Rediscovery of Routh Polynomials after Hundred Years in Obscurity

Abstract

The introductory part of the paper outlines misfortunate history of one of Routh’s most remarkable works. It points to an important distinction between infinitely many Routh polynomials forming a differential polynomial system (DPS) and their finite orthogonal subset referred to as “Romanovski-Routh” (R-Routh) polynomials (Romanovski/pseudo-Jacobi polynomials in Lesky’s classification scheme). It was shown that there are two infinite sequences of Routh polynomials without real roots (in contrast with R-Routh polynomials with all the roots located on the real axis). The factorization functions (FFs) formed by polynomials from these infinite sequences can be thus used to generate exactly solvable rational Darboux transforms of the canonical Sturm-Liouville equation (CSLE) quantized in terms of R-Routh polynomials with degree-dependent indexes. The current analysis is focused solely on the CSLE with the density function having two simple zeros on the opposite sides of the imaginary axis. A special attention is given to the limiting case of the density function with the simple poles at ±i when the given CSLE becomes translationally form-invariant, and as a result, the eigenfunctions of its rational Darboux transforms are expressible in terms of a finite number of exceptional orthogonal polynomials (EOPs).

1. Introduction

Let us first outline some groundbreaking aspects of Routh’ paper [1] overlooked by mathematicians for more than a hundred years before a brief reference to his results appeared in Ismail’s monograph [2]. Routh’ precocious discovery not properly appreciated even today (see [3] for more detailed critical remarks) was the classification of the real second-order differential eigenequations of hypergeometric type

$$\sigma \left(x\right)\frac{d^2{X}_n}{d{x}^2}+\tau \left(x\right)\frac{{\mathit{dX}}_n}{dx}+h_n{X}_n\left(x\right)=0$$E1

according to positions of zeros of the leading polynomial coefficient

$$\sigma \left(x\right)\equiv {\mathit{ax}}^2+\mathit{bx}+c,$$E2

regardless of the choice of the polynomial coefficient of the first derivative:

$$\tau \left(x\right)\equiv \mathit{fx}+g.$$E3

Namely Routh proved that the function

$$X_n\left(x\right)=\frac{\sigma \left(x\right)}{\mathrm{R}\left(x\right)}\frac{d^n}{d^{n}x}\left[{\sigma }^{n-1}\left(x\right)\mathrm{R}\left(x\right)\right]$$E4

where $\mathrm{R}\left(\mathrm{x}\right)$ is a solution of the first-order ordinary differential equation (ODE)

$$\frac{1}{\mathrm{R}\left(x\right)}\frac{d\mathrm{R}}{dx}=\frac{\tau \left(x\right)}{\sigma \left(x\right)}$$E5

(see his Art. 18, coupled with Eq. (6) in Art. 4) obeys eigeneq. (1) for

$$h_n=-n\left[f+a\left(n-1\right)\right].$$E6

(Regretfully neither Rodriguez’ [4] nor Jacobi’s [5] classic works were mentioned in this connection.) Fifty years later, Hildebrandt [6] started from the reminiscent first-order ODE

$$\frac{1}{\omega \left(x\right)}\frac{\mathit{d\omega }}{dx}=\frac{N\left(x\right)}{D\left(x\right)}$$E7

referring to it as the “Pearson differential equation” [7] and then used its analytical solutions as the weight functions for the generalized Rodrigues formula [4, 5].

$$X_n\left(x\right)=\frac{1}{\omega \left(x\right)}\frac{d^n}{d^{n}x}\left[D^n\left(x\right)\omega \left(x\right)\right],$$E8

where $D\left(x\right)\equiv \sigma \left(x\right)$ and

$$N\left(x\right)\equiv \tau \left(x\right)-{\sigma }^{\prime }\left(x\right)$$E9

in terms of [8]. Hildebrandt then proved that the polynomials generated via (8) must satisfy second-order differential eigenequation (1). Comparing (7) and (9) with (5) thus gives

$$\mathrm{R}\left(x\right)=\sigma \left(x\right)\omega \left(x\right)$$E10

so Routh’s formula (4) is nothing but the precursor of (8).

Examination of all possible solutions of first-order ODE (7) brought Hildebrandt to the classification of various types of polynomial solutions of second-order differential eigenequation (1) including different sub-cases of five cases specified by Routh:

1. discriminant of quadratic polynomial (2) is positive;

2. discriminant of quadratic polynomial (2) is negative;

3. quadratic polynomial (2) has a double root;

4. the leading coefficient function is a polynomial of the first degree;

5. the leading coefficient function is a constant.

Combining subcases i) and i’) into a single class of leading polynomial coefficient with a nonzero discriminant, we come to Bochner’s [9] four classes of second-order differential eigenequations (1) defined over complex field (i.e., Jacobi, Bessel, Laguerre, and Hermite polynomials accordingly). While case i) is nothing but the notorious Jacobi polynomials [5] leading coefficient function (2) with two complex-conjugated zeros is associated with a completely different infinite real polynomial sequence termed “Routh polynomials” below.

In following Everitt et al. [10, 11], we refer to Routh’s cases i) – v) as differential polynomial systems (DPSs), while preserving the term “orthogonal polynomial system” (OPS) used by Kwon and Littlejohn [12] in this context solely for classical orthogonal polynomials forming infinite sequences of positive definite orthogonal polynomials. As stressed by Kwon and Littlejohn [12], Bochner himself “did not mention the orthogonality of the polynomial systems that he found. The problem of classifying all classical orthogonal polynomials was handled by many authors thereafter” based on his analysis of possible polynomial solutions of complex second-order differential eigenequations.

The crucial point is that polynomial solutions of the differential eigenequation

$$\left(x^2,+,1\right)\frac{d^2{X}_n}{d{x}^2}+\left(fx+g\right)\frac{{\mathit{dX}}_n}{dx}+h_n\left(f,g\right)X_n\left(x\right)=0$$E11

referred to below as Routh polynomials represent a supplementary nontrivial real-field reduction of the complex Jacobi DPS, in addition to conventional real Jacobi polynomials. Obviously, this assertion directly follows from Bochner’s renowned paper [9]; however, it took another 40 years before Cryer [13] came up with the well-known [2] representation of Routh polynomials as Jacobi polynomials with complex conjugated indexes in an imaginary argument (see [3] for more details). Namely Cryer realized that the weight function for Jacobi polynomials:

$$w^{\left(\alpha ,\beta \right)}\left(\eta \right)\equiv {\left(1-\eta \right)}^{\alpha }{\left(1+\eta \right)}^{\beta }$$E12

formally coincides with Pearson’s distribution function of type IV [7]:

$$w^{\left(\alpha ,\alpha \ast \right)}\left(\mathit{ix}\right)=w^{\left({\alpha }_R+i{\alpha }_I\right)}\left(x\right)\equiv {\left(1+x^2\right)}^{{\alpha }_R}\mathit{exp}\left(2{\alpha }_I\mathit{arctan}x\right),$$E13

where ${\alpha }_R$ and ${\alpha }_I$ are real and imaginary parts of the complex number α. This brought him to the renowned formula

$${\Re }_m^{\left({\alpha }_R+i{\alpha }_I\right)}\left[x\right]\equiv {\left(-i\right)}^m{P}_m^{\left({\alpha }_R+i{\alpha }_I,{\alpha }_R-i{\alpha }_I\right)}\left(\mathit{ix}\right),$$E14

where

$${\Re }_m^{\left({\alpha }_R+i{\alpha }_I\right)}\left[x\right]\equiv P_m\left(x;{\alpha }_R,{\alpha }_I\right)$$E15

in Ismail’s notation [2]. As discussed more thoroughly in [3], the monic polynomials

$${\widehat{\Re }}_m^{\left({\alpha }_R+i{\alpha }_I\right)}\left[x\right]=\frac{1}{K_m\left({\alpha }_R\right)w^{\left({\alpha }_R+i{\alpha }_I\right)}\left(x\right)}\frac{d^m}{d^{m}x}\left[{\left(x^2+1\right)}^m{w}^{\left({\alpha }_R+i{\alpha }_I\right)}\left(x\right)\right]$$E16

introduced by Cryer with

$$K_m\left({\alpha }_R\right)\equiv \frac{{\left(2{\alpha }_R+2m\right)}_m}{m!2^m}$$E17

coincide with the polynomials $P_m\left(x;\nu ,N\right)$ in [14] with N = $-{\alpha }_R-1$, ν = ${\alpha }_I$:

$${\widehat{\Re }}_m^{\left({\alpha }_R+i{\alpha }_I\right)}\left[x\right]\equiv P_m\left(x;{\alpha }_I,-{\alpha }_R-1\right).$$E18

In this paper, we prefer to adopt the notation of the cited monograph [14] to make use of the formulas listed in §9.9 for the so-called “pseudo-Jacobi polynomials,” with N changed for an arbitrary real number. (A similar suggestion has been already put forward in [15] though with the lower bound of $-1/2$ for N.) Note that the term “Routh polynomials” is used by us in exactly the same sense as “pseudo-Jacobi polynomials” in [16], that is, without any limitations on sign and values of ${\alpha }_R$. The finite orthogonal subsequence of the Routh DPS discovered by Romanovsky [17]¹ is referred to by us as “Romanovski-Routh” (R-Routh) polynomials similarly to the epithet “Romanovski/pseudo-Jacobi” suggested for these polynomials by Lesky [18, 19].

