\alpha,\beta}(z)})\Gamma(a_{\alpha,-\beta,-\sigma_{\alpha,\beta}(z)})}[1+O(1+x)]$
$\displaystyle\hskip
82.51282pt+\frac{2^{\beta}\Gamma(1+\alpha)\Gamma(\beta)}{\Gamma(a_{\alpha,\beta,\sigma_{\alpha,\beta}(z)})\Gamma(a_{\alpha,\beta,-\sigma_{\alpha,\beta}(z)})}(1+x)^{-\beta}[1+O(1+x)],$
$\displaystyle\hskip
204.85974pt\alpha\in{\mathbb{R}}\backslash(-{\mathbb{N}}),\;\beta\in{\mathbb{R}}\backslash{\mathbb{Z}},$
(C.19) $\displaystyle
y_{2,\alpha,\beta,1}(z,x)\underset{x\downarrow-1}{=}\frac{2^{-\alpha}\Gamma(1-\alpha)\Gamma(-\beta)}{\Gamma(a_{-\alpha,-\beta,\sigma_{\alpha,\beta}(z)})\Gamma(a_{-\alpha,-\beta,-\sigma_{\alpha,\beta}(z)})}[1+O(1+x)]$
$\displaystyle\hskip
82.51282pt+\frac{2^{\beta-\alpha}\Gamma(1-\alpha)\Gamma(\beta)}{\Gamma(a_{-\alpha,\beta,\sigma_{\alpha,\beta}(z)})\Gamma(a_{-\alpha,\beta,-\sigma_{\alpha,\beta}(z)})}(1+x)^{-\beta}[1+O(1+x)],$
$\displaystyle\hskip
221.93158pt\alpha\in{\mathbb{R}}\backslash{\mathbb{N}},\;\beta\in{\mathbb{R}}\backslash{\mathbb{Z}}.$
(C.20)

##  Appendix D Laguerre $m$-function as a limit of the Jacobi $m$-function

The Laguerre polynomials $L_{n}^{\beta}(t)$ given by (see. [58, Ch. V])

$\displaystyle L_{n}^{\beta}(t):=\binom{n+b}{n}\ _{1}F_{1}(-n,\beta+1;t),\quad
t\in(0,\infty),\;n\in\mathbb{N}_{0},\;\beta\in(-1,\infty),$ (D.1)

can be obtained as a limit of the Jacobi polynomials (cf. [53, eq. 18.7.22],
[58, eq. 5.3.4])

$\displaystyle
L^{\beta}_{n}(t)=\lim_{\alpha\uparrow\infty}(-1)^{n}P_{n}^{\alpha,\beta}((2t/\alpha)-1),$
(D.2)

where the convergence is locally uniform in the complex $t$-plane. This is one
example of the limiting relations in the Askey scheme [3].333The Askey scheme
has been expanded by Koekoek and Swarttouw [40] (see also [59]) to include
basic orthogonal polynomials. Similarly, for the Laguerre differential
expression given by

$\displaystyle\tau_{Lag}^{\beta}=-t^{-\beta}e^{t}\frac{d}{dt}t^{\beta+1}e^{-t}\frac{d}{dt},\quad
t\in(0,\infty),\;\beta\in(-1,\infty),$ (D.3) $\displaystyle
p(t)=p_{\beta}(t)=t^{\beta+1}e^{-t},\quad q(t)=0,\quad
r(t)=r_{\beta}(t)=t^{\beta}e^{-t},$

one obtains

$\displaystyle\lim_{\alpha\uparrow\infty}\frac{1}{\alpha}\tau_{Jac}^{\alpha,\beta}=\tau_{Lag}^{\beta},\
\mbox{ with }\ x=(2t/\alpha)-1.$ (D.4)

Here convergence means local uniform convergence of the coefficients in the
complex $t$-plane. Additionally, the eigenvalues $\lambda_{n}^{Lag}$ of the
Laguerre polynomials are independent of the parameter $\beta$ and can be
obtained via

