3.1 Eigenvectors of X and $X^T$

It is directly checked that the EVP (3.18) is satisfied by

(3.20) $$ \begin{align} | z\rangle &= \gamma \sum_{k=-\infty}^{\infty} (z-1)^{-k-a}|\tau, k\rangle, \end{align} $$

(3.21) $$ \begin{align} \widetilde{| z\rangle} &= \tilde{\gamma} \sum_{k=-\infty}^{\infty} (z-1)^{k+\tilde{a}}|\tau, k\rangle, \end{align} $$

with $a, \tilde {a} \in \mathbb {R}$ and where $\gamma , \tilde {\gamma } \in \mathbb {C}$ are normalisation constants. That $| z\rangle $ and $\widetilde {| z'\rangle }$ are orthogonal can be seen as follows. We have

(3.22) $$ \begin{align} \widetilde{\langle z' |}z\rangle= \gamma \tilde{\gamma}\sum_{k, l=-\infty}^{\infty} (z'-1)^{-k-a}(z-1)^{l+\tilde{a}}\langle \tau, k | \tau, l\rangle. \end{align} $$

Now let $z=1+e^{i\phi }$ , $z'=1+e^{i\phi '}$ , so that equation (3.22) becomes

(3.23) $$ \begin{align} \widetilde{\langle z' |}z\rangle=\gamma \tilde{\gamma}e^{i\left(\tilde{a}\phi - a \phi '\right)}\sum_{k=-\infty} ^{\infty} e^{i\left(\phi - \phi '\right)k}. \end{align} $$

We then see that upon imposing

(3.24) $$ \begin{align} a=\tilde{a} + 1, \end{align} $$

we find

(3.25) $$ \begin{align} \widetilde{\langle z' |}z\rangle=-2\pi i \gamma \tilde{\gamma} \delta(z-z') \end{align} $$

with the help of the Fourier series of Dirac’s delta function and of a standard property of this distribution. Since $ \widetilde {\langle z' |}z\rangle $ is manifestly translation-invariant, equation (3.25) is preserved when the variable z lies on the unit circle centred at $z=0$ .

We also have the completeness relation

(3.26) $$ \begin{align} \frac{1}{2\pi i \gamma \tilde{\gamma}} \oint _C dz \widetilde{| z \rangle} \langle z | = 1, \end{align} $$

where the contour C consists in the unit circle infinitesimally deformed so that the singularity at $z=1$ lies inside C. Indeed,

(3.27) $$ \begin{align} \frac{1}{2\pi i \gamma \tilde{\gamma}} \oint _C dz \widetilde{| z \rangle} \langle z | = \frac{1}{2\pi i} \oint _C dz (z-1)^{k-l-a +\tilde{a}} \sum_{k, l = -\infty} ^{\infty}|\tau, k \rangle \langle \tau, l|. \end{align} $$

Again the choice (3.24) for the integration constants a and $\tilde {a}$ consistently ensures that the integral over z becomes $\frac {1}{2\pi i} \oint _C dz (z-1)^{k-l-1} = \delta _{kl}$ and hence

(3.28) $$ \begin{align} \frac{1}{2\pi i \gamma \tilde{\gamma}} \oint _C dz \widetilde{| z \rangle}\langle z |= \sum _{k=-\infty}^{\infty} |\tau, k \rangle \langle \tau, k| =1. \end{align} $$

This will play a key role in the derivation of the orthogonality relations.

3.2 GEVP bases

We shall now obtain the bases $\{| P_n\rangle \}$ and $\{| Q_n\rangle \}$ of $\mathfrak {V}(\tau )$ that satisfy the GEVP (3.16). First we need to determine the set of eigenvalues $\nu $ . From the explicit two-diagonal actions (3.5) and (3.6) of L and M on the basis vectors $\{|\tau , k\rangle \}$ , it is readily seen that the (formal) determinantal condition is

(3.29) $$ \begin{align} \det(M-\nu L) =\prod_{k=-\infty} ^{\infty} \left[ k(k + \alpha +1) - \nu ( k + \alpha + 1)\right] = 0, \end{align} $$

and hence (disregarding the degenerate case where $\alpha \in \mathbb {Z}$ ) that the spectrum consists in the following values:

(3.30) $$ \begin{align} \nu_n = n, \qquad n=0, \pm 1, \pm 2, \dotsc. \end{align} $$

Let

(3.31) $$ \begin{align} | P_n\rangle = \sum_{k=-\infty} ^{\infty} d_n(k) |\tau, k\rangle. \end{align} $$

The generalised eigenvalue equation $M| P_n\rangle = n L | P_n\rangle $ implies the following recurrence relation for the expansion coefficients $d_n(k)$ :

(3.32) $$ \begin{align} (k + 1)(k + \alpha + \beta + 1) d_n(k+1) + (k - n) (k + \alpha + 1) d_n(k) = 0. \end{align} $$

From equation (3.32), it is immediately seen that for $n\geq 0$ ,

(3.33) $$ \begin{align} d_n(k) = 0 \qquad \text{for}\ k> n\ \text{and}\ k \in \mathbb{Z}_-. \end{align} $$

The explicit expression of the nonzero coefficients $d_n(k)$ reads

(3.34) $$ \begin{align} d_n(k)= d_n(0) \frac{(-1)^k(-n)_k(\alpha +1)_k}{k!(\alpha + \beta +1)_k}, \qquad k=0, 1, 2, \dotsc, n. \end{align} $$

Turn now to the adjoint GEVP $M^T | Q_m\rangle = m L^T | Q_m\rangle $ , which imposes on the coefficients $d_m^*(k)$ in

(3.35) $$ \begin{align} | Q_m\rangle = \sum_{k=-\infty} ^{\infty} d_m^*(k) |\tau, k\rangle \end{align} $$

the recurrence relation

(3.36) $$ \begin{align} k (k + \alpha + \beta) d_m^*(k - 1) + (k - m)( k + \alpha + 1) d_m^*(k) = 0. \end{align} $$

Assuming $m\geq 0$ as previously indicated, one immediately notices that equation (3.36) implies

(3.37) $$ \begin{align} d_m^*(k) = 0 \qquad \text{for}\ k<m. \end{align} $$

In view of this fact, let

(3.38) $$ \begin{align} k=l+m, \qquad l=0, 1, \dotsc; \end{align} $$

the relation (3.36) then becomes

(3.39) $$ \begin{align} (m + l) (l + m + \alpha + \beta) d_m^*(m+l-1) + l ( l + m + \alpha + 1) d_m^*(m+l). \end{align} $$

