On a Family of Hypergeometric Sobolev Orthogonal Polynomials on the Unit Circle

Yıl 2020, Cilt: 3 Sayı: 2, 75 - 84, 01.06.2020

Sergey Zagorodnyuk

https://doi.org/10.33205/cma.690236

Cited By: 3

Öz

In this paper we study the following family of hypergeometric polynomials:
$y_n(x) = \frac{ (-1)^\rho }{ n! } x^n {}_2 F_0(-n,\rho;-;-\frac{1}{x})$, depending on a parameter $\rho\in\mathbb{N}$.
Differential equations of orders $\rho+1$ and $2$ for these polynomials are given.
A recurrence relation for $y_n$ is derived as well.
Polynomials $y_n$ are Sobolev orthogonal polynomials on the unit circle
with an explicit matrix measure.

Anahtar Kelimeler

Sobolev orthogonal polynomials, hypergeometric polynomials, unit circle, differential equation, recurrence relation

Kaynakça

• L. C. Andrews: Special functions of mathematics for engineers. Reprint of the 1992 second edition. SPIE Optical Engineering Press, Bellingham, WA; Oxford University Press, Oxford, (1998).
• H. Azad, A. Laradji and M. T. Mustafa: Polynomial solutions of differential equations. Adv. Difference Equ. 2011:58 (2011), 12 pp.
• K. Castillo: A new approach to relative asymptotic behavior for discrete Sobolev-type orthogonal polynomials on the unit circle. Appl. Math. Lett. 25 (2012), no. 6, 1000–1004.
• L. Garza, F. Marcellán and N. C. Pinzón-Cortés: (1; 1)-coherent pairs on the unit circle. Abstr. Appl. Anal. (2013), Art. ID 307974, 8 pp.
• Kh. D. Ikramov: Matrix pencils – theory, applications, numerical methods. (Russian) Translated in J. Soviet Math. 64 (1993), no. 2, 783–853. Itogi Nauki i Tekhniki, Mat. Anal., 29, Mathematical analysis, Vol. 29 (Russian), 3–106, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, (1991).
• M. E. H. Ismail: Classical and quantum orthogonal polynomials in one variable. With two chapters by Walter Van Assche.With a foreword by Richard A. Askey. Encyclopedia of Mathematics and its Applications, 98. Cambridge University Press, Cambridge, (2005).
• M. E. H. Ismail, D. R. Masson: Generalized orthogonality and continued fractions. J. Approx. Theory 83 (1995), no. 1, 1–40.
• K. H. Kim, H. K. Kwon, L. L. Littlejohn and G. J. Yoon: Diagonalizability and symmetrizability of Sobolev-type bilinear forms: a combinatorial approach. Linear Algebra Appl. 460 (2014), 111–124.
• R. Koekoek, P. A. Lesky and R. F. Swarttouw: Hypergeometric orthogonal polynomials and their q-analogues. With a foreword by Tom H. Koornwinder. Springer Monographs in Mathematics. Springer-Verlag, Berlin, (2010).
• K. H. Kwon, L. L. Littlejohn and G. J. Yoon: Ghost matrices and a characterization of symmetric Sobolev bilinear forms. Linear Algebra Appl. 431 (2009), no. 1-2, 104–119.
• F. Marcellán, Y. Xu: On Sobolev orthogonal polynomials. Expo. Math. 33 (2015), no. 3, 308–352.
• A. S. Markus: Introduction to the spectral theory of polynomial operator pencils. With an appendix by M. V. Keldysh. Translations of Mathematical Monographs, 71. American Mathematical Society, Providence, RI, (1988).
• R. Mennicken, M. Möller: Non-self-adjoint boundary eigenvalue problems. North-Holland Mathematics Studies, 192. North-Holland Publishing Co., Amsterdam, (2003).
• E. D. Rainville: Special functions. Reprint of 1960 first edition. Chelsea Publishing Co., Bronx, N.Y., (1971).
• L. Rodman: An introduction to operator polynomials. Operator Theory: Advances and Applications, 38. Birkhäuser Verlag, Basel, (1989).
• B. Simon: Orthogonal polynomials on the unit circle. Part 1. Classical theory. American Mathematical Society Colloquium Publications, 54, Part 1. American Mathematical Society, Providence, RI, (2005).
• B. Simon: Orthogonal polynomials on the unit circle. Part 2. Spectral theory. American Mathematical Society Colloquium Publications, 54, Part 2. American Mathematical Society, Providence, RI, (2005).
• A. Sri Ranga: Orthogonal polynomials with respect to a family of Sobolev inner products on the unit circle. Proc. Amer. Math. Soc. 144 (2016), no. 3, 1129–1143.
• G. Szegö: Orthogonal polynomials. Fourth edition. American Mathematical Society, Colloquium Publications, Vol. XXIII. American Mathematical Society, Providence, R.I., (1975).
• S. M. Zagorodnyuk: On some classical type Sobolev orthogonal polynomials. J. Approx. Theory 250 (2020), 105337, 14 pp.
• A. Zhedanov: Biorthogonal rational functions and the generalized eigenvalue problem. J. Approx. Theory 101 (1999), no. 2, pp. 303–329.