Pastro polynomials and Sobolev-type orthogonal polynomials on the unit circle based on a q-difference operator

References

• G.E. Andrews, R. Askey, and R. Roy, Special Functions, Encyclopedia of Mathematics and its Applications Vol. 71, Cambridge University Press, Cambridge, 1999.
   Google Scholar
• I. Area, E. Godoy, R.L. Lamblém, and A. Sri Ranga, Basic hypergeometric polynomials with zeros on the unit circle, Appl. Math. Comp. 225 (2013), pp. 622–630.
   Google Scholar
• R. Askey (ed.), Gabor Szegő: Collected Papers, Vol. 1, Contemporary Mathematics, Amer. Math. Soc., Providence, RI, 1982.
   Google Scholar
• R. Askey and M.E.H. Ismail, A generalization of ultraspherical polynomials, in Studies in Pure Mathematics, P. Erdös, ed., Birkhäuser, Basel, 1983, pp. 55–78.
   Google Scholar
• P. Baseilhac, L. Vinet, and A. Zhedanov, The q-Heun operator of big q-Jacobi type and the q-Heun algebra, Ramanujan J. 52 (2020), pp. 367–380.
   Google Scholar
• C. Bracciali, J.V. da Silva, and A. Sri Ranga, A class of Sobolev orthogonal polynomials on the unit circle and associated continuous dual Hahn polynomials: Bounds, asymptotics and zeros, J. Approx. Theory 268 (2021), Article No. 105604.
   Google Scholar
• A. Branquinho, A. Foulquié Moreno, F. Marcellán, and M.N. Rebocho, Coherent pairs of linear functionals on the unit circle, J. Approx. Theory 153(1) (2008), pp. 122–137.
   Google Scholar
• R. Chakrabarti and R. Jagannathan, A (p,q)-oscillator realization of two parameter quantum algebras, J. Phys. A: Math. Gen. 24 (1991), pp. L711–L718.
   Google Scholar
• T.S. Chihara, An Introduction to Orthogonal Polynomials, Mathematics and its Applications Series, New York, Gordon and Breach, 1978.
   Google Scholar
• K. Castillo, M.S. Costa, A. Sri Ranga, and D.O. Veronese, A Favard type theorem for orthogonal polynomials on the unit circle from a three term recurrence formula, J. Approx. Theory 184 (2014), pp. 146–162.
   Google Scholar
• M.S. Costa, E. Godoy, R.L. Lamblém, and A. Sri Ranga, Basic hypergeometric functions and orthogonal Laurent polynomials, Proc. Amer. Math. Soc. 140 (2012), pp. 2075–2089.
   Google Scholar
• T. Erdélyi, P. Nevai, J. Zhang, and J. Geronimo, A simple proof of ‘Favard's theorem’ on the unit circle, Atti Sem. Mat. Fis. Univ. Modena 39 (1991), pp. 551–556. Also in ‘Trends in functional analysis and approximation theory’ (Acquafredda di Maratea, 1989), 41–46, Univ. Modena Reggio Emilia, Modena, 1991.
   Google Scholar
• G. Gasper and M. Rahman, Basic Hypergeometric Series, 2nd ed., Encyclopedia of Mathematics and its Applications Vol. 96, Cambridge University Press, Cambridge, 2004.
   Google Scholar
• M. Hunziker, A. Martínez-Finkelshtein, T. Poe, B. Simanek, S. Poncelet-Darboux, S. Kippenhahn, and S. Szegő, Interactions between projective geometry, matrices and orthogonal polynomials, J. Math. Anal. Appl. 511(1) (2022), Paper No.126049, 35 pp.
   Google Scholar
• A. Iserles, P.E. Koch, S.P. Nørsett, and J.M. Sanz-Serna, On polynomials orthogonal with respect to certain Sobolev inner products, J. Approx. Theory 65 (1991), pp. 151–175.
   Web of Science ®Google Scholar
• M.E.H. Ismail, Classical and Quantum Orthogonal Polynomials in One Variable, Encyclopedia of Mathematics and its Applications Vol. 96, Cambridge Univ. Press, Cambridge, 2005. First paperback edition 2009.
   Google Scholar
• R. Koekoek, P.A. Lesky, and R.F. Swarttouw, Hypergeometric Orthogonal Polynomials and Their q-analogues, Springer Monographs in Mathematics, Springer-Verlag, Heidelberg, 2010.
   Google Scholar
• X. Li and R. Ranasinghe, Askey-Wilson operator on entire functions of exponential type, Proc. Amer. Math. Soc. 146 (2018), pp. 4283–4292.
   Google Scholar
• F. Marcellán and A. Sri Ranga, Sobolev orthogonal polynomials on the unit circle and coherent pairs of measures of the second kind, Results Math. 71 (2017), pp. 1127–1149.
   Google Scholar
• A. Martínez-Finkelshtein, B. Simanek, and B. Simon, Poncelet's theorem, paraorthogonal polynomials and the numerical range of compressed multiplication operators, Adv. Math. 349 (2019), pp. 992–1035.
   Google Scholar
• H.G. Meijer, Determination of all coherent pairs, J. Approx. Theory 89 (1997), pp. 321–343.
   Web of Science ®Google Scholar
• P.I. Pastro, Orthogonal polynomials and some q-beta integrals of Ramanujan, J. Math. Anal. Appl. 112 (1985), pp. 517–540.
   Google Scholar
• P.N. Sadjang, On the fundamental theorem of (p,q)-calculus and some (p,q)-Taylor formulas, Results Math. 73 (2018), Article No. 39.
   Google Scholar
• B. Simon, Orthogonal Polynomials on the Unit Circle. Part 1. Classical Theory, Amer. Math. Soc. Colloq. Publ. Vol. 54, Amer. Math. Soc., Providence, RI, 2005.
   Google Scholar
• B. Simon, Orthogonal Polynomials on the Unit Circle. Part 2. Spectral Theory, Amer. Math. Soc. Colloq. Publ. 54, Amer. Math. Soc., Providence, RI, 2005.
   Google Scholar
• L.J. Slater, Generalized Hypergeometric Functions, Cambridge Univ. Press, Cambridge, 1966.
   Google Scholar
• A. Sri Ranga, Szegő polynomials from hypergeometric functions, Proc. Amer. Math. Soc. 138 (2010), pp. 4259–4270.
   Google Scholar
• A. Sri Ranga, Orthogonal polynomials with respect to a family of Sobolev inner products on the unit circle, Proc. Amer. Math. Soc. 144 (2016), pp. 1129–1143.
   Google Scholar