It is found to have for its solution

(3.40) $$ \begin{align} d_m^*(m+l)= \frac{(-1)^l (m + 1)_l (m + \alpha + \beta + 1)_l}{l! (m + \alpha + 2)_l} d_m^*(m), \qquad l=0, 1, \dotsc. \end{align} $$

Apart from the initial condition $d_m^*(m)$ , equation (3.40) fully determines

(3.41) $$ \begin{align} | Q_m\rangle = \sum_{l=0} ^{\infty} d_m^*(m+l) |\tau, m + l\rangle. \end{align} $$

From general linear algebra considerations [Reference Tsujimoto, Vinet and Zhedanov27], [Reference Vinet and Zhedanov29], [Reference Zhedanov30], we know that the vectors $| P_n\rangle $ and $L^T | Q_m \rangle $ are biorthogonal for $n\neq m$ . We have

(3.42) $$ \begin{align} &(\langle P_n | M) | Q_m\rangle =n (\langle P_n | L) | Q_m\rangle \nonumber \\ &=\left\langle P_n | \left(M^T | Q_m\right\rangle\right)= m\left\langle P_m | \left(L^T | Q_m\right\rangle\right)= m (\langle P_n | L) | Q_m\rangle. \end{align} $$

It follows that

(3.43) $$ \begin{align} (n-m)(\langle P_n | L) | Q_m\rangle = (n-m)\left\langle P_n | \left(L^T | Q_m\right\rangle\right) = 0, \end{align} $$

which implies the asserted biorthogonality if $m \neq n$ . Since the derivation we shall provide of the biorthogonality of the Askey polynomials will rest on this property obtained formally, we shall next directly verify that it holds in the case at hand and determine the norm.

From the observations already made, we see that

(3.44) $$ \begin{align} \left\langle P_n | L^T | Q_m \right\rangle &= \sum_{k=-\infty}^n \sum_{l=0}^{\infty} d_n(k) d_m^*(l+m) \left\langle \tau, k | L^T | \tau, l+m\right\rangle \nonumber \\ &=\sum_{k=-\infty}^n \sum_{l=0}^{\infty} d_n(k) d_m^*(l+m)(m + l + \alpha + 1) \delta _{k, l+m}. \end{align} $$

We readily find that

(3.45) $$ \begin{align} \left\langle P_n | L^T | Q_m \right\rangle = 0 \qquad \text{if}\ m>n. \end{align} $$

It remains to consider the situation when $m \leq n$ . Substituting into equation (3.44) expressions (3.34) and (3.40) for $d_n(k)$ and $d_m^*(m+l)$ , using a few properties of the Pochhammer symbols such as $x(x+1)_{l-1}=(x)_l$ and $(x)_{m+l}=(x)_m(x+m)_l$ , and performing one of the sums, we arrive at

(3.46) $$ \begin{align} \left\langle P_n | L^T | Q_m \right\rangle = d_n(0) d_m^*(m) (-1)^m \frac{(-n)_m (\alpha + 1)_{m+1} }{m!(\alpha + \beta + 1)_m } \sum_{l=0} ^{n-m} \frac{(-n+m)_l}{l!}. \end{align} $$

We then recall the binomial formula

(3.47) $$ \begin{align} (1-x)^{\xi} = \sum _{k=0} ^{\infty} \frac{(-\xi)_k}{k!} x^k \end{align} $$

to conclude that

(3.48) $$ \begin{align} \left\langle P_n | L^T | Q_m \right\rangle = N_n \delta _{m,n}, \end{align} $$

with

(3.49) $$ \begin{align} N_n = d_n(0) d_n^*(n) \frac{( \alpha + 1)_{n+1}}{(\alpha + \beta + 1)_n}. \end{align} $$

3.3 Askey polynomials and their biorthogonal partners

Let us now identify some of the special functions that arise from this representation theoretic setting. In light of the completeness relation (3.26) and the orthogonality relation (3.48), we see that $\widetilde {\langle z |} P_n \rangle $ and $\left \langle z | L^T | Q_n \right \rangle $ provide two families of biorthogonal functions on the unit circle, since

(3.50) $$ \begin{align} \frac{1}{2\pi i \gamma \tilde{\gamma}} \oint _{\lvert z\rvert=1} dz \langle P_n \widetilde{| z\rangle}\left\langle z | L^T | Q_m \right\rangle = \left\langle P_n | L^T | Q_m\right\rangle = N_n \delta_{m,n}. \end{align} $$

These are explicitly obtained in the following.

3.3.1 The overlaps $\widetilde {\langle z |} P_n \rangle $

From the expansions (3.21) and (3.31) of $ \widetilde {| z\rangle }$ and $| P_n \rangle $ over the orthonormal basis vectors $|\tau , k \rangle $ , we have

(3.51) $$ \begin{align} \widetilde{\langle z |}P_n \rangle = \tilde{\gamma} \sum_{l=-\infty} ^n (z-1)^{l+\tilde{a}} d_n(l). \end{align} $$

Upon inserting equation (3.34) for $d_n(l)$ , we observe that $\widetilde {\langle z |}P_n \rangle $ is the ${_2}F_1$ polynomial

(3.52) $$ \begin{align} \widetilde{\langle z |}P_n \rangle = \tilde{\gamma}d_n(0) (z - 1)^{\tilde{a}} {}_{2}F_{1}\left({-n, \alpha + 1 \atop \alpha + \beta + 1}; 1 - z \right ). \end{align} $$

The Askey polynomials are then recognised with the help of the Pfaff formula [Reference Andrews, Askey and Roy1]:

(3.53) $$ \begin{align} {_2}F_1\left({-n, b \atop c}; z \right ) = \frac{(c-b)_n}{(c)_n}{_2}F_1\left({-n, b \atop -n + b + 1 - c}; 1 - z \right ). \end{align} $$

We find

(3.54) $$ \begin{align} \widetilde{\langle z |}P_n \rangle = \tilde{\gamma} d_n(0) \frac{(\alpha +1)_n}{(\alpha + \beta +1)_n}(z - 1)^{\tilde{a}} P_n(z; \alpha, \beta), \end{align} $$

where the polynomials $P_n(z; \alpha , \beta )$ are as defined in equation (1.1).

Proposition 3.1. The Askey polynomials $P_n(z; \alpha , \beta )$ have a natural interpretation in the representation theory of the meta-Jacobi algebra. They occur according to equation (3.54) as the overlaps between two bases of the module $\mathfrak {V}\left (\tau =\frac {1}{2} (\alpha + \beta + 1)\right )$ satisfying, respectively, equations defined in terms of the generators $X, L, M$ of $m\mathfrak {J}$ . The first basis consists in the eigenvectors of $X^T$ (the transpose of X) and the second is formed by the vectors solving the GEVP defined by L and M.

