The single-indexed exceptional Krawtchouk polynomials

References

• G.E. Andrews and R. Askey, Classical orthogonal polynomials, in Polynômes orthogonaux et applications, Springer, 1985, pp. 36–62.
   Google Scholar
• H. Bateman, Higher Transcendental Functions [volumes I–III], Vol. 1, McGraw-Hill Book Company, 1953.
   Google Scholar
• M. Bruschi, F. Calogero, and R. Droghei, Additional recursion relations, factorizations, and Diophantine properties associated with the polynomials of the Askey scheme, Advances in Mathematical Physics 2009 (2009).
   Google Scholar
• Y. Chen and M. Ismail, Hypergeometric origins of Diophantine properties associated with the Askey scheme, Proc. Am. Math. Soc. 138 (3) (2010), pp. 943–951.
   Google Scholar
• A.J. Durán, Exceptional Meixner and Laguerre orthogonal polynomials, J. Approx. Theory 184 (2014), pp. 176–208.
   Web of Science ®Google Scholar
• A.J. Durán, Exceptional orthogonal polynomials via Krall discrete polynomials, in Lectures on Orthogonal Polynomials Special Functions, Cambridge University Press, 2020. pp. 1.
   Google Scholar
• A.J. Durán and M.D. de la Iglesia, Constructing bispectral orthogonal polynomials from the classical discrete families of Charlier, Meixner and Krawtchouk, Constr. Approx. 41 (1) (2015), pp. 49–91.
   Google Scholar
• D. Gómez-Ullate, N. Kamran, and R. Milson, An extended class of orthogonal polynomials defined by a Sturm–Liouville problem, J. Math. Anal. Appl. 359 (1) (2009), pp. 352–367.
   Google Scholar
• D. Gómez-Ullate, N. Kamran, and R. Milson, An extension of Bochner's problem: Exceptional invariant subspaces, J. Approx. Theory 162 (5) (2010), pp. 987–1006.
   Google Scholar
• D. Gómez-Ullate, N. Kamran, and R. Milson, Exceptional orthogonal polynomials and the Darboux transformation, J. Phys. A Math. Theor. 43 (43) (2010), pp. 434016.
   Google Scholar
• M.E. Ismail, I. Nikolova, and P. Simeonov, Difference equations and discriminants for discrete orthogonal polynomials, Ramanujan J. 8 (4) (2005), pp. 475–502.
   Google Scholar
• V.I. Levenshtein, Krawtchouk polynomials and universal bounds for codes and designs in Hamming spaces, IEEE Trans. Inf. Theory 41 (5) (1995), pp. 1303–1321.
   Google Scholar
• H. Miki and S. Tsujimoto, A new recurrence formula for generic exceptional orthogonal polynomials, J. Math. Phys. 56 (3) (2015), pp. 033502.
   Google Scholar
• H. Miki, S. Tsujimoto, and L. Vinet, Classical and quantum walks on paths associated with exceptional Krawtchouk polynomials. J. Math. Phys. (2022), pp. 103502.
   Google Scholar
• S. Odake, New finite type multi-indexed orthogonal polynomials obtained from state-adding Darboux transformations, arXiv preprint arXiv:2201.02337.
   Google Scholar
• S. Odake and R. Sasaki, ‘Diophantine’ and factorisation properties of finite orthogonal polynomials in the Askey scheme, arXiv preprint arXiv:2207.14479.
   Google Scholar
• S. Odake and R. Sasaki, Another set of infinitely many exceptional ( Xl) Laguerre polynomials, Phys. Lett. B. 684 (2-3) (2010), pp. 173–176.
   Google Scholar
• S. Odake and R. Sasaki, Exceptional Askey–Wilson–type polynomials through Darboux–Crum transformations, J. Phys. A Math. Theor. 43 (33) (2010), pp. 335201.
   Google Scholar
• S. Odake and R. Sasaki, Multi-indexed Wilson and Askey–Wilson polynomials, J. Phys. A Math. Theor. 46 (4) (2013), pp. 045204.
   Google Scholar
• C. Quesne, Exceptional orthogonal polynomials, exactly solvable potentials and supersymmetry, J. Phys. A Math. Theor. 41 (39) (2008), pp. 392001.
   Google Scholar
• R. Sasaki, S. Tsujimoto, and Z. Alexei, Exceptional Laguerre and Jacobi polynomials and the corresponding potentials through Darboux-Crum transformations, J. Phys. A Math. Theor. 43 (31) (2010), pp. 315204.
   Google Scholar
• P.T. Yap, R. Paramesran, and S.H. Ong, Image analysis by Krawtchouk moments, IEEE. Trans. Image. Process. 12 (11) (2003), pp. 1367–1377.
   PubMed Web of Science ®Google Scholar