$\displaystyle\lambda_{n}^{Lag}=n=\lim_{\alpha\uparrow\infty}\lambda_{n}^{\alpha,\beta}/\alpha,\quad
n\in\mathbb{N}_{0}.$ (D.5)

For the weight functions we have the limiting relation

$\displaystyle
r_{\beta}(t)=t^{\beta}e^{-t}=\lim_{\alpha\uparrow\infty}2^{-\alpha}(\alpha/2)^{\beta}r_{Jac}^{\alpha,\beta}((2t/a)-1).$
(D.6)

The computation of the $m$-function for various boundary conditions has been
performed in [25, Sect. 6], we note, however, that our parameter $\beta$ is
smaller by $1$. The $m$-function for the self-adjoint extension having the
Laguerre polynomials as eigenfunctions is equal to the $m$-function
corresponding to the Friedrichs extension in the case $\beta\in(0,1)$, and
holds for $\beta\in(-1,1)\backslash\\{0\\}$

$\displaystyle
m_{Lag}^{\beta}(w)=\frac{\Gamma(-\beta)\Gamma(-w)}{\Gamma(\beta+1)\Gamma(-\beta-w)},\quad\beta\in(-1,1)\backslash\\{0\\},\;w\in{\mathbb{C}}\backslash{\mathbb{N}}_{0}.$
(D.7)

Moreover from (D.4) or (D.5) we see that the correct scaling of the new
spectral parameter $w$ as $\alpha\uparrow\infty$ should be $w=z/\alpha$, where
$z$ is the spectral parameter of the Jacobi problem.

A fundamental system for the Laguerre problem is given in terms of confluent
hypergeometric functions (see [25, Ch. 6.3]):

$\displaystyle\begin{aligned}
\widehat{y}_{1,\beta}(w,t)&={}_{1}F_{1}(-w,\beta+1;t),\\\
\widehat{y}_{2,\beta}(w,t)&=t^{-\beta}{}_{1}F_{1}(-\beta-w,1-\beta;t),\end{aligned}\quad\beta\in(-1,1)\backslash\\{0\\},\;w\in{\mathbb{C}},\;t\in(0,\infty).$
(D.8)

One can check that
$W_{Lag}(\widehat{y}_{2,\beta}(w,\cdot),\widehat{y}_{1,\beta}(w,\cdot))=\beta$.
Generalizing (D.2) we can obtain this fundamental system as a limit of the
Jacobi fundamental system found in (A.13) and (A.14). For this we need a
special case of [53, eq. 16.8.10]:

$\displaystyle{}_{1}F_{1}(a,c;t)=\lim_{b\to\infty}{}_{2}F_{1}(a,b,c;t/b),$
(D.9)

and the observation that

$\displaystyle[\alpha+\beta+1-\sigma(\alpha t)]/2$
$\displaystyle\underset{\alpha\uparrow\infty}{=}-t+O\big{(}\alpha^{-1}\big{)},$
(D.10) $\displaystyle[\alpha+\beta+1+\sigma(\alpha t)]/2$
$\displaystyle\underset{\alpha\uparrow\infty}{=}\alpha+\beta+1+t+O\big{(}\alpha^{-1}\big{)},$

as $\alpha\uparrow\infty$. Together (D.9) and (D.10) imply that

$\displaystyle\widehat{y}_{1,\beta}(w,t)$
$\displaystyle=\lim_{\alpha\uparrow\infty}y_{1,\alpha,\beta,-1}(\alpha
t,(2t/\alpha)-1),$ (D.11) $\displaystyle\widehat{y}_{2,\beta}(w,t)$
$\displaystyle=\lim_{\alpha\uparrow\infty}(2/\alpha)^{\beta}y_{2,\alpha,\beta,-1}(\alpha
t,(2t/\alpha)-1);\quad w\in{\mathbb{C}},\;t\in(0,\infty).$

Taking into account the normalization chosen in [25, Sect. 6] and in our
paper, the following relation between the two $m$-functions (8) and (D.7)
holds:

$\displaystyle
m_{Lag}^{\beta}(w)=\lim_{\alpha\uparrow\infty}2^{\alpha+1}(2/\alpha)^{\beta}\widehat{m}_{\alpha,\beta}(\alpha
w).$ (D.12)

Using [1, eq. 6.1.47] with (D.10) one can indeed check that the limit (D.12)
holds locally uniformly for $w\in{\mathbb{C}}\backslash{\mathbb{N}}_{0}$.