One may remark that the reducible module allows us to posit the eigenvalue problem for X.

3.3.2 The overlaps $\left \langle z |L^T |Q_m \right \rangle $

The biorthogonal partners to the Askey polynomials are obtained in a similar fashion. From equations (3.8), (3.40) and (3.41) we have

(3.55) $$ \begin{align} L^T |Q_m\rangle = d_m^*(m) (m+\alpha +1)\sum _{l=0}^{\infty} \frac{(-1)^l (m+1)_l (m+\alpha + \beta +1)_l}{l! (m+ \alpha + 1)_l} |\tau , l+m\rangle. \end{align} $$

Combining with equation (3.20) and using the orthonormality of the basis vectors $|\tau , k\rangle $ , we find

(3.56) $$ \begin{align} \left\langle z | L^{T} | Q_{m} \right\rangle = \gamma d_m^*(m) (m + \alpha + 1) (z-1)^{-m-a} {}_{2}F_{1}\left({m+1, m+\alpha + \beta + 1 \atop m +\alpha + 1}; \frac{1}{1-z} \right ). \end{align} $$

We may now use the fact that any three solutions of the hypergeometric equation are related by linear relations and call upon transformation formulas of ${_2}F_1$ series under homographic transformations to make the biorthogonal partners of the Askey polynomials appear in this overlap. Indeed, following the steps described in Appendix A, we arrive at the following expression:

(3.57) $$ \begin{align} \left\langle z| L^T| Q_m\right\rangle & = \gamma d_m^*(m) (m + \alpha + 1) (z - 1)^{1-a} \nonumber \\ & \quad \times \Bigg[ \frac{\Gamma (m + \alpha +1 )\Gamma (\beta + 1)}{\Gamma (m + \beta + 2)\Gamma (\alpha)}{_2}F_1\left( {m + 1, 1 - \alpha \atop m + \beta + 2}; z \right ) \nonumber \\ &\quad - \frac{\Gamma (m + \alpha + 1) \Gamma (m + \beta + 1)}{m! \Gamma (m + \alpha + \beta + 1)} (-z)^{-1-\beta} (1 - z)^{\alpha + \beta} Q_m \left(\frac{1}{z}, \alpha, \beta\right) \Bigg], \end{align} $$

where $Q_m(z)$ is defined as in equation (1.2).

Remark 3.4. Note that the first term in this expression for $\left \langle z| L^T| Q_m\right \rangle $ is a power series, while the second one, which contains the polynomial $Q_n$ in the variable $\frac {1}{z}$ , has the transcendental factor $z^{-\beta }$ . Restrictions on z could be imposed with regard to the convergence of the power series, or it could be treated formally, as the focus is on the polynomial partner $Q_n(z, \alpha , \beta )$ .

The following proposition summarises the main results of this subsection.

Proposition 3.2. The biorthogonal partners $Q_n(z, \alpha , \beta )$ of the Askey polynomials $P_n(z; \alpha , \beta )$ arise in the representation theory of the meta-Jacobi algebra in the overlaps (see equation (3.57)), between the eigenbasis vectors of the generator X and the basis vectors that obey the GEVP defined by the operators $M^T$ and $L^T$ .

3.3.3 Biorthogonality relation

The interpretation of the Askey polynomials in the framework of the meta-Jacobi algebra leads to a natural derivation of their biorthogonality. Recall equation (3.50). First observe that in multiplying the expressions of the overlaps $\widetilde {\langle z|}P_n \rangle $ and $\left \langle z | L^T |Q_m\right \rangle $ , as they are given by formulas (3.54) and (3.57), the factor $(z-1)^{1-a + \tilde {a}}$ reduces to $1$ because of equation (3.24). Furthermore, one observes that the product of the first term in equation (3.57) – a power series – with the polynomial $P_n(z, \alpha , \beta )$ will give a vanishing contribution when integrated over the circle $\lvert z\rvert =1$ . Equation (3.50) thus yields

(3.58) $$ \begin{align} d_n(0) d_m^*(m) &\frac{(m + \alpha + 1) (\alpha + 1)_n}{(\alpha + \beta + 1)_n} \frac{\Gamma (m + \alpha + 1) \Gamma (m + \beta + 1)}{m! \Gamma (m + \alpha + \beta + 1)} \nonumber\\ &\times \frac{-1}{2 \pi i} \oint_{\lvert z\rvert=1} dz (-z)^{-1 - \beta} (1-z)^{\alpha + \beta}P_n(z, \alpha, \beta) Q_m \left(\frac{1}{z}, \alpha, \beta\right) = N_n \delta_{m,n}. \end{align} $$

Recalling that $N_n$ is given by equation (3.49), we thus recover precisely the biorthogonality relation (1.3).

3.4 Eigenbases of M and $M^T$

We now undertake to show that the Jacobi polynomials can be described within the same algebraic framework. We already noted in Proposition 2.2 that the elements X and M generate the Jacobi algebra $\mathfrak {J}$ , which is thus embedded in $m\mathfrak {J}$ . We therefore expect to see the Jacobi polynomials occur in the overlaps between the eigenvectors of X, $X^T$ and $M^T$ , M, respectively. We shall hence first determine the eigenbases of $\mathfrak {V}(\tau )$ associated to M and $M^T$ .

From the two-diagonal action (3.6) of M, we see that the spectrum $\{\mu _n \}$ of this operator is of the form

(3.59) $$ \begin{align} \mu_n = n(n + \alpha + 1). \end{align} $$

Consider the EVPs (3.19). Set

(3.60) $$ \begin{align} |J_n\rangle = \sum_{k=-\infty}^{\infty} f_n (k) |\tau, k\rangle. \end{align} $$

The eigenvalue equation $M |J_n\rangle = n(n + \alpha + 1) |J_n\rangle $ yields the following two-term recurrence relation for the coefficients $f_n(k)$ :

(3.61) $$ \begin{align} (k + 1 + \alpha + \beta) f_n (k + 1) + \left[ k(k + \alpha + 1) - n(n + \alpha + 1) \right] f_n(k) = 0, \end{align} $$

which can be rewritten as

(3.62) $$ \begin{align} (k + 1) (k + 1 + \alpha + \beta) f_n (k + 1) + (k - n) (k + n + \alpha + 1) f_n (k) = 0. \end{align} $$

Here again, we shall focus on the case $n \geq 0$ . This equation is then found to imply that

