Analytic properties of some basic hypergeometric-Sobolev-type orthogonal polynomials. (English) Zbl 1405.33024

Costas-Santos, Roberto S.; Soria-Lorente, Anier

J. Difference Equ. Appl. 24, No. 11, 1715-1733 (2018).

Summary: In this contribution, we consider sequences of monic polynomials orthogonal with respect to a Sobolev-type inner product \[ \langle f,g\rangle_s := \langle\mathbf{u},fg\rangle + N(\mathcal D_q f)(\alpha)(\mathcal D_q g)(\alpha),\quad \alpha\in\mathbb R,\, N\geq 0, \] where \(\mathbf{u}\) is a \(q\)-classical linear functional and \(\mathcal D_q\) is the \(q\)-derivative operator. We obtain some algebraic properties of these polynomials such as an explicit representation, a five-term recurrence relation as well as a second order linear \(q\)-difference holonomic equation fulfilled by such polynomials. We present an analysis of the behaviour of its zeros as a function of the mass \(N\). In particular, we obtain the exact values of \(N\) such that the smallest (respectively, the greatest) zero of the studied polynomials is located outside of the support of the measure. We conclude this work by considering two examples.

MSC:

33D45	Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)
05A30	\(q\)-calculus and related topics
39A13	Difference equations, scaling (\(q\)-differences)

Keywords:

classical orthogonal polynomials; Sobolev-type orthogonal polynomials; basic hypergeometric series; zeros

References:

[1]	Al-Salam, W. A.; Carlitz, L., Some orthogonal q-polynomials, Math. Nachr., 30, 47-61 (1965) · Zbl 0135.27802 · doi:10.1002/mana.19650300105
[2]	Álvarez-Nodarse, R.; Petronilho, J., On the Krall-type discrete polynomials, J. Math. Anal. Appl., 295, 55-69 (2004) · Zbl 1051.33006 · doi:10.1016/j.jmaa.2004.02.042
[3]	Bavinck, H., On polynomials orthogonal with respect to an inner product involving differences, J. Comput. Appl. Math., 57, 17-27 (1995) · Zbl 0823.42014 · doi:10.1016/0377-0427(93)E0231-A
[4]	Bavinck, H., On polynomials orthogonal with respect to an inner product involving differences (the general case), Appl. Anal., 59, 233-240 (1995) · Zbl 0842.42015 · doi:10.1080/00036819508840402
[5]	Bavinck, H.; Koekoek, R., Difference operators with Sobolev-type Meixner polynomials as eigenfunctions, Comput. Math. Appl., 36, 163-177 (1998) · Zbl 0933.39043 · doi:10.1016/S0898-1221(98)80018-7
[6]	Bracciali, C. F.; Dimitrov, D. K.; Sri Ranga, A., Chain sequences and symmetric generalized orthogonal polynomials, J. Comput. Appl. Math., 143, 95-106 (2002) · Zbl 1006.33008 · doi:10.1016/S0377-0427(01)00499-X
[7]	Chihara, T. S., An Introduction to Orthogonal Polynomials (1978), Gordon and Breach: Gordon and Breach, New York · Zbl 0389.33008
[8]	Dimitrov, D. K.; Mello, M. V.; Rafaeli, F. R., Monotonicity of zeros of Jacobi-Sobolev-type orthogonal polynomials, Appl. Numer. Math., 60, 263-276 (2010) · Zbl 1191.33003 · doi:10.1016/j.apnum.2009.12.004
[9]	Frank, E., On the zeros of polynomials with complex coefficients, Bull. Amer. Math. Soc., 52, 144-157 (1946) · Zbl 0060.05302 · doi:10.1090/S0002-9904-1946-08526-2
[10]	Gasper, G.; Rahman, M., Basic Hypergeometric Series, 96 (2004), Cambridge University Press: Cambridge University Press, Cambridge · Zbl 1129.33005
[11]	Huertas, E. J.; Marcellán, F.; Rafaeli, F. R., Zeros of orthogonal polynomials generated by canonical perturbations of measures, Appl. Math. Comput., 218, 7109-7127 (2012) · Zbl 1246.33004
[12]	Hurwitz, A., Über die Bedingungen, unter welchen eine Gleichung nur Wurzeln mit negativen reellen Teilen besitzt, Math. Ann., 46, 273-284 (1895) · JFM 26.0119.03 · doi:10.1007/BF01446812
[13]	Jordaan, K.; Toókos, F., Interlacing theorems for the zeros of some orthogonal polynomials from different sequences, Appl. Numer. Math., 59, 2015-2022 (2009) · Zbl 1175.33003 · doi:10.1016/j.apnum.2009.04.002
[14]	Koekoek, R.; Lesky, P. A.; Swarttouw, R. F., Hypergeometric Orthogonal Polynomials and Their q-Analogues (2010), Springer-Verlag: Springer-Verlag, Berlin · Zbl 1200.33012
[15]	Krein, M. G., On an extrapolation problem due to Kolmogorov, Dokl. Akad. Nauk SSSR, 46, 306-309 (1945) · Zbl 0063.03356
[16]	Marcellán, F.; Perez, T. E.; Piñar, M. A., On zeros of Sobolev-type orthogonal polynomials, Rend. Mat. Appl., 12, 455-473 (1992) · Zbl 0768.33008
[17]	Martínez-Finkelshtein, A., Analytic aspects of Sobolev orthogonality revisited, J. Comput. Appl. Math., 127, 255-266 (2001) · Zbl 0971.33004 · doi:10.1016/S0377-0427(00)00499-4