The algebra of recurrence relations for exceptional Laguerre and Jacobi polynomials

Durán, Antonio

doi:10.1090/proc/15267

The algebra of recurrence relations for exceptional Laguerre and Jacobi polynomials
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by Antonio J. Durán PDF

Proc. Amer. Math. Soc. 149 (2021), 173-188 Request permission

Abstract:

Exceptional Laguerre and Jacobi polynomials $p_n(x)$ are bispectral, in the sense that as functions of the continuous variable $x$, they are eigenfunctions of a second order differential operator and as functions of the discrete variable $n$, they are eigenfunctions of a higher order difference operator (the one defined by any of the recurrence relations they satisfy). In this paper, under mild conditions on the sets of parameters, we characterize the algebra of difference operators associated to the higher order recurrence relations satisfied by the exceptional Laguerre and Jacobi polynomials.

References

S. Bochner, Über Sturm-Liouvillesche Polynomsysteme, Math. Z. 29 (1929), no. 1, 730–736 (German). MR 1545034, DOI 10.1007/BF01180560
Niels Bonneux, Exceptional Jacobi polynomials, J. Approx. Theory 239 (2019), 72–112. MR 3892887, DOI 10.1016/j.jat.2018.11.002
Niels Bonneux and Arno B. J. Kuijlaars, Exceptional Laguerre polynomials, Stud. Appl. Math. 141 (2018), no. 4, 547–595. MR 3879969, DOI 10.1111/sapm.12204
Niels Bonneux and Marco Stevens, Recurrence relations for Wronskian Hermite polynomials, SIGMA Symmetry Integrability Geom. Methods Appl. 14 (2018), Paper No. 048, 29. MR 3802659, DOI 10.3842/SIGMA.2018.048
N. Bonneux and M. Stevens, Recurrence relations for Wronskian Laguerre polynomials, preprint, arXiv:1905.12312.
Guillermo P. Curbera and Antonio J. Durán, Invariance properties of Wronskian type determinants of classical and classical discrete orthogonal polynomials, J. Math. Anal. Appl. 474 (2019), no. 1, 748–764. MR 3912926, DOI 10.1016/j.jmaa.2019.01.078
J. J. Duistermaat and F. A. Grünbaum, Differential equations in the spectral parameter, Comm. Math. Phys. 103 (1986), no. 2, 177–240. MR 826863, DOI 10.1007/BF01206937
Antonio J. Durán, Exceptional Charlier and Hermite orthogonal polynomials, J. Approx. Theory 182 (2014), 29–58. MR 3195377, DOI 10.1016/j.jat.2014.03.004
Antonio J. Durán, Exceptional Meixner and Laguerre orthogonal polynomials, J. Approx. Theory 184 (2014), 176–208. MR 3218798, DOI 10.1016/j.jat.2014.05.009
Antonio J. Durán, Higher order recurrence relation for exceptional Charlier, Meixner, Hermite and Laguerre orthogonal polynomials, Integral Transforms Spec. Funct. 26 (2015), no. 5, 357–376. MR 3312610, DOI 10.1080/10652469.2015.1009455
Antonio J. Durán, Exceptional Hahn and Jacobi orthogonal polynomials, J. Approx. Theory 214 (2017), 9–48. MR 3588528, DOI 10.1016/j.jat.2016.11.003
Antonio J. Durán, Corrigendum to the papers on Exceptional orthogonal polynomials: J. Approx. Theory 182 (2014) 29–58, 184 (2014) 176–208 and 214 (2017) 9–48 [3195377; 3218798; 3588528], J. Approx. Theory 253 (2020), 105349, 5. MR 4073426, DOI 10.1016/j.jat.2019.105349
A. J. Durán, A proof of the Veselov Conjecture for segments, arXiv:2001.08464 [math.CA], 2020.
Antonio J. Durán and Mario Pérez, Admissibility condition for exceptional Laguerre polynomials, J. Math. Anal. Appl. 424 (2015), no. 2, 1042–1053. MR 3292716, DOI 10.1016/j.jmaa.2014.11.035
D. Dutta and P. Roy, Information entropy of conditionally exactly solvable potentials, J. Math. Phys. 52 (2011), no. 3, 032104, 7. MR 2814692, DOI 10.1063/1.3566977
G. Felder, A. D. Hemery, and A. P. Veselov, Zeros of Wronskians of Hermite polynomials and Young diagrams, Phys. D 241 (2012), no. 23-24, 2131–2137. MR 2998116, DOI 10.1016/j.physd.2012.08.008
F. R. Gantmacher, The theory of matrices. Vols. 1, 2, Chelsea Publishing Co., New York, 1959. Translated by K. A. Hirsch. MR 0107649
Ma Ángeles García-Ferrero, David Gómez-Ullate, and Robert Milson, A Bochner type characterization theorem for exceptional orthogonal polynomials, J. Math. Anal. Appl. 472 (2019), no. 1, 584–626. MR 3906391, DOI 10.1016/j.jmaa.2018.11.042