(3.63) $$ \begin{align} f_n(k) = 0 \qquad \text{for}\ k>n\ \text{and}\ k \in \mathbb{Z}_- \end{align} $$

and is solved by

(3.64) $$ \begin{align} f_n(k) = f_n (0) (-1)^k\frac{(-n)_k (n + \alpha + 1)_k}{k! (\alpha + \beta + 1)_k}, \qquad k = 0, 1, \dotsc, n. \end{align} $$

Similarly, let

(3.65) $$ \begin{align} \widetilde{|J_n\rangle} = \sum_{k=-\infty}^{\infty} \tilde{f}_n (k) |\tau, k\rangle. \end{align} $$

The EVP $M^T \widetilde {|J_n \rangle } = n (n + \alpha + 1) \widetilde {|J_n \rangle } $ is readily seen to give

(3.66) $$ \begin{align} k ( k + \alpha + \beta) \tilde{f}_n (k - 1) + (k - n) (k + n + \alpha + 1) \tilde{f}_n (k)= 0. \end{align} $$

In this instance, for $m \geq 0$ , we observe that

(3.67) $$ \begin{align} \tilde{f}_n (k) = 0 \qquad \text{when}\ k<n. \end{align} $$

We set $k = n + l , l = 0, 1, 2, \dotsc ,$ and convert equation (3.66) into

(3.68) $$ \begin{align} (n + l) (n + l + \alpha + \beta) \tilde{f}_n (n + l - 1) + l ( l + 2n + \alpha + 1) \tilde{f}_n (n + l) = 0 \end{align} $$

to find

(3.69) $$ \begin{align} \tilde{f}_n (n + l) = \tilde{f}_n (n) (-1)^l \frac{(n + 1)_l (n +1 + \alpha + \beta)_l}{l! (2n + \alpha + 2)_l}. \end{align} $$

We can now verify that $|J_n\rangle $ and $\widetilde {|J_m \rangle }$ are orthogonal when $m \neq n$ . This proceeds in a way similar to the computation of $\left \langle P_n | L^T | Q_m \right \rangle $ carried out before. Clearly, $\langle J_n \widetilde {| J_m \rangle } = 0$ if $m> n$ . If $m \leq n$ , after some algebraic simplifications we see that

(3.70) $$ \begin{align} \langle J_n \widetilde{| J_m \rangle} = f_n(0) \tilde{f}_m(n) (-1)^m\frac{(-n)_m (n + \alpha + 1)_m}{m! (\alpha + \beta + 1)_m } {_2}F_1\left({m - n, n + m + \alpha + 1 \atop 2m + \alpha + 2}; 1 \right ). \end{align} $$

From the Vandermonde formula [Reference Gasper and Rahman8]

(3.71) $$ \begin{align} {_2}F_1\left({-n, b \atop c}; 1 \right ) = \frac{(c-b)_n}{(c)_n}, \end{align} $$

we may then conclude that because of the factor $(m - n +1)_{n-m}$ that appears, $\langle J_n \widetilde {| J_m \rangle } = 0$ unless $n=m$ , in which case

(3.72) $$ \begin{align} \langle J_n \widetilde{| J_m \rangle} = \mathcal{N}_n \delta_{m,n}, \end{align} $$

with

(3.73) $$ \begin{align} \mathcal{N}_n = f_n(0) \tilde{f}_n(n) \frac{(n + \alpha + 1)_n}{(\alpha + \beta + 1)_n}. \end{align} $$

3.5 Jacobi polynomials

We will now observe how the Jacobi polynomials emerge in this framework and indicate how this allows for another derivation of their orthogonality relation.

3.5.1 The overlaps

Let us now look at the overlaps $\widetilde {\langle z |} J_n \rangle $ and $\langle z \widetilde {| J_m \rangle }$ . From equations (3.21), (3.60) and (3.64) we obtain

(3.74) $$ \begin{align} \widetilde{\langle z |} J_n \rangle = \tilde{\gamma} f_n(0) (z - 1 )^{\tilde{a}} {}_{2}F_1\left({-n, n + \alpha + 1 \atop \alpha + \beta + 1}; 1-z \right ). \end{align} $$

Using equation (A.1) – or equivalently equation (3.53) – we arrive at

(3.75) $$ \begin{align} \widetilde{\langle z |} J_n \rangle = \tilde{\gamma} (z - 1)^{\tilde{a}} \frac{(-1)^n \Gamma (1 + \alpha + \beta) \Gamma (\beta) \Gamma (2n + \alpha + 1) \Gamma (1 - \beta)}{\Gamma (-n + \beta) \Gamma (n + \alpha + \beta + 1)\Gamma (n + \alpha + 1) \Gamma (n + 1 - \beta)} \hat{P}_n^{\left(\alpha, \beta\right)}(z), \end{align} $$

where $\hat {P}_n^{\left (\alpha , \beta \right )}(z)$ are the Jacobi polynomials defined in equation (1.4) extended to the complex plane.

The second overlap is recovered from equations (3.20), (3.65) and (3.69). We find

(3.76) $$ \begin{align} \langle z \widetilde{| J_m \rangle} = \gamma {\tilde{f}}_m(m) {(z-1)^{-m - a}}{} {_2}{F_1}\left({m + 1, m + \alpha + \beta + 1 \atop 2m +\alpha + 2}; \frac{1}{1-z} \right ). \end{align} $$

At this point, by performing the transformations described in Appendix B that make use of identities involving gamma functions and solutions of the hypergeometric equation, the following formula is discovered:

(3.77) $$ \begin{align} \langle z \widetilde{| J_m \rangle} & = (-1)^{m + 1}\gamma {\tilde{f}}_m(m) (z-1)^{1 - a}\nonumber \\ & \quad \times\Bigg[ \frac{\Gamma (2m + \alpha +2 )\Gamma (-\beta)}{\Gamma (m + \alpha + 1)\Gamma (m -\beta +1)} {}{_2}F_1\left( {m + 1, -m - \alpha \atop 1 + \beta}; z \right ) \nonumber \\ & \quad + \frac{(-1)^m \Gamma (2m + \alpha + 2) \Gamma (\beta) \Gamma (2m + \alpha + 1)\Gamma (1 - \beta)}{m! \Gamma (m + \alpha + \beta + 1) \Gamma (m + \alpha + 1) \Gamma (m + 1 - \beta)} (-z)^{-\beta} (1 - z)^{\alpha + \beta} \hat{P}_m^{\left(\alpha, \beta\right)}(z) \Bigg]. \end{align} $$

Remark 3.5. As already encountered in expression (3.57) of $\langle z| L^T| Q_m\rangle $ , we see that the first term in equation (3.77) is a power series, and the second, which involves the Jacobi polynomials, contains the transcendental term $z^{-\beta }$ .