##  Appendix E Special Cases

###  E.1. The Case of Gegenbauer or Ultraspherical Polynomials

In this section, we look at some of the special cases of the Jacobi
differential expression in more detail beginning with the Gegenbauer (or
ultraspherical) case (see, e.g., [1, Ch. 22], [53, Ch. 18], [58, Ch. IV]).
This can be realized by choosing the parameters $\alpha=\beta=\mu-1/2$ so that

$\displaystyle p(x)=(1-x^{2})^{\mu+1/2},\quad r(x)=(1-x^{2})^{\mu-1/2},\quad
q(x)=0,\quad x\in(-1,1),\quad\mu\in{\mathbb{R}}.$ (E.1)

Thus, one considers the differential expression

$\displaystyle\tau_{\mu}=-(1-x^{2})^{1/2-\mu}(d/dx)\big{(}(1-x^{2})^{\mu+1/2}\big{)}(d/dx),\quad
x\in(-1,1),\ \mu\in{\mathbb{R}},$ (E.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 $\mu\in{\mathbb{R}}\backslash(-1/2,3/2)$. Furthermore,
choosing $\mu=1/2$ one arrives 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 $\mu\in(-1/2,3/2)$, whereas Theorem 7.1 describes
$\mu\in{\mathbb{R}}\backslash(-1/2,3/2)$.

###  E.2. The Case of Chebyshev Polynomials

The Chebyshev cases of the first and second kind are two more special cases.

The Chebyshev case of the first kind is realized by choosing $\mu=0$ in the
Gegenbauer case, or $\alpha=\beta=-1/2$ in the Jacobi case (see, e.g., [1, Ch.
22], [53, Ch. 18], [58, Ch. IV]). Thus, one considers the differential
expression

$\displaystyle\tau_{0}=-(1-x^{2})^{1/2}(d/dx)\big{(}(1-x^{2})^{1/2}\big{)}(d/dx),\quad
x\in(-1,1),$ (E.3)

which is regular at $x=\pm 1$. By (5.4) one can write

$\displaystyle\begin{split}m_{\gamma,\delta,-1/2,-1/2}(z)&=\big{[}-2^{-1/2}\cos(\gamma)(\cos(\delta)\widetilde{y}_{1}(z,1)+\sin(\delta)\widetilde{y}_{1}^{\,\prime}(z,1))\\\
&\quad-\sin(\gamma)(\cos(\delta)\widetilde{y}_{2}(z,1)+\sin(\delta)\widetilde{y}_{2}^{\,\prime}(z,1))\big{]}\\\
&\quad\times\big{[}-2^{-1/2}\sin(\gamma)(\cos(\delta)\widetilde{y}_{1}(z,1)+\sin(\delta)\widetilde{y}_{1}^{\,\prime}(z,1))\\\
&\quad\quad+\cos(\gamma)(\cos(\delta)\widetilde{y}_{2}(z,1)+\sin(\delta)\widetilde{y}_{2}^{\,\prime}(z,1))\big{]}^{-1},\\\
&\quad\hskip 52.63777pt\gamma,\delta\in[0,\pi),\
z\in\rho(T_{\gamma,\delta,-1/2,-1/2}),\end{split}$ (E.4)