I. M. Gel′fand, M. M. Kapranov, and A. V. Zelevinsky, Discriminants, resultants, and multidimensional determinants, Mathematics: Theory & Applications, Birkhäuser Boston, Inc., Boston, MA, 1994. MR 1264417, DOI 10.1007/978-0-8176-4771-1
David Gómez-Ullate, Niky Kamran, and Robert Milson, An extension of Bochner’s problem: exceptional invariant subspaces, J. Approx. Theory 162 (2010), no. 5, 987–1006. MR 2610341, DOI 10.1016/j.jat.2009.11.002
David Gómez-Ullate, Yves Grandati, and Robert Milson, Rational extensions of the quantum harmonic oscillator and exceptional Hermite polynomials, J. Phys. A 47 (2014), no. 1, 015203, 27. MR 3146977, DOI 10.1088/1751-8113/47/1/015203
David Gomez-Ullate, Yves Grandati, and Robert Milson, Extended Krein-Adler theorem for the translationally shape invariant potentials, J. Math. Phys. 55 (2014), no. 4, 043510, 30. MR 3390608, DOI 10.1063/1.4871443
D. Gómez-Ullate, A. Kasman, A. B. J. Kuijlaars, and R. Milson, Recurrence relations for exceptional Hermite polynomials, J. Approx. Theory 204 (2016), 1–16. MR 3460215, DOI 10.1016/j.jat.2015.12.003
Alex Kasman, Darboux transformations and the bispectral problem, The bispectral problem (Montreal, PQ, 1997) CRM Proc. Lecture Notes, vol. 14, Amer. Math. Soc., Providence, RI, 1998, pp. 81–91. MR 1611024, DOI 10.1090/crmp/014/06
Choon-Lin Ho, Dirac(-Pauli), Fokker-Planck equations and exceptional Laguerre polynomials, Ann. Physics 326 (2011), no. 4, 797–807. MR 2771725, DOI 10.1016/j.aop.2010.12.006
I. Marquette and C. Quesne, Two-step rational extensions of the harmonic oscillator: exceptional orthogonal polynomials and ladder operators, J. Phys. A 46 (2013), no. 15, 155201, 14. MR 3043880, DOI 10.1088/1751-8113/46/15/155201
Hiroshi Miki and Satoshi Tsujimoto, A new recurrence formula for generic exceptional orthogonal polynomials, J. Math. Phys. 56 (2015), no. 3, 033502, 13. MR 3390927, DOI 10.1063/1.4914334
Satoru Odake, Recurrence relations of the multi-indexed orthogonal polynomials, J. Math. Phys. 54 (2013), no. 8, 083506, 18. MR 3135488, DOI 10.1063/1.4819255
Satoru Odake, Recurrence relations of the multi-indexed orthogonal polynomials. III, J. Math. Phys. 57 (2016), no. 2, 023514, 24. MR 3457929, DOI 10.1063/1.4941087
Satoru Odake and Ryu Sasaki, Infinitely many shape invariant potentials and new orthogonal polynomials, Phys. Lett. B 679 (2009), no. 4, 414–417. MR 2569488, DOI 10.1016/j.physletb.2009.08.004
Satoru Odake and Ryu Sasaki, Exactly solvable quantum mechanics and infinite families of multi-indexed orthogonal polynomials, Phys. Lett. B 702 (2011), no. 2-3, 164–170. MR 2822724, DOI 10.1016/j.physletb.2011.06.075
Sarah Post, Satoshi Tsujimoto, and Luc Vinet, Families of superintegrable Hamiltonians constructed from exceptional polynomials, J. Phys. A 45 (2012), no. 40, 405202, 10. MR 2974065, DOI 10.1088/1751-8113/45/40/405202
C. Quesne, Exceptional orthogonal polynomials, exactly solvable potentials and supersymmetry, J. Phys. A 41 (2008), no. 39, 392001, 6. MR 2439200, DOI 10.1088/1751-8113/41/39/392001
Ryu Sasaki, Satoshi Tsujimoto, and Alexei Zhedanov, Exceptional Laguerre and Jacobi polynomials and the corresponding potentials through Darboux-Crum transformations, J. Phys. A 43 (2010), no. 31, 315204, 20. MR 2665675, DOI 10.1088/1751-8113/43/31/315204
Axel Schulze-Halberg and Barnana Roy, Darboux partners of pseudoscalar Dirac potentials associated with exceptional orthogonal polynomials, Ann. Physics 349 (2014), 159–170. MR 3244615, DOI 10.1016/j.aop.2014.06.016
Vyacheslav Spiridonov, Luc Vinet, and Alexei Zhedanov, Bispectrality and Darboux transformations in the theory of orthogonal polynomials, The bispectral problem (Montreal, PQ, 1997) CRM Proc. Lecture Notes, vol. 14, Amer. Math. Soc., Providence, RI, 1998, pp. 111–122. MR 1611027, DOI 10.1090/crmp/014/09
R. Yadav, A. Khare, and B.P. Mandal, The scattering amplitude for one parameter family of shape invariant potentials related to Jacobi polynomials, Phys. Lett. B 732 (2013), 433–435.