Proposition 3.3. The Jacobi polynomials $\hat {P}_m^{\left (\alpha , \beta \right )}(z)$ over $\mathbb {C}$ also arise in the context of the meta-Jacobi algebra $m\mathfrak {J}$ . They occur as per equations (3.75) and (3.77) in two overlaps between eigenbases of the module $\mathfrak {V}\left (\frac {1}{2} (\alpha + \beta + 1)\right )$ : on the one hand between the eigenstates of M and $X^T$ and on the other hand between those of $M^T$ and X.

3.5.2 Orthogonality

This interpretation of the Jacobi polynomials in the framework of the algebra $m\mathfrak {J}$ entails a derivation of their orthogonality. Owing to the completeness relation (3.28), we have

(3.78) $$ \begin{align} \frac{1}{2 \pi i \gamma \tilde{\gamma}} \oint _{C_{\lvert z\rvert=1} }\langle J_n \widetilde{| z \rangle} \langle z \widetilde{| J_m \rangle} = \langle J_n \widetilde{| J_m \rangle} = \mathcal{N}_n \delta_{m,n}, \end{align} $$

where $\mathcal {N}_n$ is given by equation (3.73). When substituting expressions (3.75) and (3.77) for $\widetilde {\langle z |} J_n \rangle $ and $\langle z \widetilde {| J_m \rangle }$ , we first observe anew that the resulting factor $(z-1)^{1-a + \tilde {a}} = 1$ , since $1 - a + \tilde {a} = 0$ . Then we note that the product of $\widetilde {\langle z |} J_n \rangle $ with the first term of equation (3.77) is a power series that will integrate to $0$ over the unit circle. Taking into account formula (3.73) for $\mathcal {N}_n$ and after some simplifications, equation (3.78) thus amounts to

(3.79) $$ \begin{align} - \frac{1}{2 \pi i} \left(\frac{\pi}{\sin \pi \beta}\right) \oint _{C_{\lvert z\rvert=1}} dz (-z)^{-\beta} (1 - z) ^{\alpha + \beta} \hat{P}_n^{\left(\alpha, \beta\right)}(z)\hat{P}_m^{\left(\alpha, \beta\right)}(z) = h_n \delta _{m,n}, \end{align} $$

with $h_n$ given in equation (1.6). In obtaining equation (3.79) we have used the identity $\Gamma (x) \Gamma (1-x) = \frac {\pi }{\sin \pi x}$ , and in particular

(3.80) $$ \begin{align} \Gamma (-n + \beta) \Gamma (n + 1 - \beta) = \frac{\pi}{\sin \pi (-n + \beta)} = (-1)^n \dfrac{\pi}{\sin \pi \beta}. \end{align} $$

Finally, the orthogonality of the Jacobi polynomials on the interval $[0, 1]$ is recovered by using the contour depicted in Figure 1 and computations carried out in [Reference Hendriksen and van Rossum14]. Schematically the contour $\Xi = C_{\lvert z\rvert =1} + [1, 0] + C_{\epsilon } + [0, 1]$ , it is composed of the unit circle (short of crossing the branch cut), the segment from $x=1$ to $x=0$ below the branch cut, a circle of radius $\epsilon $ around $z=0$ and the segment from $x=0$ to $x=1$ above the branch cut. Consider the integral in equation (3.79) with the contour $C_{\lvert z\rvert =1}$ replaced by the contour $\Xi $ of Figure 1. Since no singularities are enclosed by $\Xi $ , that integral is equal to $0$ .

Figure 1 The contour $\Xi $ .

If we restrict $\beta $ to be smaller than $1$ – that is, if we take $\beta < 1$ as in the standard definition of the Jacobi polynomials – it is readily seen that

(3.81) $$ \begin{align} \lim_{\epsilon \to 0} \frac{1}{2 \pi i} \oint _{C_{\epsilon}}dz (-z)^{-\beta} (1 - z) ^{\alpha + \beta} \hat{P}_n^{\left(\alpha, \beta\right)}(z)\hat{P}_m^{\left(\alpha, \beta\right)}(z) = 0, \qquad \text{for}\ \beta < 1. \end{align} $$

It follows that the integral over $C_{\lvert z\rvert =1}$ must be the negative of the sum of the integrals over and above the real axis. Hence, recalling the choice of branch $(-z)^{-\beta } = \lvert z\rvert ^{-\beta }$ when $\arg z = \pi $ , we have

(3.82) $$ \begin{align} &- \frac{1}{2 \pi i} \left(\frac{\pi}{\sin \pi \beta}\right) \oint _{C_{\lvert z\rvert=1}} dz (-z)^{-\beta} (1 - z) ^{\alpha + \beta} \hat{P}_n^{\left(\alpha, \beta\right)}(z)\hat{P}_m^{\left(\alpha, \beta\right)}(z) = \nonumber \\ & \frac{1}{2i} \left(\frac{1}{\sin \pi \beta}\right) e^{i\pi \beta} \left(1 - e^{-2 \pi i \beta}\right) \int_0^1 x^{-\beta} (1 - x) ^{\alpha + \beta} \hat{P}_n^{\left(\alpha, \beta\right)}(x)\hat{P}_m^{\left(\alpha, \beta\right)}(x) dx. \end{align} $$

The factors before the integral sign in the last expression cancel, and this gives the orthogonality relation (1.5) of the Jacobi polynomials in view of equation (3.79).

4.1 Action of L and $R=XL$ in the basis $\{|P_n\rangle \}$

We shall now show that the generator L and the product $XL$ act in a two-diagonal fashion in the basis $\{|P_n\rangle , n=0, 1, \dots \}$ . We have

(4.6) $$ \begin{align} L |P_n(\alpha, \beta)\rangle = \frac{(\alpha + \beta + 1)_n}{(\alpha + 1)_n} \sum_{k=0}^n (-1)^k \frac{(-n)_k (\alpha + 1)_k}{k! (\alpha + \beta + 1)_k} L|\tau, k\rangle. \end{align} $$

From equation (3.5) and the identity

(4.7) $$ \begin{align} (-n)_k (k + \alpha + 1) &= (-n)_k [n + \alpha + 1 + (-n + k) ]\nonumber \\ &= (n + \alpha + 1) (-n)_k - n(-n + 1)_k, \end{align} $$

we see that

(4.8) $$ \begin{align} L |P_n(\alpha, \beta)\rangle = (n + \alpha + 1) |P_n(\alpha, \beta)\rangle - \frac{n (n + \alpha + \beta)}{(n + \alpha)} |P_{n-1} (\alpha, \beta) \rangle. \end{align} $$