2. Rational Sturm-Liouville problem solvable via R-Routh polynomials with degree-dependent indexes

Author’s own interest in Routh polynomials was stimulated by examination of the Routh-reference (RRef) canonical Sturm-Liouville equation (CSLE) [20, 21].

$$\left\{\frac{d^2}{d{\eta }^2}+{}_{i}I\left[\eta ;h_o;\kappa ;\epsilon \right]\right\}{}_i\mathrm{\Phi }\left[\eta ;h_o;\kappa ;\epsilon \right]=0,$$E19

where the real “Bose invariant” [22].

$${}_{i}I\left[\eta ;h_o;\kappa ;\epsilon \right]\equiv {{}_{i}I}^o\left[\eta ;h_o\right]+\epsilon {}_i\mathit{\rho }\left[\eta ;\kappa \right]$$E20

in Milson’s terms [20] represents a superposition of the reference polynomial fraction (RefPF)

$${{}_{i}I}^o\left[\eta ;h_o\right]=-\frac{h_o}{4{\left(\eta +i\right)}^2}-\frac{h_o^{\ast }}{4{\left(\eta -i\right)}^2}+\frac{2h_{o;R}+1}{4\left({\eta }^2+1\right)}$$E21

$$=\frac{h_{o;R}-h_{o;I}\eta }{{\left({\eta }^2+1\right)}^2}+\frac{1}{4\left({\eta }^2+1\right)}$$E22

and the density function

$${}_i\mathit{\rho }\left[\eta ;\kappa \right]=\frac{T_2\left[\eta ;\kappa \right]}{{\left({\eta }^2+1\right)}^2}$$E23

dependent accordingly on a complex parameter

$$h_o\equiv h_{o;R}+i{h}_{o;I}$$E24

and on the second-degree monic “tangent polynomial” (TP)

$${{}_{i}T}_2\left[\eta ;\kappa \right]\equiv {\eta }^2+\kappa \equiv \left(\eta -i\sqrt{\kappa }\right)\left(\eta +i\sqrt{\kappa }\right)$$E25

with the negative discriminant -κ. The energy reference point for RCSLE (19) was chosen in such a way that the indicial equation for the pole at infinity has real (complex-conjugated) roots at any negative (any positive) energy. The density function with a constant TP in numerator, giving rise to the translationally form-invariant (TFI) CSLE of Group B with the trigonometric Liouville potential [23], requires a special consideration.

It has been proven in [21] that the eigenfunctions of CSLE (19) can be expressed in terms of R-Routh polynomials:

$${{}_i\varphi }_n\left[\eta ;h_{o;R}+i{h}_{o;I};\kappa \right]\propto {\left(1-i\eta \right)}^{{{}_i\rho }_n}{\left(1+i\eta \right)}^{{{}_i\rho }_n^{\ast }}{R}_n^{\left(2{\lambda }_{n;I},{\lambda }_{n;R}+1\right)}\left(\eta \right)$$E26

with

$${{}_i\lambda }_n\equiv {\lambda }_{n;R}+i{\lambda }_{n;I}\equiv 2{{}_i\rho }_n-1$$E27

which are defined via (46) and (47) in [24]:²

$$R_n^{\left(2{\alpha }_I,{\alpha }_R+1\right)}\left(\eta \right)={\left(-i\right)}^n{P}_n^{\left({\alpha }_R+i{\alpha }_I,{\alpha }_R-i{\alpha }_I\right)}\left(\mathit{i\eta }\right)\equiv {\Re }_n^{\left({\alpha }_R+i{\alpha }_I\right)}\left(\eta \right)$$E28

for

$$n\le N_0\equiv ⌊-{\alpha }_R-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.⌋.$$E29

One can easily verify that eigenfunctions (26) are square integrable with density function (23) as far as condition (29) holds. To prove that the Sturm-Liouville problem in question is exactly solvable, the author [21] took advantage of Stevenson’s idea [27] to express an analytically continued solution in terms of hypergeometric polynomials in complex argument. It was just confirmed that the latter formally complex polynomials can be converted into real R-Routh polynomials (28).

Examination of the exponent differences for three (including ∞) poles of CSLE (19) reveals that the sought-for real parameters ${\lambda }_{n;R}$ and ${\lambda }_{n;I}$ are unambiguously determined by the following set of the algebraic equations [21]:

$${{}_i\lambda }_n^2=h_{o;R}+i{h}_{o;I}+1+\left(1-\kappa \right){{}_i\cancel{\rho }}_{\infty ;n}^2$$E30

and

$${{}_i\cancel{\rho }}_{\infty ;n}=-{\lambda }_{n;R}-n-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.>0,$$E31

with the square of real parameter (31) determining the magnitude of the corresponding eigenvalue

$${{}_i\epsilon }_n\left(h_o;\kappa \right)=-{{}_i\cancel{\rho }}_{\infty ;n}^2<0.$$E32

Keeping in mind that

$$2{\lambda }_{n;R}{\lambda }_{n;I}=h_{o;I}$$E33

we come to the quartic eq. [21]

$${\lambda }_{n;R}^4-\left[h_{o;R}+1+\left(1-\kappa \right)\left({\lambda }_{n;R}+n+\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right)\right]{\lambda }_{n;R}^2-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.h_{o;I}^2=0$$E34

with the positive leading coefficient $\kappa$ and the negative free term (except the limiting case $h_{o;I}=0$ [28] associated with the symmetric Ginocchio potential [29]). This implies that quartic eq. (34) necessarily has at least two real roots ${\lambda }_{n;R}\equiv {\lambda }_{\mathbf{c},n;R}<0$ and ${\lambda }_{\mathbf{d},n;R}>0$ if CSLE (19) has at minimum n+1 discrete energy levels. We thus assert that each eigenfunction (26) must be accompanied by the second quasi-rational solution (q-RS)

$${{}_i\varphi }_{\mathbf{d},n}\left[\eta ;h_o;\kappa \right]={\left(1+\mathit{i\eta }\right)}^{{{}_i\rho }_{\mathbf{d},n}}{\left(1-\mathit{i\eta }\right)}^{{{}_i\rho }_{\mathbf{d},n}^{\ast }}{\Re }_n^{\left({{}_i\lambda }_{\mathbf{d},n}\right)}\left(\eta \right),$$E35

$$\equiv {\left(1+{\eta }^2\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\left({\lambda }_{\mathbf{d},\mathit{n};R}+1\right)}\mathit{exp}\left(\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.{\lambda }_{\mathbf{d},\mathit{n};\mathit{I}}\mathit{arctan}\eta \right){\Re }_n^{\left({{}_i\lambda }_{\mathbf{d},\mathit{n}}\right)}\left(\eta \right)$$E36

at the energy

$${{}_i\epsilon }_{\mathbf{d},n}=-{{}_i\cancel{\rho }}_{\infty ;\mathbf{d},n}^2,$$E37

where we set

$${{}_i\lambda }_{\mathbf{d},n}={\lambda }_{\mathbf{d},n;R}+\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\left(h_{o;I}/{\lambda }_{\mathbf{d},n;R}\right)i,$$E38

$${{}_i\rho }_{\mathbf{d},n}=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\left({{}_i\lambda }_{\mathbf{d},n}+1\right),$$E39

and

$${{}_i\cancel{\rho }}_{\infty ;\mathbf{d},n}=-{\lambda }_{\mathbf{d},\mathrm{n};R}-n-1.$$E40

Here, in following the classification of the “factorization functions” (FFs) suggested by us for Darboux transformations of radial potentials [30] (years before birth of supersymmetric quantum mechanics [31, 32]), we use the labels c and d to specify the eigenfunctions of the given Sturm-Liouville problem and their counterparts infinite at both ends (type III in Quesne’s terms [24]).

My own interest in the q-RSs of type d was stimulated by Quesne’s conjecture [24] concerning the existence of Routh polynomials with no real roots, which can be thereby used to construct nodeless FFs for rational Darboux transformations giving rise to new exactly solvable rational potentials.

3. Two infinite sequences of nodeless q-RSs composed of Routh polynomials

Rewriting quartic eq. (34) as

$${\lambda }_{\mathbf{†},m;R}^4-[h_{o;R}+1+\left(1-\kappa \right){\left({\lambda }_{\mathbf{†},m;R}+m+\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right)}^2{\lambda }_{\mathbf{†},m;R}^2-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.h_{o;I}^2=0$$E41

and keeping in mind that its leading coefficient $\kappa$ and free term have opposite signs (again except the mentioned symmetric case), we assert the given quartic equation always has at least two real roots and therefore a pair of unbounded q-RSs

$${{}_i\varphi }_{\mathbf{d},\mathrm{m}}\left[\eta ;h_o;\kappa \right]=\sqrt{{\eta }^2+1}{\left(1+\mathit{i\eta }\right)}^{\frac{1}{2}{{}_i\lambda }_{\mathbf{d},\mathrm{m}}}{\left(1-\mathit{i\eta }\right)}^{\frac{1}{2}{{}_i\lambda }_{\mathbf{d},\mathrm{m}}^{\ast }}{\Re }_m^{\left({{}_i\lambda }_{\mathbf{d},\mathrm{m}}\right)}\left[\eta \right]\text{with}{\lambda }_{\mathbf{d},\mathrm{m};\mathrm{R}}>0$$E42

and

$${{}_i\varphi }_{{\mathbf{†}}^{\prime },\mathrm{m}}\left[\eta ;h_o;\kappa \right]=\sqrt{{\eta }^2+1}{\left(1+\mathit{i\eta }\right)}^{\frac{1}{2}{{}_i\lambda }_{{\mathbf{†}}^{\prime },\mathrm{m}}}{\left(1-\mathit{i\eta }\right)}^{\frac{1}{2}{{}_i\lambda }_{{\mathbf{†}}^{\prime },\mathrm{m}}^{\ast }}{\Re }_m^{\left({{}_i\lambda }_{{\mathbf{†}}^{\prime },\mathrm{m}}\right)}\left[\eta \right]\text{with}{\lambda }_{{\mathbf{†}}^{\prime },\mathrm{m};\mathrm{R}}<0$$E43

formed by Routh polynomials exists for any nonnegative integer m, with t’ = c or d’.