with

$\displaystyle\widetilde{y}_{1}(z,1)$
$\displaystyle=\frac{\Gamma(1/2)\Gamma(1/2)}{\Gamma((1/2)+z^{1/2})\Gamma((1/2)-z^{1/2})}=\frac{\pi}{\Gamma((1/2)+z^{1/2})\Gamma((1/2)-z^{1/2})},$
$\displaystyle\begin{split}\widetilde{y}_{1}^{\,\prime}(z,1)&=\frac{\Gamma(1/2)\Gamma(1/2)}{\Gamma(z^{1/2})\Gamma(-z^{1/2})}=\frac{\pi}{\Gamma(z^{1/2})\Gamma(-z^{1/2})},\\\\[8.53581pt]
\widetilde{y}_{2}(z,1)&=\frac{2^{1/2}\Gamma(3/2)\Gamma(1/2)}{\Gamma(1+z^{1/2})\Gamma(1-z^{1/2})}=\frac{2^{-1/2}\pi}{\Gamma(1+z^{1/2})\Gamma(1-z^{1/2})},\end{split}$
(E.5) $\displaystyle\widetilde{y}_{2}^{\,\prime}(z,1)$
$\displaystyle=\frac{2^{1/2}\Gamma(3/2)\Gamma(1/2)}{\Gamma((1/2)+z^{1/2})\Gamma((1/2)-z^{1/2})}=\frac{2^{-1/2}\pi}{\Gamma((1/2)+z^{1/2})\Gamma((1/2)-z^{1/2})}.$

Here we have used (cf. [1, Eqs. 6.1.8, 6.1.9]) $\Gamma(1/2)=\pi^{1/2}$,
$\Gamma(3/2)=2^{-1}\pi^{1/2}$, and the fact that

$\displaystyle\sigma_{-1/2,-1/2}(z)=2z^{1/2}.$ (E.6)

######  Example E.1 (Neumann boundary conditions).

Considering the Neumann boundary conditions, one obtains

$\displaystyle m_{\pi/2,\pi/2,-1/2,-1/2}(z)$
$\displaystyle=2^{1/2}\frac{\widetilde{y}_{2}^{\,\prime}(z,1)}{\widetilde{y}_{1}^{\,\prime}(z,1)}$
(E.7)
$\displaystyle=\frac{\Gamma(z^{1/2})\Gamma(-z^{1/2})}{\Gamma((1/2)+z^{1/2})\Gamma((1/2)-z^{1/2})},\
z\in\rho(T_{\pi/2,\pi/2,-1/2,-1/2}),$

which has simple poles at

$\displaystyle z_{n}=n^{2},\quad n\in{\mathbb{N}}_{0},$ (E.8)

as expected since the Chebyshev polynomials of the first kind,
$T_{n}(\,\cdot\,)$, satisfy Neumann boundary conditions because of the
relation $($cf. [1, eq. 22.5.31]$)$

$\displaystyle
T_{n}(x)=\frac{n!\pi^{1/2}}{\Gamma(n+(1/2))}P_{n}^{-1/2,-1/2}(x),\quad
n\in{\mathbb{N}}_{0},\;x\in(-1,1).$ (E.9)

The Chebyshev case of the second kind is realized by choosing $\mu=1$ in the
Gegenbauer case, or $\alpha=\beta=1/2$ in the Jacobi case (see, e.g., [1, Ch.
22], [53, Ch. 18], [58, Ch. IV]). Thus, one obtains the differential
expression

$\displaystyle\tau_{1}=-(1-x^{2})^{-1/2}(d/dx)\big{(}(1-x^{2})^{3/2}\big{)}(d/dx),\quad
x\in(-1,1),$ (E.10)