Alternatively, using

(4.9) $$ \begin{align} (k + \alpha + 1) (\alpha + 1)_k = (\alpha + 1)_{k+1} = ((\alpha +1) + 1)_k (\alpha + 1) \end{align} $$

we note that L also has the effect of shifting the parameters:

(4.10) $$ \begin{align} L |P_n(\alpha, \beta)\rangle = ( n + \alpha + 1) |P_n(\alpha + 1, \beta -1)\rangle. \end{align} $$

Consider now the action of the operator $R=XL$ . Knowing that L acts diagonally as per equation (3.5) on the basis vectors $|\tau , k\rangle $ , and according to equation (3.7), we have

(4.11) $$ \begin{align} R |P_n(\alpha, \beta)\rangle = \frac{(\alpha + \beta + 1)_n}{(\alpha + 1)_n} \sum_{k=0}^n (-1)^k \frac{(-n)_k (\alpha + 1)_k (k + \alpha + 1)}{k! (\alpha + \beta + 1)_k} [|\tau, k\rangle + |\tau, k + 1\rangle]. \end{align} $$

Collecting the factors of the vectors $|\tau , k\rangle $ , $k=0, \dotsc , n+1$ , we find

(4.12) $$ \begin{align} R |P_n(\alpha, \beta)\rangle &= \frac{(\alpha + \beta + 1)_n}{(\alpha + 1)_n} \Bigg[ (\alpha + 1) |\tau, 0\rangle \nonumber\\ &\quad + \sum_{k=1}^n \frac{(-1)^k (\alpha + 1)_k}{k! (\alpha + \beta + 1)_k} [ (-n)_k (k + \alpha + 1) - (-n)_{k-1} k(k + \alpha + \beta) ] |\tau,k \rangle \nonumber\\ & \quad + \frac{(\alpha + 1)_{n+1}}{(\alpha + \beta + 1)_n} |\tau, n + 1 \rangle \Bigg]. \end{align} $$

The two relations

(4.13) $$ \begin{align} &k (-n)_{k-1} = (-n)_k - (-n-1)_k, \end{align} $$

(4.14) $$ \begin{align} &(-n)_k (-n-1) - (-n - 1)_k (n + 1 -k)=0 \end{align} $$

come in handy in deriving the following identity:

(4.15) $$ \begin{align} &(-n)_k (k + \alpha + 1) - k (-n)_{k-1} (k + \alpha + \beta) \nonumber \\ &\quad = (-n)_k (k + \alpha + 1) - [ (-n)_k - (-n-1)_k ] (k + \alpha + \beta) \nonumber \\ & \quad = (-n)_k (1 - \beta) + (-n - 1)_k (k + \alpha + \beta) \nonumber \\ &\quad = (-n)_k (-n - \beta) + (-n-1)_k ( n + \alpha + \beta + 1). \end{align} $$

Clearly, equation (4.13) has been used in getting the second line and equation (4.14) has been added to the third line to obtain the end result. Upon inserting this relation (4.15) into equation (4.12), we recognise easily that R is a two-diagonal raising operator:

(4.16) $$ \begin{align} R |P_n(\alpha, \beta)\rangle = (n + \alpha + 1) |P_{n+1}(\alpha, \beta)\rangle - (\beta + n) |P_n(\alpha, \beta)\rangle. \end{align} $$

In the following we shall also consider the element

(4.17) $$ \begin{align} \tilde{R} = XM. \end{align} $$

4.2 A differential realisation

A differential model of the meta-Jacobi algebra is directly obtained. With the choices in equation (4.2), we have

(4.18) $$ \begin{align} \widetilde{\langle z |} \tau, k\rangle \equiv f(z, k) = (z - 1)^k. \end{align} $$

We can dually define an operator acting on the variable z as follows:

(4.19) $$ \begin{align} \mathcal{O}_z \widetilde{\langle z |} \tau, k\rangle = \widetilde{\langle z |} O |\tau, k\rangle = \mathrm{O}_k f(z, k), \end{align} $$

where $\mathcal {O}_z$ corresponds to the operator O acting on the module $\mathfrak {V}\left (\frac {1}{2} (\alpha + \beta + 1)\right )$ and $\mathrm {O}_k$ as in Remark 3.2, acts on the components $f(z,k)$ of the vector $\widetilde {|z\rangle }$ . With $O = L, M, X$ we find the following:

Proposition 4.1. The differential operators $\mathcal {L}$ , $\mathcal {M}$ and $\mathcal {X}$ provide a realisation of the commutation relations (2.1), (2.2) and (2.3) of the meta-Jacobi algebra:

(4.20) $$ \begin{align}\begin{aligned} \mathcal{L} = (z-1) \partial_z + (\alpha+1) \mathcal{I}, \end{aligned}\end{align} $$

(4.21) $$ \begin{align}\begin{aligned} \mathcal{M}= z(z-1) \partial_z^2 + \left[ (\alpha+2)z + \beta-1 \right] \partial_z; \end{aligned}\end{align} $$

(4.22) $$ \begin{align}\begin{aligned} \mathcal{X}=z. \end{aligned}\end{align} $$

It follows that $R=XL$ and $\tilde {R}=XM$ are realised by

(4.23) $$ \begin{align}\begin{aligned} \mathcal{R} = z(z-1)\partial _z + (\alpha + 1) z, \end{aligned}\end{align} $$

(4.24) $$ \begin{align}\tilde{\mathcal{R}} = z^2 (z-1) \partial _z ^2 + z[(\alpha+2)z+\beta-1 ] \partial_z. \end{align} $$

Remark 4.2. One may also take X acting on the left on $\widetilde {\langle z |}$ and giving the eigenvalue z so that

(4.25) $$ \begin{align} \mathcal{R} \widetilde{\langle z |}P_n(\alpha, \beta)\rangle = \widetilde{\langle z |}XL |P_n(\alpha, \beta)\rangle= z \widetilde{\langle z |}L |P_n(\alpha, \beta)\rangle = z\mathcal{L} \widetilde{\langle z |}P_n(\alpha, \beta)\rangle, \end{align} $$

and similarly for $\tilde {R} = XM$ .

Remark 4.5. This differential model for $m\mathfrak {J}$ can also be retrieved by using the Barut–Ghirardello realisation of $\mathfrak {su}(1,1)$ ,

(4.26) $$ \begin{align} J_0&=(z-1)\frac{d}{dz} + \tau, \nonumber \\ J_+&=(z-1),\nonumber \\ J_-&=(z-1)\frac{d^2}{dz^2}+2\tau\frac{d}{dz}, \qquad \tau = \frac{1}{2}(\alpha + \beta + 1), \end{align} $$

in formulas (2.10), (2.11) and (2.12), giving L, M and X in terms of the $\mathfrak {su}(1,1)$ generators. Note that the variable z is here translated by $1$ with respect to the usual Barut–Ghirardello formulas.