Dividing quartic eq. (41) by m⁴, setting

$$C_{\mathbf{†}}=\underset{m\to \infty }{\mathit{lim}}\left({\lambda }_{\mathbf{†},m;R}/m\right),$$E44

and making m tend to ∞, we come to the quadratic equation

$$C_{\mathbf{†}}^2+\left(\kappa -1\right){\left(C_{\mathbf{†}}+1\right)}^2=0$$E45

in $C_{\mathbf{†}}$, which has the positive leading coefficient $\kappa$ and negative (positive) free term for $0<\kappa <1$ ($\kappa >1$), so its two roots always have opposite signs:

$$C_{\mathbf{d}}>0,C_{{\mathbf{d}}^{\prime }}<0\text{for}\mathrm{any}\kappa <1.$$E46

One can directly verify that $C_{{\mathbf{d}}^{\prime }}$ necessarily differs from −1 and therefore the magnitudes of the intrinsic characteristic exponents (ChExps) at infinity

$${{}_i\cancel{\rho }}_{\infty ;\mathbf{†},m}=-{\lambda }_{\mathbf{†},m;R}-m-1<0\left(\mathbf{†}=\mathbf{d}\text{or}{\mathbf{d}}^{\text{'}}\right)$$E47

monotonically grow with m for sufficiently large m iff $\kappa <1.$ Under the latter condition, the energies of both q-RSs (42) and (43),

$${{}_i\epsilon }_{\mathbf{†},m}=-{{}_i\cancel{\rho }}_{\infty ;\mathbf{†},m}^2,$$E48

must thus lie below the ground energy level ${{}_i\epsilon }_{\mathbf{c}0}$ for m > > 1. As a direct consequence of the disconjugacy theorem recently brought into the theory of rationally extended potentials by Grandati et al. [33, 34, 35, 36, 37], we conclude that the solutions in question may not have more than one node. Taking into account that solutions of any SLE may have only a simple zero at any regular point and also that the monic polynomial of an even degree is necessarily positive for sufficiently large absolute values of its argument, we conclude that q-RSs (42) and (43) with $\kappa$ between 0 and 1 must be nodeless if m = $2\mathrm{j}$ > > 1 assuming that leading coefficient (17) of Routh polynomial (14) differs from zero. The latter constraint is always valid for any q-RS (42) since the real part of the index ${\lambda }_{\mathbf{d},m;R}$ is positive by definition. The cited constraint also holds for q-RS (43) unless 2 ${\lambda }_{{\mathbf{d}}^{\prime },m;R}$ is a negative integer smaller than 1–m.

As discussed in next section, each of infinitely many nodeless q-RSs (42) and (43) can be used as an FF to construct the RDT of CSLE (19) exactly solvable by polynomials. The latter constitute a new type of polynomials originally discovered by the author [38] while analyzing a structure of eigenfunctions for the Cooper-Ginocchio-Khare potential [39]. The distinctive common feature of these polynomials [40] is that they satisfy Heine-type [41, 42] differential equations with the first-derivative coefficient function dependent on the polynomial degree. The polynomial sequences in question turn into finite or infinite sequences of exceptional orthogonal polynomials (EOPs) in Quesne’s terms [24] if the exponent differences (ExpDiffs) for the SLE poles in the finite plane become energy-independent [43], and as a result, the corresponding Liouville potential belongs to Group A of translationally shape-invariant (TSI) potentials in Odake and Sasaki’s [44, 45] classification scheme. The RDTs of the translationally form-invariant (TFI) CSLEs from Group B [46] represent a more typical case of rational CSLEs (RCSLEs) converted by gauge transformations to Heine-type differential equations with the first-derivative coefficient function dependent on the polynomial degree (see [34, 37, 45, 47, 48] for examples). Again we deal with the polynomial solutions of a new type (referred to by us as Gauss-seed Heine polynomials [40]), which are in general not expressible in terms of EOPs, in contrast with the statement made in [37].

Before going to the discussion of RDTs of CSLE (19), it is convenient to reformulate the corresponding spectral problem by introducing the (algebraic) “prime” SLE [49].

$$\left\{\frac{d}{\mathit{d\eta }}{\left({\eta }^2+1\right)}^{\frac{1}{2}}\frac{d}{\mathit{d\eta }}-{}_i\cancel{q}[\eta ;h_o]+\epsilon {}_i\cancel{w}[\eta ;\kappa ]\right\}{}_i\cancel{\mathrm{\Psi }}\left[\eta ;h_o;\kappa ;\epsilon \right]=0$$E49

obtained from CSLE (19) by the gauge transformation

$${}_i\cancel{\mathrm{\Psi }}\left[\eta ;h_o;\kappa ;\epsilon \right]={\left({\eta }^2+1\right)}^{-\frac{1}{4}}{}_i\mathrm{\Phi }\left[\eta ;h_o;\kappa ;\epsilon \right]$$E50

so

$${}_{i}w\left[\eta ;\kappa \right]\equiv {\left({\eta }^2+1\right)}^{\frac{1}{2}}{}_i\mathit{\rho }\left[\eta ;\kappa \right].$$E51

The particular form of the energy-free term ${}_i\cancel{q}\left[\eta ;h_o\right]$ is nonessential for our discussion. and we refer the reader to [3] for all the details.

The main advantage of converting CSLE (19) to its prime form with respect to the regular singular point at infinity comes from our observation [49] that the ChExps for this pole have opposite signs, and therefore, the corresponding principal Frobenius solution is unambiguously selected by the Dirichlet boundary conditions (DBCs):

$$\underset{\eta \to \pm \infty }{\mathit{lim}}{{}_i\cancel{\psi }}_{\mathbf{c},n}\left[\eta ;h_o;\kappa \right]=0\text{for}\mathrm{n}=0,\dots ,{\mathrm{n}}_{\max }$$E52

imposed on its eigenfunctions

$${{}_i\cancel{\psi }}_{\mathbf{c},n}\left[\eta ;h_o;\kappa \right]\equiv {}_i\cancel{\mathrm{\Psi }}\left[\eta ;h_o;\kappa ;{{}_i\epsilon }_{\mathbf{c},n}\left(h_o;\kappa \right)\right].={\left({\eta }^2+1\right)}^{-\frac{1}{4}}{{}_i\varphi }_{\mathbf{c},n}\left[\eta ;h_o;\kappa \right].$$E53

(Remember that the energy reference point was chosen by us in such a way that the indicial equation for the pole at infinity has real roots at any negative energy.) Reformulating the given spectral problem in such a way allows us to take advantage of powerful theorems proven in [50] for zeros of principal solutions of SLEs solved under the DBCs at singular ends.

4. Exactly solvable rational Liouville-Darboux transforms of RRef CSLE

As discussed in previous section, any q-RS

$${{}_i\varphi }_{\mathbf{†},2\mathrm{j}}\left[\eta ;h_o;\kappa \right]=\sqrt{{\eta }^2+1}{\left(1+\mathit{i\eta }\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.{{}_i\lambda }_{\mathbf{†},2\mathrm{j}}}{\left(1-\mathit{i\eta }\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.{{}_i\lambda }_{{\mathbf{†}}^{\prime },2\mathrm{j}}^{\ast }}{\Re }_{2\mathrm{j}}^{\left({{}_i\lambda }_{\mathbf{†},2\mathrm{j}}\right)}\left[\eta \right]$$E54

at the energy

$${{}_i\epsilon }_{\mathbf{†},2\mathrm{j}}\left(h_o;\kappa \right)<{{}_i\epsilon }_{\mathbf{c}0}\left(h_o;\kappa \right)\left(\mathbf{†}=\mathbf{d}\text{or}{\mathbf{d}}^{\prime }\right)$$E55

is nodeless for $\mathrm{j}>>1$ unless 2${\lambda }_{{\mathbf{d}}^{\prime },2\mathrm{j};R}$ is a negative integer smaller than 1–$2\mathrm{j}$ for $\mathbf{†}={\mathbf{d}}^{\prime }$. (In the latter case, the degree of the Routh polynomial is smaller than $2\mathrm{j}$ and may happen to be odd.) Each of these solutions can be thus used as an FF for the rational Liouville-Darboux transformation (RLDT) such that Rudyak and Zahariev’s reciprocal function [51].

$${{}_i\varphi }_{\mathbf{c},0}\left[\eta ;h_o;\kappa |\mathbf{†},2\mathrm{j}\right]=\frac{{{}_i\mathit{\rho }}^{-\frac{1}{2}}\left[\eta ;\kappa \right]}{{{}_i\varphi }_{\mathbf{†},2\mathrm{j}}\left[\eta ;h_o;\kappa \right]}\left(\mathbf{†}=\mathbf{d}\text{or}{\mathbf{d}}^{\prime }\right)$$E56

represents a q-RS of the transformed RCSLE:

$$\left\{\frac{d^2}{d{\eta }^2}+{{}_{i}I}^o\left[\eta ;h_o;\kappa |\mathbf{†},2\mathrm{j}\right]+{{}_i\epsilon }_{\mathbf{†},2\mathrm{j}}(h_o;\kappa \left){}_i\mathit{\rho }\right[\eta ;\kappa ]\right\}{{}_i\varphi }_{\mathbf{c},0}\left[\eta ;h_o;\kappa |\mathbf{†},2\mathrm{j}\right]=0$$E57

at same energy (55), where

$${{}_{i}I}^o\left[\eta ;h_o;\kappa |\mathbf{†},2\mathrm{j}\right]=-{\mathit{ld}}^2{{}_i\varphi }_{\mathbf{c},0}\left[\eta ;h_o;\kappa |\mathbf{†},2\mathrm{j}\right]-{\mathit{ld}}^{•}{{}_i\varphi }_{\mathbf{c},0}\left[\eta ;h_o;\kappa |\mathbf{†},2\mathrm{j}\right]$$

$$-{{}_i\epsilon }_{\mathbf{†},2\mathrm{j}}\left(h_o;\kappa \right){}_i\mathit{\rho }\left[\eta ;\kappa \right]$$E58

with ld and dot standing for the logarithmic derivative and the derivative with respect to η, respectively. It will be proven below that q-RS (56) represents the lowest-energy eigenfunction of the given Sturm-Liouville problem as indicated by its label.

We [40] introduced the term “Liouville-Darboux transformation” (LDT) to stress that we deal with the three-step operation:

1. the Liouville transformation [20, 22] from the given SLE to the conventional Schrödinger equation;

2. the Darboux deformation of the corresponding Liouville potential;

3. the inverse Liouville transformation from the Schrödinger equation to the new SLE preserving both leading coefficient function and weight.

