1. Related Sequences of Multi-Index Orthogonal Eigenpolynomials
Single-Source Nets of Fuschian RCSLEs
with Common Simple-Poles Density Functions
and
Related Sequences of Multi-Index Orthogonal Eigenpolynomials
Gregory Natanson

2. Outline
Rational Darboux-Crum transformations (RDCTs) with almost- everywhere holomorphic (AEH) seed solutions
Single-source nets of Fuschian differential equations of Heine type
Algebraic Schrödinger equation (ASE) -- RCSLE written in self-adjoint form
Bochner eigenproblem for the simple-pole density function
Sequences of complex Jacobi-seed exceptional (JSX) eigenpolynomials
Dirichlet boundary conditions and resultant Gauss-seed exceptional (GSX) orthogonal eigenpolynomials
Possible choices of nodeless AEH factorization functions (FFs)
Concluding remarks

3. Real-field reduction of complex seed eigenpolynomials
Complex Jacobi polynomials (Kuijlaars, Martinez-Finkelshtein, and Orive 2005) Real Jacobi polynomials Routh polynomials (Routh 1885) Classical Jacobi (cl-Jacobi) Romanovski-Routh (R-Routh) and polynomials (Romanovski 1928, 1929) Romanovski-Jacobi (R-Jacobi) polynomials (Romanovski 1928, 1929) definite-parity R-Routh or Masjedjamei (M-) polynomials (Masjedjamei 2002)

4. Orthogonal Rational Darboux-Crum Transforms (ORDCTs)
Complex JSX eigenpolynomials
real JSX eigenpolynomials RSX eigenpolynomials
ORDCTs ORDCTs of R-Routh polynomials
of cl-Jacobi polynomials
and
R-Jacobi polynomials
ORDCTs of M-polynomials

5. Fuschian RCSLE with 3 singular points (including infinity)
with ray identifies (RIs)
PFr beam

6. GS solutions Cooper, Ginocchio and Khare (1987) used ground-energy eigenfunction
to construct a new exactly solvable rational potential
Gomez-Ullate, Kamran, Milson (2004) gave examples of nodeless GS solutions for some shape-invariant potentials:
Shape-invariant potentials quantized by orthogonal Xm-Jacobi polynomials (Quesne 2008, 2009; Odake and Sasaki 2009) represent the particular case with energy-independent exponent differences (ExpDiffs) at 1 (Natanson 2013)

7. Generalized Darboux-Crum transformations (GDCTs)
Schulze-Halberg (2012) extended the Darboux-Crum theorem to the generic SLE using the Crum Wroskian.
The author (2017) replaced Crum Wroskian (CW) for Krein determinant (KD) formed by seed solutions of the generic CSLE:
(fully got rid of the Liouville transformation in my arguments)
The simplifications for multi-indexed X-Jacobi and X-Laguerre orthogonal polynomials (Odake and Sasaki 2017) are apparently equivalent to use of KDs instead of CWs:
Natanson (2017)
where are solutions of the Schrödinger equation
expressed in terms of

8. Rational Darboux-Crum transformations (RDCTs) GDCTs using GS solutions as seed functions
canonical form of RSLEs
conditionally exactly
solvable by polynomials
PFr beam
RefPFr:
= polynomial fraction (PFr) with 2nd-order poles
All the singular points other than the fixed poles of the density function depend on the RIs specifying the forefather GRef PFr beam

9. Algebraic Schrödinger equation (ASE)
is the Schwarzian derivative
expressed in terms of
ASE is solved under the Dirichlet boundary conditions. It generally has apparent singularities at TP zeros (except shape-invariant potentials!) which disappear when ASE is converted to its canonical form. In the particular case of shape-invariant potentials it has been implicitly used in works of Odake and Sasaki (2011, 2013)

10. Major conjecture AEH factorization functions constructed using GS solutions,
are irregular at all apparent singular points.
Justifies my intuitive use of the term ‘almost-everywhere holomorphic’ (AEH) solution, with reference to Lambe and Ward (1934)!
Four gauge transformations of the RCSLE result in Heine-type equations:

11. Complex PFr beams with the first-order pole density function
and related complex Jacobi-seed X-eigenpolynomials
The gauge transformations result in four complex Fuschian eigenequations
of Bochner type with a polynomial of non-zero degree as the weight
n  n0 > 0
Gomez-Ullate, Kamran, and Milson (2010) over the real field
Krein determinant (KD)
polynomial determinant (PD)

12. Polynomial Determinants (PDs)
PDs satisfy eigenequations with irregular singular points at 1 !
Supplementary seed polynomials

13. Complex JSX eigenpolynomials
PDs satisfy eigenequations with irregular singular points at 1 !
Complex JSX eigenpolynomials
JSX eigenpolynomial
JSX eigenpolynomials satisfy Fuschian eigenequations!

