On the limit of discrete q-Hermite I polynomials

Year 2019, Volume: 68 Issue: 2, 2272 - 2282, 01.08.2019

Rezan Sevinik Adıgüzel Sakina Alwhishi Mehmet Turan

https://doi.org/10.31801/cfsuasmas.529703

Cited By: 1

Abstract

Bu çalışmanaın ana konusu ayrık q-Hermite I ve Hermite polinomları arasındaki limit ilişkilerini ele almaktır. Söz konusu limit durumunda ortogonal ilişkileri ve üç terim bağıntıları da sağlanmaktadır. Ayrık q-Hermite I polinomları klasik ortogonal polinomların önemli bir sınıfı olan Hermite polinomlarının q-analoğudur. Bahsi geçen limit durumunda hipergeometrik tipte q-fark denklemi, Rodrigues formülü ve üretici fonksiyonlar da ele alınmıştır.

Keywords

Discrete q-Hermite I polynomials, Hermite polynomials, Rodrigues formula

References

• Alvarez Nodarse, R., On characterizations of classical polynomials, J. Comput. Appl. Math. 196 (2006), 320--337.
• Andrews G. E., Askey, R. and Roy, R., Special functions, Encyclopedia of Mathematics and Its Applications, The University Press, Cambridge, 1999.
• Askey, R. A., Atakishiyev, N. M. and Suslov, S.K., An analog of the fourier transformation for a q-harmonic oscillator, Symmetries in Science VI (1993), 57--63. (Bregenz, 1992).
• Askey, R. A. and Suslov, S. K., The q-harmonic oscillator and an analogue of the Charlier polynomials, J. Phys. A: Math. and Gen. 26 (1993), L693--L698.
• Askey, R. A. and Suslov, S. K., The q-harmonic oscillator and the Al-salam and Carlitz polynomials, Let. Math. Phys. 29 (1993), 123--132.
• Atakishiyev, N. M. and Suslov, S. K., Difference analogs of the harmonic oscillator, Theoretical and Mathematical Physics 85 (1991), 442--444.
• Atakishiyev, N. M. and Suslov, S. K., Realization of the q-harmonic oscillator, Theoretical and Mathematical Physics 87 (1991), 1055--1062.
• Bannai, E., Orthogonal polynomials in coding theory and algebraic combinatorics, Theory and Practice (P. G. Nevai, ed.) (1990), 25--54. NATO, ASI Series.
• Berg, C. and Ismail, M. E. H., q-Hermite polynomials and classical orthogonal polynomials, arXiv:math/9405213v1 [math.CA] 24.
• Bochner, S., Über sturm-liouvilleesche polynomsysteme, Math. Z. 29 (1929), 730--736.
• Borzov, V. V. and Damaskinsky, E. V., Generalized coherent states for q-oscillator connected with q-Hermite polynomials, Journal of Mathematical Sciences 132 (1), 26--36.
• Boyadzhiev, K. N. and Dil, A., Series with Hermite polynomials and applications, Publ. Math. Debrecen 80, No. 3-4 (2012), 385--404.
• Chihara, T. S., An introduction to orthogonal polynomials, Gordon and Breach, New York, London, Paris, 1978.
• Cryer, C. W., Rodrigues formula and the classical orthogonal polynomials, Bol. Un. Mat. Ital. 25 (3) (1970), 1--11.
• Dattoli, G., Incomplete 2D Hermite polynomials: properties and applications, J. Math. Anal. Appl. 284 (2003), 447--454.
• Fine, N. J., Basic hypergeometric series and applications, American Mathematical Society 27 (1988), Providence, RI.
• Gasper, G. and Rahman, M., Basic hypergeometric series (2nd Ed.), Encyclopedia of Mathematics and its Applications 96, Cambridge University Press, Cambridge, 2004.
• Hahn, W., Über die jacobischen polynome und zwei verwandte polynomklassen, Math. Z. 39 (1935), 634--638.
• Koekoek, R., Lesky, P. A. and Swarttouw R.F., Hypergeometric orthogonal polynomials and their q-analogues, Springer Monographs in Mathematics, Springer-Verlag, Berlin-Heidelberg, 2010.
• Koekoek, R. and Swarttouw, R.F., The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue, Reports of the Faculty of Technical Mathematics and Informatics, 1998.
• Koornwinder, T. H., Orthogonal polynomials in connection with quantum groups, in: P. Nevai (ed.), orthogonal polynomials, theory and practice, 257--292.
• Koornwinder, T. H., Compact quantum groups and q-special functions, in: V. baldoni, m. a. picardello (eds.), representations of lie groups and quantum groups, 46--128.
• Macfarlane, A. J., On q-analogues of the quantum harmonic oscillator and the quantum group suq(2), J. Phys. A: Math. and Gen. 22 (1989), 4581--4588.
• Marcellan, F., Branquinho A. and Petronilho J., Classical orthogonal polynomials: A functional approach, Acta Appl. Math. 34 (1994), 283--303.
• Nikiforov, A., Suslov S. K. and Uvarov V., Classical orthogonal polynomials of a discrete variable, Springer-Verlag, Berlin Heidelberg, 1991.
• Nikiforov, A. and Uvarov, V., Classical orthogonal polynomials of a discrete variable on nonuniform lattices, Lett. Math. Phys. 11 (1) (1986), 27--34.
• Nikiforov, A. and Uvarov, V., Special Functions of Mathematical Physics, Birkhäuser, Basel, 1988.
• Routh, R. J., On some properties of certain solutions of a differential equation of the second order, Proc. London Math. Soc. 16 (1885), 245--261.
• Szegö, G., Orthogonal Polynomials, American Mathematical Society Colloquium Publication, Volume XXIII, 1939.
• Tricomi, F., Vorlesungen über orthogonalreihen, Grundlehren der mathematischen wissenschaften 76 (1955).
• Vilenkin, N. J. and Klimyk, A. U., Representations of lie groups and special functions, Bulletin (new series) of the American Mathematical Society 35 No. 3 (1998), 265--270.
• Yuan, Y. and Tung, C. C., Application of hermite polynomial to wave and wave force statistics, Ocean Engineering 11 No. 6 (1984), 593--607.