Similarly to the discussion presented in previous Section 2 for RRef CSLE (19), the gauge transformation

$$\begin{array}{cc}{}_i\cancel{\mathrm{\Psi }}\left[\eta ;h_o;\kappa ;\epsilon |\mathbf{†},2\mathrm{j}\right]={\left({\eta }^2+1\right)}^{-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}& {}_i\mathrm{\Phi }\left[\eta ;h_o;\kappa ;\epsilon |\mathbf{†},2\mathrm{j}\right]\end{array}$$E59

converts the CSLE

$$\left\{\frac{d^2}{d{\eta }^2}+{{}_{i}I}^o\left[\eta ;h_o;\kappa |\mathbf{†},2\mathrm{j}\right]+\epsilon {}_i\mathit{\rho }[\eta ;\kappa ]\right\}{}_i\mathrm{\Phi }\left[\eta ;h_o;\kappa ;\epsilon |\mathbf{†},2\mathrm{j}\right]=0\left(\mathbf{†}=\mathbf{d}\text{or}{\mathbf{d}}^{\prime }\right)$$E60

into the prime SLE

$$\left\{\frac{d}{\mathit{d\eta }}{\left({\eta }^2+1\right)}^{\frac{1}{2}}\frac{d}{\mathit{d\eta }}-{}_i\cancel{q}[\eta ;h_o|\mathbf{†},2\mathrm{j}]+\epsilon {}_i\cancel{w}[\eta ;\kappa ]\right\}{}_i\cancel{\mathrm{\Psi }}\left[\eta ;h_o;\kappa ;\epsilon |\mathbf{†},2\mathrm{j}\right]=0$$E61

analytically solved under the DBCs

$$\underset{\eta \to \pm \infty }{\mathit{lim}}{}_i\cancel{\mathrm{\Psi }}\left[\eta ;h_o;\kappa ;\epsilon |\mathbf{†},2\mathrm{j}\right]=0\left(\mathbf{†}=\mathbf{d}\text{or}{\mathbf{d}}^{\prime }\right).$$E62

Setting

$$\begin{array}{cc}{{}_i\cancel{\psi }}_{\mathbf{†},m}\left[\eta ;h_o;\kappa \right]={\left({\eta }^2+1\right)}^{-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}& {{}_i\varphi }_{\mathbf{†},m}\left[\eta ;h_o;\kappa \right]\left(\mathbf{†}=\mathbf{c},\mathbf{d},\text{or}{\mathbf{d}}^{\prime }\right)\end{array}$$E63

we assert that the RLDT in question inserts the new ground energy level associated with the nodeless eigenfunction

$${{}_i\cancel{\psi }}_{\mathbf{c}0}\left[\eta ;h_o;\kappa |\mathbf{†},2\mathrm{j}\right]=\sqrt{\frac{{\eta }^2+1}{T_2\left[\eta ;\kappa \right]}}{{}_i\cancel{\psi }}_{\mathbf{†},2\mathrm{j}}^{-1}\left[\eta ;h_o;\kappa \right]$$E64

$$\propto {\eta }^{-2\mathrm{j}-1}{\left(1+{\eta }^2\right)}^{-\frac{1}{2}{{}_i\lambda }_{\mathbf{†},2\mathrm{j};\mathrm{R}}\left(h_o;\kappa \right)+\frac{1}{4}}\text{for}\left|\eta \right|>>1\left(\mathbf{†}=\mathbf{d}\text{or}{\mathbf{d}}^{\prime }\right)$$E65

at the energy

$${{}_i\epsilon }_{\mathbf{c},0}\left(h_o;\kappa |\mathbf{†},2\mathrm{j}\right)={{}_i\epsilon }_{\mathbf{†},2\mathrm{j}}\left(h_o;\kappa \right)<{{}_i\epsilon }_{\mathbf{c}0}\left(h_o;\kappa \right)\left(\mathbf{†}=\mathbf{d}\text{or}{\mathbf{d}}^{\prime }\right)$$E66

for sufficiently large $\mathrm{j}$. (Remember that ${{}_i\lambda }_{\mathbf{d},2\mathrm{j};\mathrm{R}}\left(h_o;\kappa \right)$ is necessarily positive in this case, whereas the function ${{}_i\cancel{\psi }}_{{\mathbf{d}}^{\prime },2\mathrm{j}}\left[\eta ;h_o;\kappa \right]$ in (64) for $\mathbf{†}={\mathbf{d}}^{\prime }$ is not an eigenfunction of the prime SLE by definition, so it must infinitely grow as $\eta \to \pm \infty$).

To obtain explicit formula for higher-energy eigenfunctions formed by the RDTs of the eigenfunctions of SLE (49),

$${{}_i\cancel{\psi }}_{\mathbf{c},n}\left[\eta ;h_o;\kappa \right]={\left({\eta }^2+1\right)}^{-\frac{1}{4}}{{}_i\varphi }_{\mathbf{c},n}\left[\eta ;h_o;\kappa \right],$$E67

we take advantage of Rudyak and Zahariev’s conventional formula [51]:

$$\mathrm{\Phi }\left[\eta ;h_o;\kappa ;\epsilon |\mathbf{†},2\mathrm{j}\right]=\frac{W\left\{{{}_i\varphi }_{\mathbf{†},2\mathrm{j}}\left[\eta ;h_o;\kappa \right],\mathrm{\Phi }\left[\eta ;h_o;\kappa ;\epsilon \right]\right\}}{{{}_i\mathit{\rho }}^{\frac{1}{2}}\left[\eta ;\kappa \right]{{}_i\varphi }_{\mathbf{†},2\mathrm{j}}\left[\eta ;h_o;\kappa \right]}$$E68

for the general solution of the generic CSLE and represent the mentioned RDTs as

$$\cancel{\mathrm{\Psi }}\left[\eta ;h_o;\kappa ;{{}_i\epsilon }_{\mathbf{c},n}\left(h_o;\kappa \right)|\mathbf{†},2\mathrm{j}\right]=\frac{W\left\{{{}_i\cancel{\psi }}_{\mathbf{†},2\mathrm{j}}\left[\eta ;h_o;\kappa \right],{{}_i\cancel{\psi }}_{\mathbf{c},n}\left[\eta ;h_o;\kappa \right]\right\}}{{{}_i\mathit{\rho }}^{\frac{1}{2}}\left[\eta ;\kappa \right]{{}_i\cancel{\psi }}_{\mathbf{†},2\mathrm{j}}\left[\eta ;h_o;\kappa \right]}\text{for}\mathrm{n}=0,\dots ,{\mathrm{n}}_{\max }\left(\mathbf{†}=\mathbf{d}\text{or}{\mathbf{d}}^{\prime }\right)$$E69

or, which is equivalent,

$$\cancel{\mathrm{\Psi }}\left[\eta ;h_o;\kappa ;{{}_i\epsilon }_{\mathbf{c},n}\left(h_o;\kappa \right)|\mathbf{†},2\mathrm{j}\right]={{}_i\mathit{\rho }}^{-\frac{1}{2}}\left[\eta ;\kappa \right]{{}_i\cancel{\psi }}_{\mathbf{c},n}\left[\eta ;h_o;\kappa \right]$$E70

$$\times \left\{\mathit{ld}{{}_i\cancel{\psi }}_{\mathbf{c},n}\left[\eta ;h_o;\kappa \right]-\mathit{ld}{{}_i\cancel{\psi }}_{\mathbf{†},2\mathrm{j}}[\eta ;h_o;\kappa ]\right\}.$$

Keeping in mind that

$$\underset{\eta \to \pm \infty }{\mathit{lim}}\left|{{}_i\mathit{\rho }}^{-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\left[\eta ;\kappa \right]\mathit{ld}{\cancel{\psi }}_{\mathbf{c},n}[\eta ;h_o;\kappa ]\right|<\infty \text{for}\mathrm{n}=0,\dots ,{\mathrm{n}}_{\max }$$E71

and

$$\underset{\eta \to \pm \infty }{\mathit{lim}}\left|{{}_i\mathit{\rho }}^{-\frac{1}{2}}\left[\eta ;\kappa \right]\mathit{ld}{{}_i\cancel{\psi }}_{\mathbf{†},2\mathrm{j}}[\eta ;h_o;\kappa ]\right|<\infty$$E72

coupled with (52), we conclude that q-RSs (70) satisfy DBCs (62) and therefore represent an eigenfunction of SLE (61) with the eigenvalues ${{}_i\epsilon }_{\mathbf{c},n}\left(h_o;\kappa \right).$ We thus proved that the RLDT in question keeps unchanged all n_max + 1 eigenvalues of prime SLE (49):

$${{}_i\epsilon }_{\mathbf{c},{\tilde{n}}_n}\left(h_o;\kappa |\mathbf{†},2\mathrm{j}\right)={{}_i\epsilon }_{\mathbf{c},n}\left(h_o;\kappa \right)\text{for}\mathrm{n}=0,\dots ,{\mathrm{n}}_{\max }\left(\mathbf{†}=\mathbf{d}\text{or}{\mathbf{d}}^{\prime }\right).$$E73

If prime SLE (61) has no additional eigenvalues between ${{}_i\epsilon }_{\mathbf{†},m}\left(h_o;\kappa \right)$ and ${{}_i\epsilon }_{\mathbf{c},n_{\max }}\left(h_o;\kappa \right)$_, then

$${\tilde{n}}_n=n+1$$E74

and

$${{}_i\cancel{\mathrm{\Psi }}}_{\mathbf{c},n+1}\left[\eta ;h_o;\kappa |\mathbf{†},2\mathrm{j}\right]=\cancel{\mathrm{\Psi }}\left[\eta ;h_o;\kappa ;{{}_i\epsilon }_{\mathbf{c},n}\left(h_o;\kappa \right)|\mathbf{†},2\mathrm{j}\right].$$E75

Let us now prove that eigenvalues (73) cover all the energy spectrum above the lowest-energy level ${{}_i\epsilon }_{\mathbf{†},2\mathrm{j}}\left(h_o;\kappa \right)$ or, in other words, that the given Dirichlet problem is exactly solvable. Indeed suppose that prime SLE (61) has another eigenfunction

${{\cancel{\mathrm{\Psi }}}^{˜}}_{\mathbf{c},\tilde{n}}\left[\eta ;h_o;\kappa |\mathbf{†},2\mathrm{j}\right]$ at an energy

$$\tilde{\epsilon }\left(h_o;\kappa |\mathbf{†},2\mathrm{j}\right)\ne {{}_i\epsilon }_{\mathbf{c},n}\left(h_o;\kappa |\mathbf{†},2\mathrm{j}\right)\mathit{for}\mathit{any}n\ge 0,$$E76

so

$$\underset{\eta \to \pm \infty }{\mathit{lim}}{{\cancel{\mathrm{\Psi }}}^{˜}}_{\mathbf{c},\tilde{n}}\left[\eta ;h_o;\kappa |\mathbf{†},2\mathrm{j}\right]=0.$$E77

Since solution (64) is nodeless, the eigenfunction in question must have at least one node.