14. Liouville transformation to Schrödinger equation (weighted Dirichlet boundary conditions to Dirichlet boundary conditions)
i) finite interval
by classical Jacobi polynomials
Darboux/Pöschl-Teller potential
ii) semi-axis
hyperbolic Pöschl-Teller potential
by Romanovski-Jacobi polynomials
iii) real axis
Gendenshtein (Scarf II) potential
and in particular
sech-squared potential by Masjedjamei polynomials
by Romanovski-Routh polynomials

15. Nodeless AEH factorization functions
- Pairs of juxtaposed eigenfunctions
infinite orthogonal sets of jux-Seed X-Jacobi eigenpolynomials (Odake and Sasaki 2013)
ii) finite orthogonal sets of jux-Seed X-R/J eigenpolynomials
iii) finite orthogonal sets of jux-Seed X-R/R eigenpolynomials
- In addition, I use nodeless regular seed (reg-S ) solutions (types a and b) to construct multi-step SUSY partners of the D/PT and h-PT potentials which are quantized by reg-S X-Jacobi and reg-S X-R/J eigenpolynomials;
- And also even seed solutions below the ground energy for A sech 2x

16. Future developments
i) Major conjecture of my paper:
RDC transform of the Jacobi-seed solution in question is irregular at all the apparent singular points
On other hand, Ho, Sasaki, and Takemura (2013) have discussed existence of AEH solutions of algebraic Schrödinger equation (with the D/PT potential) which are regular at apparent singularities
ii) Analysis of the limit-circle region
iii) Multi-step rational SUSY partners of the Gendenshtein
(Scarf II) potential with upper eigenstates expressible
in terms of RS orthogonal eigenpolynomials
And just a reminder

17. BACKUP SLIDES

18. Four energy-dependent gauge transformations and associated sequences of GS Heine equations with only 3 non-apparent singular points (including infinity)

19. Reduction to the real field
1st-order pole density function

20. Bochner theorem for an orthogonal subset of Routh polynomials
discovered by Romanovsky (1928, 1929) using
Rodrigues formula with Pearson’s distribution of type IV:
reclaimed by Pearson (1895) 11 years after Routh’paper
Romanovski-Routh polynomials:

21. Degrees of complex Heine eigenpolynomials
Real part of the Jacobi index of any JS polynomial must exceed if the RDCT decreases the appropriate ExpDiff by n -- the common deficiency of defining orthogonal exceptional polynomials via the Dirichlet boundary conditions imposed on solutions of the RSLE.
In the particular case of Xm-Jacobi polynomials
for
Gomez-Ullate, Marcellan, and Milson (2012)

22. Regular-Seed X-Jacobi orthogonal polynomials
(Odake and Sasaki 2011)
where and stand for sums of the polynomial degrees in the sets and , respectively. But my constraints
are less restrictive than
(Odake and Sasaki 2011)
The degrees of the JS Heine eigenpolynomials
where and
under constraint:

23. Multi-step symmetric rational SUSY partners of the sech2 potential
Multi-step symmetric rational SUSY partners of the sech2 potential with upper eigenfunctions expressible in terms of orthogonal e-RS Heine eigenpolynomials
The most important corollary of the aforementioned dualism is that the symmetric GRef potential is exactly quantized by both ultraspherical and definite-parity Romanovski-Routh polynomials (also referred to by us as ‘Masjedjamei polynomials’).
RDCTs of Masjedjamei polynomials are mutually orthogonal because the ExpDiffs at the singular points 1 are energy-independent along the sech2 potential curve
One can construct the symmetric rational potentials quantized by orthogonal definite-parity Heine eigenpolynomials using either pairs of the juxtaposed eigenfunctions or irregular even RS solutions below the ground energy level (Natanson 2015). An infinite net of such potentials quantized by Gegenbauer-seed (GS) Heine polynomials (in our terms) was recently discussed by Odake and Sasaki (2013b)