Given expression (4.4) of $P_n(z; \alpha , \beta )$ as $\widetilde {\langle z |}P_n(\alpha , \beta ) \rangle $ , in view of the actions (4.8) and (4.16) of L and R on the vectors $|P_n\rangle $ , of Remark 4.1 and of the realisations of these operators already given (equations (4.20)–(4.24)), we have the following:

Proposition 4.2. The biorthogonal Askey polynomials $P_n(z; \alpha , \beta )$ on the unit circle satisfy the following differential identities:

(4.27) $$ \begin{align} \mathcal{L} P_n(z; \alpha, \beta) &= (n+\alpha+1) P_n(z; \alpha, \beta) - \frac{n (\alpha+\beta+n)}{\alpha+n} P_{n-1}(z; \alpha, \beta), \end{align} $$

(4.28) $$ \begin{align} \mathcal{M} P_n(z; \alpha, \beta) &= n(n+\alpha+1) P_n(z; \alpha, \beta) - \frac{n^2 (\alpha+\beta+n)}{\alpha+n} P_{n-1}(z; \alpha, \beta), \end{align} $$

(4.29) $$ \begin{align} \mathcal{R} P_n(z; \alpha, \beta) &= (n+\alpha+1) P_{n+1}(z; \alpha, \beta) - (\beta+n) P_n(z; \alpha, \beta), \end{align} $$

(4.30) $$ \begin{align} \tilde{\mathcal{R}} P_n(z; \alpha, \beta) &= n(n+\alpha+1) P_{n + 1}(z; \alpha, \beta) - n(\beta+n) P_n(z; \alpha, \beta). \end{align} $$

4.3 Bispectrality

The bispectral equations of the Askey polynomials can now easily be identified and interpreted in terms of generalised eigenvalue problems.

4.3.1 The differential equation

The GEVP $M |P_n(\alpha , \beta )\rangle = n L |P_n(\alpha , \beta )\rangle $ translates after projection on $\widetilde {\langle z |}$ into the second-order differential equation

(4.31) $$ \begin{align} \mathcal{M} P_n(z; \alpha, \beta) = n \mathcal{L} P_n(z; \alpha, \beta), \end{align} $$

with eigenvalue n and where the operators $\mathcal {M}$ and $\mathcal {L}$ are respectively given by equations (4.28) and (4.27).

4.3.2 The recurrence relation

The recurrence relation is obtained by considering the GEVP

(4.32) $$ \begin{align} \mathcal{R} P_n(z; \alpha, \beta) = z \mathcal{L} P_n(z; \alpha, \beta), \end{align} $$

which is satisfied by construction (see equation (4.25) in Remark 4.2). Expressing in equation (4.32) the two-diagonal actions (4.27) and (4.29) of $\mathcal {L}$ and $\mathcal {R}$ , one arrives at the recurrence relation

(4.33) $$ \begin{align} P_{n+1}(x) + b_n P_n(x) = x \left(P_n(x) + g_n P_{n-1}(x) \right), \end{align} $$

where

(4.34) $$ \begin{align} b_n = -\frac{\beta+n}{\alpha+n+1}, \qquad g_n = -\frac{n(n+\alpha+\beta)}{(\alpha+n)(\alpha+n+1)}. \end{align} $$

This recurrence relation was obtained by Hendriksen and van Rossum in [Reference Hendriksen and van Rossum14]. It was derived in [Reference Grünbaum, Vinet and Zhedanov12] by considering linear pencils in $\mathfrak {sl}_2$ . It is also constructed through a gluing procedure by Kim and Stanton in their recent study of $R_I$ polynomials [Reference Kim and Stanton17].

Remark 4.6. It is manifest from this recurrence relation of $R_I$ -type [Reference Ismail and Masson16] that z (resp., X) is a lower Hessenberg matrix on the space of Askey polynomials (resp., in the basis $\{|P_n(\alpha , \beta )\rangle \}$ ). This feature of the representation theory of $m\mathfrak {J}$ was also observed in a study of the meta-Hahn algebra [Reference Vinet and Zhedanov29].

Proposition 4.3. The biorthogonal Askey polynomials defined on the unit circle are bispectral. They satisfy the differential equation (4.31) and the recurrence relation of $R_I$ -type (4.33) with coefficients (4.34). Both spectral equations are of GEVP type.

4.4 Contiguity relations

Some contiguity relations for the Askey polynomials also arise naturally in the meta-Jacobi algebra framework. Indeed, we already observed in equation (4.10) that the generator L has the effect of performing the shifts $\alpha \rightarrow \alpha + 1$ , $\beta \rightarrow \beta - 1$ when acting on $|P_n (\alpha , \beta ) \rangle $ . That M has a similar effect follows from the fact that $M=nL$ in the GEVP basis $|P_n (\alpha , \beta ) \rangle $ . This translates into the following for the polynomials $P_n(\alpha , \beta ) = \widetilde {\langle z |} P_n (\alpha , \beta )\rangle $ :

Proposition 4.4. The Askey polynomials $P_n(\alpha , \beta )$ verify the following contiguity equations:

(4.35) $$ \begin{align} \mathcal{L} P_n(z;\alpha,\beta) &= (\alpha+n+1) P_n(z; \alpha+1, \beta-1), \end{align} $$

(4.36) $$ \begin{align} \mathcal{M} P_n(z;\alpha,\beta) &= n(\alpha+n+1) P_n(z; \alpha+1, \beta-1), \end{align} $$

where $\mathcal {L}$ and $\mathcal {M}$ are the differential operators (4.20) and (4.21), respectively.

4.5 Solutions of the generalised eigenvalue problems in the differential realisation

We shall finally examine how solving the GEVP $\mathcal {M}f(z) = n \mathcal {L} f(z)$ and the adjoint problem compares to the representation-theoretic computations that were performed for the overlaps $\widetilde {\langle z|}P_n(\alpha , \beta )\rangle $ and $\left \langle z |L^T |Q_m(\alpha , \beta )\right \rangle $ . A first look shows that the GEVPs in the differential model will be of hypergeometric nature, as is confirmed by the expressions of the overlaps. Let us focus on this more closely.