Taking into account that RCSLE (60) has the second-order pole at infinity, we also affirm that this eigenfunction must obey the asymptotic formula

$$\underset{\eta \to \pm \infty }{\mathit{lim}}\left|\eta \mathit{ld}{{\cancel{\mathrm{\Psi }}}^{˜}}_{\mathbf{c},\tilde{n}}\left[\eta ;h_o;\kappa |\mathbf{†},2\mathrm{j}\right]\right|<\infty ,$$E78

similar to (71) and (72).

Combining (78) with a similar limit for lowest-energy eigenfunction (64):

$$\underset{\eta \to \pm \infty }{\mathit{lim}}\left|\eta \mathit{ld}{{}_i\cancel{\mathrm{\Psi }}}_{\mathbf{c}0}\left[\eta ;h_o;\kappa |\mathbf{†},2\mathrm{j}\right]\right|<\infty$$E79

and making use of (77) we conclude that the function

$${\cancel{\mathrm{\Psi }}}^{˜}\left[\eta ;h_o;\kappa \right]\equiv {{}_i\mathit{\rho }}^{-\frac{1}{2}}\left[\eta ;\kappa \right]{{\cancel{\mathrm{\Psi }}}^{˜}}_{\mathbf{c},\tilde{n}}\left[\eta ;h_o;\kappa |\mathbf{†},2\mathrm{j}\right]$$E80

$$\times \left\{\mathit{ld}{{\cancel{\mathrm{\Psi }}}^{˜}}_{\mathbf{c},\tilde{n}}\left[\eta ;h_o;\kappa |\mathbf{†},2\mathrm{j}\right]-\mathit{ld}{{}_i\cancel{\psi }}_{\mathbf{†},2\mathrm{j}}\left[\eta ;h_o;\kappa \right]\right\}$$

satisfies the DBCs at $\eta =\pm \infty$_, and therefore, it must coincide with one of eigenfunctions of SLE (49), in contradiction with the assumption that eigenvalue (76) differs from any eigenvalue of the original Sturm-Liouville problem.

Setting

$${{}_i\cancel{\psi }}_{\mathbf{†},\mathrm{m}}\left[\eta ;h_o;\kappa \right]=\cancel{\psi }\left[\eta ;{{}_i\lambda }_{\mathbf{†},\mathrm{m}}\right]{\Re }_m^{\left({{}_i\lambda }_{\mathbf{†},\mathrm{m}}\right)}\left[\eta \right],$$E81

where

$$\cancel{\psi }\left[\eta ;\lambda \right]\equiv {\left({\eta }^2+1\right)}^{-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}\varphi \left[\eta ;\lambda \right]=\sqrt[4]{{\eta }^2+1}{\left(1-\mathit{i\eta }\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\lambda }{\left(1+\mathit{i\eta }\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\lambda \ast }$$E82

one can directly verify that eigenfunctions (75) have a quasi-rational form

$${{}_i\cancel{\mathrm{\Psi }}}_{\mathbf{c},n+1}\left[\eta ;h_o;\kappa |\mathbf{†},2\mathrm{j}\right]={{}_i\cancel{\psi }}_{\left[\eta ;{{}_i\lambda }_{\mathbf{c},n}\right]}\frac{{{{}_i\mathrm{D}}_n}_{+2\mathrm{j}+1}\left[\eta ;h_o;\kappa |\mathbf{†},2\mathrm{j};n\right]}{\sqrt{T_2\left[\eta ;\kappa \right]}{\Re }_{2\mathrm{j}}^{\left({{}_i\lambda }_{\mathbf{†},2\mathrm{j}}\right)}\left(\eta \right)},$$E83

where the polynomial numerator of the fraction on the right represents the Routh-seed ($\mathcal{R}$S) “polynomial determinant” (PD)

$${{{}_i\mathrm{D}}_n}_{+2\mathrm{j}+1}\left[\eta ;h_o;\kappa |\mathbf{†},2\mathrm{j};n\right]=\left|\underset{{{}_{i}S}_{m+1}^{\left({{}_i\lambda }_{\mathbf{†},2\mathrm{j}}\right)}\left[\eta \right]{{}_{i}S}_{n+1}^{\left({{}_i\lambda }_{\mathbf{c},n}\right)}\left[\eta \right]}{\overset{{\Re }_m^{\left({{}_i\lambda }_{\mathbf{†},2\mathrm{j}}\right)}\left[\eta \right]{\Re }_n^{\left({{}_i\lambda }_{\mathbf{c},n}\right)}\left[\eta \right]}{}}\right|,$$E84

with ${{}_{i}S}_{m+1}^{\left(\lambda \right)}\left[\eta \right]$ standing for the “$\mathcal{R}$S polynomial supplement” [49].

$${{}_{i}S}_{m+1}^{\left(\lambda \right)}\left[\eta \right]\equiv \left({\eta }^2+1\right){\cancel{\psi }}^{-1}\left[\eta ;\lambda \right]\frac{d}{\mathit{d\eta }}\left({\cancel{\psi }}^{-1}\left[\eta ;\lambda \right]{\Re }_m^{\left(\lambda \right)}\left[\eta \right]\right)$$E85

$$=\left[\left(\mathit{Re}\lambda +1\right)\eta +\mathit{Im}\lambda \right]{\Re }_m^{\left(\lambda \right)}\left[\eta \right]+\left({\eta }^2+1\right){{\Re }^{•}}_m^{\left(\lambda \right)}\left[\eta \right].$$E86

It will be demonstrated in next section that PDs (84) satisfy the Heine-type differential equations with degree-dependent linear coefficient functions so we refer to its polynomial solutions as “$\mathcal{R}$S Heine polynomials” despite the fact that they belong to different sets of (n + 1)(n + 2)/2 Heine polynomials depending on the choice of n.

5. $\mathcal{R}$S Heine polynomials

The gauge transformation

$${}_i\mathrm{\Phi }\left[\eta ;h_o;\kappa ;\epsilon |\mathbf{†},2\mathrm{j}\right]=\varphi \left[\eta ;\kappa ;{{}_i\lambda }_{\mathbf{†},2\mathrm{j}};\lambda \left(h_o;\kappa ;\epsilon \right)\right]\times F\left[\eta ;h_o;\kappa ;\epsilon |\mathbf{†},2\mathrm{j}\right],$$E87

where

$$\lambda \left(h_o;\kappa ;\epsilon \right)\equiv \sqrt{h_o+1+4\left(\kappa -1\right)\epsilon }$$E88

is the root with the positive real part

$$\mathit{Re}\lambda \left(h_o;\kappa ;\epsilon \right)>0$$E89

and

$$\varphi \left[\eta ;\kappa ;{{}_i\lambda }_{\mathbf{†},2\mathrm{j}};\lambda \right]=\frac{\varphi \left[\eta ;-\lambda \right]}{\sqrt{T_2\left[\eta ;\kappa \right]}{\Re }_{2\mathrm{j}}^{\left({{}_i\lambda }_{\mathbf{†},2\mathrm{j}}\right)}\left(\eta \right)},$$E90

with the numerator defined via (82), converts RCSLE (60) into the ODE

$${\mathbf{D}}_{\mathbf{†},2\mathrm{j}}\left[\eta ;h_o;\kappa ;\epsilon \right]F\left[\eta ;h_o;\kappa ;\epsilon |\mathbf{†},2\mathrm{j}\right]=0,$$E91

where

$${\mathbf{D}}_{\mathbf{†},2\mathrm{j}}\left[\eta ;h_o;\kappa ;\epsilon \right]\equiv A_{2\mathrm{j}+4}\left[\eta ;h_o;\kappa |\mathbf{†},2\mathrm{j}\right]\frac{d^2}{d{\eta }^2}$$E92

$$+2B_{2\mathrm{j}+3}\left[\eta ;h_o;\kappa ;\epsilon |\mathbf{†},2\mathrm{j}\right]\frac{d}{\mathit{d\eta }}+C_{2\mathrm{j}+2}\left[\eta ;h_o;\kappa ;\epsilon |\mathbf{†},2\mathrm{j}\right]$$

is the second-order differential operator with polynomial coefficient functions, namely

$$A_{2\mathrm{j}+4}\left[\eta ;h_o;\kappa |\mathbf{†},2\mathrm{j}\right]\equiv T_2\left[\eta ;\kappa \right]\left({\eta }^2+1\right){\Re }_{2\mathrm{j}}^{\left({{}_i\lambda }_{\mathbf{†},2\mathrm{j}}\right)}\left[\eta \right],$$E93

and

$$B_{2\mathrm{j}+3}\left[\eta ;h_o;\kappa ;\epsilon |\mathbf{†},2\mathrm{j}\right]\equiv T_2\left[\eta ;\kappa \right]\left({\eta }^2+1\right){\Re }_{2\mathrm{j}}^{\left({{}_i\lambda }_{\mathbf{†},2\mathrm{j}}\right)}\left[\eta \right]$$E94

$$\times \left(\frac{1-\lambda \left(h_o;\kappa ;\epsilon \right)}{2\left(\eta -i\right)}+c.c.-\sum _{l=1}^{2\mathrm{j}}\frac{1}{\eta -{\eta }_{2\mathrm{j};l}\left({{}_i\lambda }_{\mathbf{†},2\mathrm{j}}\right)}-\frac{\eta }{T_2\left[\eta ;\kappa \right]}\right)$$

with ${\eta }_{2\mathrm{j};l}\left(\lambda \right)$ standing for $2\mathrm{j}$ zeros of the Routh polynomial ${\Re }_{2\mathrm{j}}^{\left(\lambda \right)}\left[\eta \right]$. The explicit form of the free-term of ODE (91) is nonessential in the current context, and we simply refer the reader to [3] for the specifics. The only important detail is that the polynomial coefficient function (94) is energy-dependent, and as a result, the polynomial coefficient function of the first derivative in Heine-type ODE (95) below depends on the polynomial degree.