which is in the limit circle case at $x=\pm 1$. Thus by (5.4) one can write

$\displaystyle\begin{split}m_{\gamma,\delta,1/2,1/2}(z)&=\big{[}2^{1/2}\sin(\gamma)(\cos(\delta)\widetilde{y}_{1}(z,1)+\sin(\delta)\widetilde{y}_{1}^{\,\prime}(z,1))\\\
&\quad+\cos(\gamma)(\cos(\delta)\widetilde{y}_{2}(z,1)+\sin(\delta)\widetilde{y}_{2}^{\,\prime}(z,1))\big{]}\\\
&\quad\times\big{[}2^{1/2}\cos(\gamma)(\cos(\delta)\widetilde{y}_{1}(z,1)+\sin(\delta)\widetilde{y}_{1}^{\,\prime}(z,1))\\\
&\quad\quad+\sin(\gamma)(\cos(\delta)\widetilde{y}_{2}(z,1)+\sin(\delta)\widetilde{y}_{2}^{\,\prime}(z,1))\big{]}^{-1},\\\
&\quad\hskip 64.01869pt\gamma,\delta\in[0,\pi),\
z\in\rho(T_{\gamma,\delta,1/2,1/2}),\end{split}$ (E.11)

with

$\displaystyle\begin{split}\widetilde{y}_{1}(z,1)&=\frac{-4\Gamma(3/2)\Gamma(3/2)}{\Gamma(1+(1+z)^{1/2})\Gamma(1-(1+z)^{1/2})}\\\
&=\frac{-\pi}{\Gamma(1+(1+z)^{1/2})\Gamma(1-(1+z)^{1/2})},\\\\[2.84526pt]
\widetilde{y}_{1}^{\,\prime}(z,1)&=\frac{\Gamma(3/2)\Gamma(-1/2)}{\Gamma((1/2)+(1+z)^{1/2})\Gamma((1/2)-(1+z)^{1/2})}\\\
&=\frac{-\pi}{\Gamma((1/2)+(1+z)^{1/2})\Gamma((1/2)-(1+z)^{1/2})},\\\\[8.53581pt]
\widetilde{y}_{2}(z,1)&=\frac{-2^{3/2}\Gamma(3/2)\Gamma(1/2)}{\Gamma((1/2)+(1+z)^{1/2})\Gamma((1/2)-(1+z)^{1/2})}\\\
&=\frac{-2^{1/2}\pi}{\Gamma((1/2)+(1+z)^{1/2})\Gamma((1/2)-(1+z)^{1/2})},\\\\[2.84526pt]
\widetilde{y}_{2}^{\,\prime}(z,1)&=\frac{2^{-1/2}\Gamma(3/2)\Gamma(1/2)}{\Gamma((1+z)^{1/2})\Gamma(-(1+z)^{1/2})}=\frac{2^{-3/2}\pi}{\Gamma((1+z)^{1/2})\Gamma(-(1+z)^{1/2})}.\end{split}$
(E.12)

Here we have used (cf. [1, Eqs. 6.1.8, 6.1.9]) $\Gamma(-1/2)=-2\pi^{1/2}$,
$\Gamma(1/2)=\pi^{1/2}$, $\Gamma(3/2)=2^{-1}\pi^{1/2}$, and the fact that

$\displaystyle\sigma_{1/2,1/2}(z)=2(1+z)^{1/2}.$ (E.13)

######  Example E.2 (The Friedrichs extension).

Considering the Friedrichs extension, one obtains

$\displaystyle m_{0,0,1/2,1/2}(z)$
$\displaystyle=2^{-1/2}\frac{\widetilde{y}_{2}(z,1)}{\widetilde{y}_{1}(z,1)}$
(E.14)
$\displaystyle=\frac{\Gamma(1+(1+z)^{1/2})\Gamma(1-(1+z)^{1/2})}{\Gamma((1/2)+(1+z)^{1/2})\Gamma((1/2)-(1+z)^{1/2})},\
z\in\rho(T_{0,0,1/2,1/2}),$

which has simple poles at

$\displaystyle z_{n}=n(n+2),\quad n\in{\mathbb{N}}_{0},$ (E.15)

as expected since the Chebyshev polynomials of the second kind,
$U_{n}(\,\cdot\,)$, satisfy Friedrichs boundary conditions as they satisfy the
relation $($cf. [1, eq. 22.5.32]$)$