Given expressions (4.20) and (4.21) for $\mathcal {L}$ and $\mathcal {M}$ , we see that $\mathcal {M}f(z) = n \mathcal {L} f(z)$ takes the form of the hypergeometric equation [Reference Erdelyi, Magnus, Oberhettinger and Tricomi5]

(4.37) $$ \begin{align} z (1-z) \frac{d^2 f}{dz^2} + [c - (a + b + 1)z] \frac{df}{dz} - ab f = 0, \end{align} $$

with parameters

(4.38) $$ \begin{align} a = -n, \qquad b = \alpha + 1, \qquad c= 1 - n - \beta. \end{align} $$

In the following we shall use Bateman’s nomenclature [Reference Erdelyi, Magnus, Oberhettinger and Tricomi5] for the $24$ Kummer solutions; these are arranged in six sets such that the four elements in each set represent the same function. The representatives $u_1, u_2, \dotsc , u_6$ of the sets are in general different, although equation (3.53) is a case where $u_1 \propto u_2$ . With the parameters given by equation (4.38), it is immediate to see that the solution

(4.39) $$ \begin{align} u_1 = {_2}F_1 \left( {a, b\atop c}; z\right) \end{align} $$

will yield directly (up to a constant) the Askey polynomials $P_n(z; \alpha , \beta )$ .

Consider now the adjoint operators

(4.40) $$ \begin{align}\begin{aligned} \mathcal{L}^T = (1-z) \partial_z + \alpha \mathcal{I}, \end{aligned}\end{align} $$

(4.41) $$ \begin{align}\begin{aligned} \mathcal{M}^T = z(z-1) \partial_z^2 + [(2-\alpha)z -\beta-1 ] \partial_z - \alpha \mathcal{I}. \end{aligned}\end{align} $$

The adjoint GEVP $\mathcal {M}^T f^*(z) = m \mathcal {L} f^*(z)$ also turns out to yield the hypergeometric equation (4.37), but this time with parameters

(4.42) $$ \begin{align} a = m + 1, \qquad b = - \alpha, \qquad c = 1 + \beta + m. \end{align} $$

Recall that $\mathcal {L}^T f^*(z)$ will provide a solution orthogonal to $f(z)$ . Selecting $u_1$ for $f^*$ also will lead to functions trivially orthogonal over the unit circle. Consider instead

(4.43) $$ \begin{align} f^*(z) = u_3 = {(1 - z)^{-a}}{}{_2}F_1 \left( {a, c - b\atop a + 1 - b}; \frac{1}{1 - z}\right). \end{align} $$

Using

(4.44) $$ \begin{align} \frac{(k + m + \alpha + 1)}{(m + \alpha + 2)_k} = \frac{(m + \alpha + 1)}{(m + \alpha + 1)_k}, \end{align} $$

it is easy to find that in that case,

(4.45) $$ \begin{align} \mathcal{L}^T f^*(z) = ( m + \alpha + 1){(1-z)^{-m - 1}}{} {_2}F_1 \left( {m + 1, m + \alpha + \beta + 1\atop m + \alpha + 1}; \frac{1}{1 - z}\right). \end{align} $$

We thus see that choosing the solution $u_3$ yields the result (3.56) obtained algebraically for $\left \langle z |L^T |P_m (\alpha , \beta ) \right \rangle $ . (Recall that the constant a in this expression is here set equal to $1$ .) The reader is reminded of equation (3.57), where this function is seen to be composed of two parts: one a power series in z and the other the function orthogonal to the Askey polynomial dressed with the hypergeometric weight.

Let us point out that within the differential realisation, it is possible to pick a solution of $\mathcal {M}^T f^*(z) = m \mathcal {L} f^*(z)$ that will solely give the biorthogonal partner $Q_m\left (\frac {1}{z}, \alpha , \beta \right )$ multiplied by the weight. Indeed, take $f^*(z)$ to be given by the solution

(4.46) $$ \begin{align} u_4 = {(1 - z)^{-b}}{} {_2}F_1 \left( {b, c - a\atop b + 1 - a}; \frac{1}{1 - z}\right). \end{align} $$

Substituting the parameters (4.42), we have in this instance

(4.47) $$ \begin{align} f^*(z) &= {(1 - z)^{\alpha}}{} {_2}F_1 \left( {-\alpha, \beta\atop -\alpha - m}; \frac{1}{1 - z}\right) \nonumber\\ &= \sum_{k=0}^{\infty} \frac{(-\alpha)_k (\beta)_k}{(-\alpha - m)_k} \frac{(1 - z) ^{-k + \alpha}}{k!}. \end{align} $$

The action of $\mathcal {L}^T$ is again readily computed when the argument is a function of $(1 - z)$ :

(4.48) $$ \begin{align} \mathcal{L}^Tf^*(z) &= \sum_{k=1}^{\infty} \frac{(-\alpha)_k (\beta)_k}{(-\alpha - m)_k} \frac{(1 - z) ^{-k + \alpha}}{(k-1)!} \nonumber \\ &=\sum_{l=0}^{\infty} \frac{(-\alpha)_{l+1} (\beta)_{l+1}}{(-\alpha - m)_{l+1}} \frac{(1 - z) ^{-l + \alpha -1}}{l!} \nonumber \\ & = \frac{\alpha \beta}{(\alpha +m)} {(1 - z)^{\alpha -1}}{} {_2}F_1 \left( {-\alpha + 1, \beta + 1 \atop 1 -\alpha - m}; \frac{1}{1 - z}\right), \end{align} $$

where we have used $(x)_{l+1} = x (x + 1)_l$ . We thus observe that the action of $\mathcal {L}^T$ is to effect $ \alpha \rightarrow \alpha - 1, \; \beta \rightarrow \beta + 1$ ; in view of the parameter identification (4.42), the action of $\mathcal {L}^T$ on $u_4$ yields again $u_4$ , with the following parameters:

(4.49) $$ \begin{align} a = m + 1, \qquad b = -\alpha + 1, \qquad c = 2 + \beta + m. \end{align} $$

Now another expression for $u_4$ is

(4.50) $$ \begin{align} u_4 = (-z)^{a - c} {(1 - z)^{c-a-b}}{} {_2}F_1 \left( {1 - a, c - a\atop b + 1 - a}; \frac{1}{z}\right). \end{align} $$

Using the parameters (4.49), we then find that choosing $u_4$ as solution of the hypergeometric equation stemming from the adjoint GEVP $\mathcal {M}^T f^*(z) = m \mathcal {L} f^*(z)$ leads to

(4.51) $$ \begin{align} \mathcal{L}^Tf^*(z) \propto (-z) ^{-1 - \beta} (1 - z)^{\alpha + \beta} Q_m\left(\frac{1}{z}, \alpha, \beta\right). \end{align} $$

That is, we obtain as unique term, up to a factor, the orthogonal partner of $P_n(z; \alpha , \beta )$ multiplied by the weight.