Making solution (87) at $\epsilon ={{}_i\epsilon }_{\mathbf{c},n}\left(h_o;\kappa \right)$ coincide with eigenfunction (75) shows that PD (84) satisfies the Heine-type ODEs

$${\mathbf{D}}_{\mathbf{†};2\mathrm{j}}\left[\eta ;h_o;\kappa ;{{}_i\epsilon }_{\mathbf{c},n}\left(h_o;\kappa \right)\right]{{{}_i\mathrm{D}}_n}_{+2\mathrm{j}+1}\left[\eta ;h_o;\kappa |\mathbf{†},2\mathrm{j};n\right]=0.$$E95

The sequences of Gauss-seed Heine polynomials turn into finite or infinite sequences of EOPs if the ExpDiffs for poles in the finite plane are energy-independent [43], and as a result, the corresponding Liouville potential belongs to Group A of translationally shape-invariant (TSI) potentials in Odake-Sasaki’s [44, 45] classification scheme. One of such “exceptional” cases will be discussed in next section.

Note that

$$\lambda \left(h_o;\kappa ;{{}_i\epsilon }_{\mathbf{d},2\mathrm{j}}\right)={{}_i\lambda }_{\mathbf{d},2\mathrm{j}}>0$$E96

which implies that the free term of ODE (91) with $\mathbf{†}=\mathbf{d}$ vanishes at $\epsilon ={{}_i\epsilon }_{\mathbf{d},2\mathrm{j}}$, and therefore, the ODE has a constant solution at this energy. The polynomial sequence specifying eigenfunctions of prime SLE (61) solved under the DBCs thus starts from a constant but lacks $2\mathrm{j}$ polynomials of higher degrees. It will we proven in Section 7 that these polynomials turn into EOPs at κ = 1.

6. Translational form-invariance of RRef CSLE with simple pole density function

In the limit κ → 1, the density function

$${{}_i\mathit{\rho }}_{\diamond }\left[\eta \right]\equiv {}_i\mathit{\rho }\left[\eta ;1\right]=\frac{1}{{\eta }^2+1}$$E97

has simple poles at ±i. As a result of such a very specific choice of the density function the corresponding CSLE

$$\left\{\frac{d^2}{d{\eta }^2}+{{}_{i}I}^o\left[\eta ;h_o\right]+\epsilon {{}_i\mathit{\rho }}_{\diamond }\left[\eta \right]\right\}{}_i\mathrm{\Phi }\left[\eta ;h_o;1;\epsilon \right]=0$$E98

becomes TFI [46], that is, it has two basic solutions

$${{}_i\varphi }_{\pm ,0}\left[\eta ;{\lambda }_o\right]={{}_i\mathit{\rho }}_{\diamond }^{-\frac{1}{2}}\left[\eta \right]{\left(1+\mathit{i\eta }\right)}^{\pm \frac{1}{2}{\lambda }_o}{\left(1-\mathit{i\eta }\right)}^{\pm \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.{\lambda }_o^{\ast }}$$E99

which satisfy the generic translational form-invariance condition

$${{}_i\varphi }_{-,0}\left[\eta ;a+1+\mathit{ib}\right]{{}_i\varphi }_{+;0}\left[\eta ;a+\mathit{ib}\right]={{}_i\mathit{\rho }}_{\diamond }^{-\frac{1}{2}}\left[\eta \right]\equiv {{}_i\sigma }^{\frac{1}{2}}\left[\eta \right],$$E100

where we use labels “+” and “-” instead of $\mathbf{d}$ and ${\mathbf{d}}^{\prime }$ (or $\mathbf{c}$) accordingly and also define the TFI parameters $a$ and $b$ as the real and imaginary parts of the square root

$${\lambda }_o\equiv a+\mathit{ib}=\sqrt{h_o+1}\left(\mathit{Re}{\lambda }_o>0\right).$$E101

It is remarkable that the cubic term in quartic eq. (41) disappears so the latter equation turns into the quadratic equation

$${\lambda }_{\pm ,m;R}^4-\left(a^2-b^2\right){\lambda }_{\pm ,m;R}^2-a^2{b}^2=0$$E102

with respect to ${\lambda }_{\pm ,m;R}^2$ [21]. The crucial point is that the coefficients of this quadratic equation are independent of m, and as a result, the indexes of the Routh polynomials in the right-hand side of (42) and (43) become independent of the polynomial degree m. As originally pointed to by the author [43], the energy independence of the ExpDiffs for the poles in the finite complex plane is the necessary and sufficient condition for the RLDTs of the given RCSLE to be quantized in terms of EOPs. Concurrently with [43], Odake and Sasaki [44, 45] grouped all the TSI potentials of this kind into the so-called “Group A” associated with TFI SLEs with energy-independent ExpDiffs for the poles in the finite complex plane. The same year, Quesne [24] scrupulously studied rational SUSY partners of the Scarf II potential (the Gendenshtein potential [52, 53] in our terms) speculating that there exist nodeless FFs of type $\mathbf{d}$, which can be thereby used to construct finite sequences of EOPs formed by RDTs of R-Routh polynomials (referred to by her simply as “Romanovski polynomials”). Her conjecture stimulated our intense interest in this issue [21].

Keeping in mind that the sought-for root ${{}_i\lambda }_{\pm ,\mathrm{m};R}^2$ of the quadratic equation must be positive, one finds

$${{}_i\lambda }_{\pm ,\mathrm{m};R}^2=a^2,$$E103

that is,

$${{}_i\lambda }_{\pm ,\mathrm{m};R}=\pm a,$$E104

where we use labels “+” and “-” instead of $\mathbf{d}$ and $\mathbf{d}$‘(or $\mathbf{c}$) accordingly. Substituting.

(104) into (47) thus gives

$${{}_i\cancel{\rho }}_{\infty ;\pm ,m}=\mp a-m-1$$E105

so (37) takes the elementary form:

$${{}_i\epsilon }_{\pm ,m}\left(a\right)=-{\left(\pm a+m+\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right)}^2$$E106

Since the ExpDiffs for the poles of CSLE (98) in the finite complex plane are energy-independent, it is the CSLE of Group A [45, 46], and as a result, ChExps of q-RSs (42) and (43) for these poles become independent of the polynomial degree:

$${{}_i\varphi }_{\pm ,\mathrm{m}}\left[\eta ;{\lambda }_o\right]={{}_i\mathrm{C}}_{\pm ,m}{{}_i\varphi }_{\pm ,0}\left[\eta ;{\lambda }_o\right]{\Re }_m^{\left(\pm {\lambda }_o\right)}\left[\eta \right],$$E107

where the constant factors [3].

$${{}_{i}C}_m=2^{m}m!$$E108

are selected in such a way that q-RSs (107) satisfy the generic ladder relations

$${}_{i}w\left[\eta ;a,b|\mp ,0;\pm ,m\right]={{}_i\varphi }_{\pm ,m+1}\left[\eta ;a\mp 1,b\right]/{{}_i\varphi }_{\pm ,0}\left[\eta ;a\mp 1,b\right]$$E109

derived by us [46] for TFI SLEs, with

$${}_{i}w\left[\eta ;a,b|\pm ,m;\mp ,m^{\prime }\right]\equiv W\left\{{{}_i\varphi }_{\pm ,m}\left[\eta ;a,b\right],{{}_i\varphi }_{\mp ,m^{\prime }}\left[\eta ;a,b\right]\right\}$$E110

The sequences start from basic solutions (99) respectively.

The gauge transformations

$${}_i\mathrm{\Phi }\left[\eta ;a+\mathit{ib};\epsilon \right]\equiv {}_i\mathrm{\Phi }\left[\eta ;{\left(a+\mathit{ib}\right)}^2-1;1;\epsilon \right]={{}_i\varphi }_{\pm ;0}\left[\eta ;a+\mathit{ib}\right]{{}_{i}F}_{\pm }\left[\eta ;a+\mathit{ib};\epsilon \right]$$E111

convert CSLE (98) into a pair of Bochner-type eigenequations

$$\left\{\left({\eta }^2+1\right)\frac{d^2}{d{\eta }^2}+{{}_i\tau }_{\pm }[\eta ;a,b]\frac{d}{\mathit{d\eta }}+\epsilon -{{}_i\epsilon }_{\pm ,0}\left(a\right)\right\}{{}_{i}F}_{\pm }\left[\eta ;a+\mathit{ib};\epsilon \right]=0,$$E112

where

$${{}_i\tau }_{\pm }\left[\eta ;a,b\right]\equiv 2\mathit{ld}{{}_i\varphi }_{\pm ,0}\left[\eta ;a+ib\right]=2\left(1\pm a\right)\eta \pm 2b.$$E113

Taking into account that

$${{}_i\epsilon }_{\pm ,m}\left(a\right)-{{}_i\epsilon }_{\pm ,0}\left(a\right)=m\left(1\pm 2a\right)$$E114

and comparing (112) and (113) with (9.9.5) in [14], we find that eigenequations (112) have polynomial solutions at energies (106):

$$\left\{\left({\eta }^2+1\right)\frac{d^2}{d{\eta }^2}+{{}_i\tau }_{\pm }[\eta ;a,b]\frac{d}{\mathit{d\eta }}+m\left(1\pm 2a\right)\right\}{P}_n\left(x;b,\mp a-1\right)=0.$$E115

The basic solution (99) is thus nothing but constant solution of these eigenequations converted back via gauge transformation (111). It is also worth mentioning that the solutions ${}_i\mathrm{R}\left[\eta ;\mp a,\pm b\right]$ of Routh ODE (6) written as

$$\mathit{ld}{{}_i\mathrm{R}}^{\pm }\left[\eta ;a,b\right]={{}_i\tau }_{\pm }\left[\eta ;a,b\right]/\left({\eta }^2+1\right)$$E116

coincide with the squares of basic solutions (99).