$\displaystyle
U_{n}(x)=\frac{(n+1)!\sqrt{\pi}}{2\Gamma(n+(3/2))}P_{n}^{1/2,1/2}(x),\quad
n\in{\mathbb{N}}_{0},\;x\in(-1,1).$ (E.16)

###  E.3. The Case of Zernike Polynomials

The Zernike polynomials are a sequence of orthogonal polynomials on the unit
disk that play important roles in various branches of optics and optometry
(see, e.g., [46], [50]). The even and odd Zernike polynomials are defined,
respectively, as

$\displaystyle\begin{split}Z_{n}^{\ell}(\rho,\varphi)&=R_{n}^{\ell}(\rho)\cos(\ell\varphi),\\\
Z_{n}^{-\ell}(\rho,\varphi)&=R_{n}^{\ell}(\rho)\sin(\ell\varphi);\quad\ell,n\in{\mathbb{N}}_{0},\
n\geqslant\ell\geqslant 0,\ 0\leqslant\rho\leqslant 1,\end{split}$ (E.17)

with the radial polynomials, $R_{n}^{\ell}(\rho)$, defined by hypergeometric
functions as

$\displaystyle
R_{n}^{\ell}(\rho)=\begin{cases}\frac{(-1)^{(n-\ell)/2}((n+\ell)/2)!}{\ell!((n-\ell)/2)!\rho^{-\ell}}F\big{(}1+[(n+\ell)/2],-(n-\ell)/2;1+\ell;\rho^{2}\big{)},\\\
\hfill n-\ell\text{ even},\\\ 0,\hfill n-\ell\text{ odd}.\end{cases}$ (E.18)

They are related to the Jacobi polynomials by (cf. [46, eq. (14)])

$\displaystyle
R_{n}^{\ell}(\rho)=(-1)^{(n-\ell)/2}\rho^{\ell}P_{(n-\ell)/2}^{\ell,0}\big{(}1-\rho^{2}\big{)},\quad\ell,n\in{\mathbb{N}}_{0},\
n\geqslant\ell\geqslant 0,\ 0\leqslant\rho\leqslant 1.$ (E.19)

This implies that letting $\alpha=\ell\in{\mathbb{N}}_{0},\ \beta=0$ in the
Jacobi expression yields a representation of the radial Zernike differential
expression as

$\displaystyle\tau_{\ell}=-(1-x)^{-\ell}(d/dx)\big{(}(1+x)(1-x)^{\ell+1}\big{)}(d/dx),\quad\ell\in{\mathbb{N}}_{0},\
x\in(-1,1),$ (E.20)

which is always in the limit circle case at $x=-1$, but only in the limit
circle case at $x=1$ when $\ell=0$ and in the limit point case otherwise, that
is, for $\ell\in{\mathbb{N}}$. In particular, when $\ell=0$, this actually
coincides with the Legendre case discussed in Example 5.3. The case when
$\ell\in{\mathbb{N}}$ was covered in Section 6. Replacing $n$ by $(n-\ell)/2$
and $\alpha$ by $\ell$ in (6.9), $n,\ell\in{\mathbb{N}}_{0}$,
$n\geqslant\ell\geqslant 1$, one obtains for the Friedrichs extension

$\displaystyle\begin{split}m_{F,\ell,0}(z)=-2^{-\ell-1}[2\gamma_{E}+\psi([\ell+1+\sigma(z)]/2)+\psi([\ell+1-\sigma(z)]/2)],&\\\
\ell\in{\mathbb{N}}_{0},\ z\in\rho(T_{F,\ell,0}),&\end{split}$ (E.21)

with simple poles at

$\displaystyle
z_{n,\ell}=\frac{n-\ell}{2}\left(\frac{n+\ell}{2}+1\right),\quad
n,\ell\in{\mathbb{N}}_{0},\ n\geqslant\ell\geqslant 1,$ (E.22)

as expected from relation (E.19).

Acknowledgments. We are indebted to Dale Frymark, Conni Liaw, Andrei Martinez-
Finkelshtein, Roger Nichols, and Brian Simanek for very helpful discussions.

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