It is crucial all q-RSs (107) with the upper index lie below the lowest-energy level:

$${{}_i\epsilon }_{+,m}\left(a\right)=-{\left(a+m+\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right)}^2<{{}_i\epsilon }_{-,0}\left(a\right)=-{\left(a-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right)}^2$$E117

and therefore, according to the disconjugacy theorem, any Routh polynomial ${\Re }_{\mathrm{m}}^{\left(a+\mathit{ib}\right)}\left[\eta \right]$ of an even (odd) degree m does not have real roots (has one and only one real root) if the real part of its index is positive. The latter assertion also holds for Routh polynomials ${\Re }_m^{\left(-a-\mathit{ib}\right)}\left[\eta \right]$ provided that

$${{}_i\epsilon }_{-,m}\left(a\right)=-{\left(m+\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.-a\right)}^2<{{}_i\epsilon }_{-,0}\left(a\right),$$E118

that is, if $m>2a-1>0$.

7. Quantization of rational Darboux transforms of Gendenshtein potential by finite sequences of EOPs

The RLDT with the nodeless FF ${{}_i\varphi }_{\pm ,2\mathrm{j}}\left[\eta ;{\lambda }_o\right]$ brings us to the RCSLE

$$\left\{\frac{d^2}{d{\eta }^2}+{{}_{i}I}^o\left[\eta ;{\lambda }_o|\pm ,2\mathrm{j}\right]+\epsilon {{}_i\mathit{\rho }}_{\diamond }\left[\eta \right]\right\}{}_i\mathrm{\Phi }\left[\eta ;{\lambda }_o;\epsilon |\pm ,2\mathrm{j}\right]=0.$$E119

Representing the given RefPF

$${{}_{i}I}^o\left[\eta ;{\lambda }_o|\pm ,2\mathrm{j}\right]={{}_{i}I}^o\left[\eta ;{\lambda }_o\right]+2\sqrt{{{}_i\mathit{\rho }}_{\diamond }\left[\eta \right]}\frac{d}{\mathit{d\eta }}\frac{\mathit{ld}{{}_i\varphi }_{\pm ,2\mathrm{j}}\left[\eta ;{\lambda }_o\right]}{\sqrt{{{}_i\mathit{\rho }}_{\diamond }\left[\eta \right]}}+\mathrm{J}\left\{{{}_i\mathit{\rho }}_{\diamond }\left[\eta \right]\right\}$$E120

where the so-called [49] “universal correction” is defined via the generic formula

$$\mathrm{J}\left\{\mathrm{f}\left[\eta \right]\right\}\equiv \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\sqrt{\mathrm{f}\left[\eta \right]}\frac{d}{\mathit{d\eta }}\frac{\mathit{ld}\mathrm{f}\left[\eta \right]}{\sqrt{\mathrm{f}\left[\eta \right]}},$$E121

one finds [24].

$${{}_{i}I}^o\left[\eta ;{\lambda }_o|\pm ,2\mathrm{j}\right]={{}_{i}I}^o\left[\eta ;|{\lambda }_o\pm 1|\right]+2l\dot{d}{\Re }_{2\mathrm{j}}^{\left(\pm {\lambda }_o\right)}\left[\eta \right]+\frac{2\eta }{{\eta }^2+1}\mathit{ld}{\Re }_{2\mathrm{j}}^{\left(\pm {\lambda }_o\right)}\left[\eta \right]$$E122

that is, the RLDTs in question change by ±1 the ExpDiffs for the poles at +i and -i.

Let us set

$${{}_i\mathrm{D}}_{m+2\mathrm{j}+1}^{\left(-{\lambda }_o\right)}\left[\eta |+,2\mathrm{j};-,m\right]=\left|\underset{{{}_i\mathrm{S}}_{2\mathrm{j}+1}^{\left({\lambda }_o\right)}\left[\eta \right]{{}_i\mathrm{S}}_{m+1}^{\left(-{\lambda }_o\right)}\left[\eta \right]}{\overset{{\Re }_{2\mathrm{j}}^{\left({\lambda }_o\right)}\left[\eta \right]{\Re }_m^{\left(-{\lambda }_o\right)}\left[\eta \right]}{}}\right|,$$E123

where m is an arbitrary non-negative integer, and demonstrate that the polynomials in question, coupled with a constant, form an “exceptional” [54] DPS (X-DPS), which lacks $2\mathrm{j}$ sequential polynomial degrees starting from 1 and thereby does not obey the prerequisites of the Bochner theorem [9]. To prove this assertion, first note that the power exponents of the q-RS

$${}_i\varphi \left[\eta ;{\lambda }_o|+,2\mathrm{j};-,m\right]={{}_i\varphi }_{\mathbf{c},0}\left[\eta ;{\lambda }_o+1\right]\frac{{{}_i\mathrm{D}}_{m+2\mathrm{j}+1}^{\left(-{\lambda }_o\right)}\left[\eta |+,2\mathrm{j};-,m\right]}{{\Re }_{2\mathrm{j}}^{\left({\lambda }_o\right)}\left(\eta \right)}$$E124

for the singular points ± i coincide with the ChExps for the respective poles of RCSLE (119) and therefore the numerator of the PF in the right-hand side of (124) may not have zeros at these singular points (at least as far as a is not a positive integer). Rewriting (124) as

$${}_i\varphi \left[\eta ;{\lambda }_o|+,2\mathrm{j};-,m\right]={{}_i\varphi }_{\mathbf{c},0}\left[\eta ;{\lambda }_o|+,2\mathrm{j}\right]{{}_i\mathrm{D}}_{m+2\mathrm{j}+1}^{\left(-{\lambda }_o\right)}\left[\eta |+,2\mathrm{j};-,m\right],$$E125

and making gauge transformation (87) but now with the energy-independent gauge function

$${{}_i\varphi }_{\mathbf{c},0}\left[\eta ;{\lambda }_o|+,2\mathrm{j}\right]=\frac{{{}_i\varphi }_{\mathbf{c},0}\left[\eta ;{\lambda }_o+1\right]}{{\Re }_{2\mathrm{j}}^{\left({\lambda }_o\right)}\left[\eta \right]}$$E126

representing the lowest-energy eigenfunction of RCSLE (119), we come to the Bochner-type equation with energy-independent polynomial coefficient functions of both first and second derivatives. As a direct corollary of this result, we conclude that PDs (123) satisfy the Bochner-type equation

$${\mathbf{D}}_{+;2\mathrm{j}}\left[\eta ;{\lambda }_o;{{}_i\epsilon }_{-,m}\left({\lambda }_o\right)\right]{{}_i\mathrm{D}}_{m+2\mathrm{j}+1}^{\left(-{\lambda }_o\right)}\left[\eta |+,2\mathrm{j};-,m\right]=0,$$E127

where

$${\mathbf{D}}_{+;2\mathrm{j}}\left[\eta ;{\lambda }_o;\epsilon \right]\equiv \left({\eta }^2+1\right){\Re }_{2\mathrm{j}}^{\left({\lambda }_o\right)}\left[\eta \right]\frac{d^2}{d{\eta }^2}+{{}_i\tau }_{2\mathrm{j}+1}\left[\eta ;{\lambda }_o|+,2\mathrm{j}\right]\frac{d}{\mathit{d\eta }}+C_{2\mathrm{j}}\left[\eta ;{\lambda }_o;\epsilon |+,2\mathrm{j}\right]$$E128

is the second-order differential operator with the coefficient function of the first derivative

$${{}_i\tau }_{2\mathrm{j}+1}\left[\eta ;a+\mathit{ib}|+,2\mathrm{j}\right]={{}_i\tau }_{-}\left[\eta ;a+1,b\right]{\Re }_{2\mathrm{j}}^{\left(a+\mathit{ib}\right)}\left[\eta \right]$$

$$-2\left({\eta }^2+1\right){{\Re }^{•}}_{2\mathrm{j}}^{\left(a+\mathit{ib}\right)}\left[\eta \right].$$E129

The important new feature of differential operator (128), compared with (92), is that polynomial coefficient (129) is energy-independent and therefore the free term of differential operator (128) necessarily coincides with one of the “characteristic polynomials” [55] of the Heine equation under consideration while PD (123) represents the corresponding Heine polynomial. It directly follows from our choice (126) of the gauge function that

$$C_{2\mathrm{j}}\left[\eta ;{\lambda }_o;{{}_i\epsilon }_{+,2\mathrm{j}}\left({\lambda }_o\right)|+,2\mathrm{j}\right]=0.$$E130

One can directly verify that the q-RSs of the associate prime equation

$${{}_i\cancel{\psi }}_{\mathbf{c},\mathrm{n}+1}\left[\eta ;{\lambda }_o|+,2\mathrm{j}\right]={{}_i\cancel{\psi }}_{\mathbf{c},0}\left[\eta ;{\lambda }_o+1\right]\frac{{{}_i\mathrm{D}}_{n+2\mathrm{j}+1}^{\left(-2b,1-a\right)}\left[\eta |+,2\mathrm{j};-,n\right]}{{\Re }_{2\mathrm{j}}^{\left({\lambda }_o\right)}\left(\eta \right)},$$E131

where

$${{}_i\mathrm{D}}_{n+2\mathrm{j}+1}^{\left(-2b,1-a\right)}\left[\eta |+,2\mathrm{j};-,n\right]\equiv {{}_i\mathrm{D}}_{n+2\mathrm{j}+1}^{\left(-a-\mathit{ib}\right)}\left[\eta |+,2\mathrm{j};-,n\right]$$E132

$$=\left|\underset{{{}_i\mathrm{S}}_{2\mathrm{j}+1}^{\left(a+\mathit{ib}\right)}\left[\eta \right]{{}_i\mathrm{S}}_{n+1}^{\left(-a-\mathit{ib}\right)}\left[\eta \right]}{\overset{{\Re }_{2\mathrm{j}}^{\left(a+\mathit{ib}\right)}\left[\eta \right]R_n^{\left(-2b,1-a\right)}\left(\eta \right)}{}}\right|\text{for}0\le n<a-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.,$$

vanish at infinity and thereby the RDT of the n^th eigenfunction of RRef CSLE (98) is itself the eigenfunction of RCSLE (119) with the eigenvalue ${{}_i\epsilon }_{\mathbf{c},n}\left({\lambda }_o\right)$. Since the lowest-energy eigenfunction is represented by nodeless solution (126) with the eigenvalue ${{}_i\epsilon }_{+,2\mathrm{j}}\left({\lambda }_o\right)$, there is a jump in the degrees of polynomials forming a complete set of $⌈a+\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.⌉$ eigenfunctions of RCSLE (119).

Keeping in mind that eigenfunctions (131) are orthogonal with the weight

$${}_i\cancel{w}\left[\eta \right]={\left({\eta }^2+1\right)}^{-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$$E133

PDs (132) must be orthogonal with the weight

$$\mathrm{W}\left[\eta ;{\lambda }_o|+,2\mathrm{j}\right]\equiv \frac{{{}_i\cancel{\psi }}_{\mathbf{c},0}^2\left[\eta ;{\lambda }_o+1\right]}{{\left|{\Re }_{2\mathrm{j}}^{\left(-{\lambda }_o\right)}\left(\eta \right)\right|}^2}{}_i\cancel{w}\left[\eta \right].$$E134

All PDs (132) also obey the linear orthogonality condition:

$$\underset{-\infty }{\overset{-\infty }{\int }}{{}_i\mathrm{D}}_{n+2\mathrm{j}+1}^{\left(-2b,1-a\right)}\left[\eta |+,2\mathrm{j};-,n\right]\mathrm{W}\left[\eta ;{\lambda }_o|+,2\mathrm{j}\right]\mathit{d\eta }=0$$E135

for 0 ≤ n ≤ $⌊a-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.⌋$.

We thus deal with a finite sequence of EOPs, which starts from a constant but lacks $2\mathrm{j}$ polynomials of higher degree starting from the first-degree polynomial so the prerequisites of the Bochner theorem [9] do not hold.

Coming back to RCSLE (119), first note that its q-RSs

$${}_i\varphi \left[\eta ;{\lambda }_o|-,2\mathrm{j};\mathrm{n}\right]=\frac{W\{{{}_i\varphi }_{-,2\mathrm{j}}\left[\eta ;{\lambda }_o\right],{{}_i\varphi }_{\mathbf{c},n}\left[\eta ;{\lambda }_o\right]\}}{{{}_i\varphi }_{-,2\mathrm{j}}\left[\eta ;{\lambda }_o\right]},$$E136

where n is an arbitrary non-negative integer, can be expressed in terms of the Wronskians of two Routh polynomials:

$${{}_i\mathrm{W}}_{2\mathrm{j}+n-1}^{\left(-{\lambda }_o\right)}\left[\eta |2\mathrm{j},n\right]\equiv W\left\{{\Re }_{2\mathrm{j}}^{\left(-{\lambda }_o\right)}\left[\eta \right],{\Re }_n^{\left(-{\lambda }_o\right)}\left[\eta \right]\right\}$$E137

as follows

$${}_i\varphi \left[\eta ;{\lambda }_o|-,2\mathrm{j};\mathrm{n}\right]=-g\left[\eta ;{\lambda }_o|2\mathrm{j}\right]{{}_i\mathrm{W}}_{2\mathrm{j}+n-1}^{\left(-{\lambda }_o\right)}\left[\eta |n,2\mathrm{j}\right],$$E138

where

$$g\left[\eta ;{\lambda }_o|2\mathrm{j}\right]\equiv \frac{{}_i\varphi \left[\eta ;{\lambda }_o-1\right]}{{\Re }_{2\mathrm{j}}^{\left(-{\lambda }_o\right)}\left(\eta \right)}.$$E139

Since the power exponents of this solution for the singular points ± i coincide with the ChExps for the respective poles of RCSLE (119) polynomial Wronskian (137) may not have zeros at these singular points (again as far as the parameter $a$ is not a positive integer). As a direct corollary of this assertion, we conclude that the polynomial Wronskians in question must satisfy the Bochner-type differential equation with the polynomial coefficient of the first derivative independent of the polynomial degree and thereby form an X-DPS starting from the polynomial of degree $2\mathrm{j}-1$. In contrast with the X-DPS formed by PDs (123), the X-DPS composed of the polynomial Wronskians does not contain a polynomial of zero degree.

One can verify that the quasi-rational solution of the corresponding prime SLE,

$${}_i\cancel{\psi }\left[\eta ;{\lambda }_o|-,2\mathrm{j};n\right]=-{\left({\eta }^2+1\right)}^{-\frac{1}{4}}g\left[\eta ;{\lambda }_o|2\mathrm{j}\right]{{}_i\mathrm{W}}_{2\mathrm{j}+n-1}^{\left(-{\lambda }_o\right)}\left[\eta |n,2\mathrm{j}\right],$$E140

vanishes at infinity iff n ≤ $⌊a-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.⌋$ and therefore represents the (n + 1)-th eigenfunction

$${{}_i\cancel{\psi }}_{\mathbf{c},\mathrm{n}+1}\left[\eta ;a+\mathit{ib}|-,2\mathrm{j}\right]=-{\left({\eta }^2+1\right)}^{-\frac{1}{4}}g\left[\eta ;a+\mathit{ib}|2\mathrm{j}\right]{{}_i\mathrm{W}}_{n+2\mathrm{j}-1}^{\left(-2b,1-a\right)}\left[\eta \right]$$E141

where

$${{}_i\mathrm{W}}_{n+2\mathrm{j}-1}^{\left(-2b,1-a\right)}\left[\eta \right]\equiv W\left\{{\Re }_{2\mathrm{j}}^{\left(-{\lambda }_o\right)}\left[\eta \right],{\mathrm{R}}_n^{\left(-2b,1-a\right)}\left(\eta \right)\right\}.$$E142

Since eigenfunctions (141) are orthogonal with weight (133) polynomial Wronskians (142) must be orthogonal with the weight

$$\mathrm{W}\left[\eta ;a+\mathit{ib}|-,2\mathrm{j}\right]\equiv g^2\left[\eta ;a+\mathit{ib}|2\mathrm{j}\right]/\left({\eta }^2+1\right).$$E143

We thus deal with a finite sequence of EOPs, which starts from a polynomial of degree $2\mathrm{j}-1\ge 1$

$${{}_i\mathrm{W}}_{2\mathrm{j}-1}^{\left(-2b,1-a\right)}\left[\eta \right]=-{{\Re }^{•}}_{2\mathrm{j}}^{\left(-{\lambda }_o\right)}\left[\eta \right]$$E144

so the prerequisites of the Bochner theorem [9] do not hold. Since lowest-energy eigenfunction

$${{}_i\varphi }_{\mathbf{c},0}\left[\eta ;{\lambda }_o|-,2\mathrm{j}\right]=\frac{{\left(1-\mathit{i\eta }\right)}^{\frac{1}{2}{\lambda }_o}{\left(1+\mathit{i\eta }\right)}^{\frac{1}{2}{\lambda }_o^{\ast }}}{{\Re }_{2\mathrm{j}}^{\left(-{\lambda }_o\right)}\left[\eta \right]}$$E145

represents a solution of RCSLE (119) with the opposite ChExp (compared with higher-energy eigenfunctions), the given sequence of EOPs does not start from a constant in contrast with the finite EOP sequence composed of PDs.

8. Conclusions

The presented analysis points to the important distinction between Routh polynomials [1] and its finite orthogonal subset revealed by Romanovsky [17] a few decades later (with no connection to Routh’ paper ignored by mathematicians for more than a century). We refer to the latter subset as “Romanovski-Routh” polynomials (similarly to the terms “Romanovski-Jacobi” and “Romanovski-Bessel” polynomials suggested by Lesky [18, 19] for two other finite sequences of the orthogonal polynomials discovered by Romanovsky). It was shown that there are two infinite sequences of Routh polynomials without real roots (in contrast with R-Routh polynomials with all the roots located on the real axis). The FFs formed by polynomials from these sequences can be thus used to generate new rational Sturm-Liouville equations exactly solvable under the DBCs. This observation opens a novel area of applications for the Routh DPS—a little known real-field reduction of the complex Jacobi DPS left without proper attention for decades [12] in the shadow of its powerful sibling composed of real Jacobi polynomials.

The current analysis is focused solely on RRef CSLE (19) with the density function having two simple zeros on the opposite sides of the imaginary axis [20, 21]. (We refer the reader to [3] for the general case of the density function with two complex conjugated zeros away from the imaginary axis.) A special attention was given to the limiting case of the density function with the simple poles at ±i when the RRef CSLE becomes TFI. Based on the disconjugacy theorem [33, 34, 35, 36, 37], it was proved that each Routh polynomial ${\Re }_{2j}^{\left(a+\mathit{ib}\right)}\left[\eta \right]$ of even degree with a > 0 may not have real roots. Similarly all other Routh polynomials from this sequence may have only one (necessarily simple) real root. Any of the mentioned Routh polynomials of even degree can be used for constructing a finite EOP sequence [24], which starts from a constant and lacks as minimum two polynomials of the first and second degrees.

There also exists the second infinite family of the RDTs of R-Routh polynomials, which is composed of finite EOP sequences of Wronskians of Routh and R-Routh polynomials.

As originally proven in [46] and scrupulously examined in [56], one can construct the complete net of finite EOP sequences formed by rational Darboux-Crum transforms (RDCTs) of R-Routh polynomials using only the FFs composed of Wronskians of Routh polynomials ${\Re }_{2\mathrm{j}}^{\left(-a-\mathit{ib}\right)}\left[\eta \right]$ and “juxtaposed” [57, 58, 59] pairs of R-Routh polynomials. The RDTs of R-Routh polynomials using the FFs ${{}_i\varphi }_{-,2\mathrm{j}}\left[\eta ;{\lambda }_o\right]$ represent the simplest example of the finite EOP sequences constructed in such a way. On other hand, as discussed in detail in [3], the infinite sequence of RCSLEs obtained by means of the FFs ${{}_i\varphi }_{+,2\mathrm{j}}\left[\eta ;{\lambda }_o\right]$ can be alternatively constructed using $2\mathrm{j}$ seed functions ${{}_i\varphi }_{-,\mathrm{n}}\left[\eta ;{\lambda }_o\right]$ composed of R-Routh polynomials of degrees n = 1,…, $2\mathrm{j}$.

Acknowledgments

I am grateful to Mariana Kirchbach for numerous thoughtful comments on Routh’s paper, which notably shaped my current perception of the subject. Christiane Quesne’s conjecture on existence of Routh polynomials without real zeros played a crucial role in this research. I am especially thankful to Mourad Ismail for bringing my attention to quasi-definite orthogonality of Routh polynomials, which helped to rectify my terminology.