On (p,q)-Rogers-Szegö matrices

In the present article, we have discussed the (p,q)-numbers, the Rogers-Szegő polynomial and the (p,q)-Rogers-Szegő polynomial and have defined the (p,q)-matrices and the (p,q)-Rogers-Szegő matrices. We have presented some algebraic properties of these matrices and have proved them. In particular, we have obtained the factorization of these matrices, their inverse matrices, as well as the matrix representations of the (p,q)-numbers, the Rogers-Szegő polynomials and the (p,q)-Rogers-Szegő polynomials.

Keywords:

(pq) analogue, (pq)-Rogers-Szegö Polynomials (pq)-Rogers-Szegö matrix,

PDF

___

[1] Burban, I.M., Klimyk, A.U., P,Q-differentiation, P,Q-integration, and P,Q-hypergeometric functions related to quantum groups, Integral Transforms and Special Functions, 2, 15-36, 1994.
[2] Chaichian, M., Demichev, A., Introduction to Quantum Groups, World Scientific, Singapore, 1996.
[3] Chari, V., Pressley, A., A Guide to Quantum Groups, Cambridge Univ. Press, Cambridge, 1994.
[4] Chakrabarti, R., Jagannathan, R.A., (p, q)-oscillator realization of two-parameter quantum algebras, Journal of Physics A: Mathematical and General, 24, L711, 1991.
[5] Jannussis, A., Brodimas, G., Mignani, L., Quantum groups and Lie-admissible time evolution, Journal of Physics A: Mathematical and General, 24(14), L775, 1991.
[6] Arik, M., Demircan, E., Turgut, E., Ekinci, L., Mungan, M., Fibonacci oscillators, Zeitschrift für Physik C Particles and Fields, 55(1), 89-95, 1992.
[7] Katriel, J., Kibler, M., Normal ordering for deformed boson operators and operator-valued deformed Stirling numbers, Journal of Physics A: Mathematical and General, 25(9), 2683, 1992.
[8] Jagannathan, R., Srinivasa Rao, K., Two-parameter quantum algebras, twin-basic numbers, and associated generalized hypergeometric series, arXiv:math/0602613, 2006.
[9] Smirnov, Y.F., Wehrhahn, R.F., The Clebsch-Gordan coefficients for the two-parameter quantum algebra SUp,q(2) in the Lowdin-Shapiro approach, Journal of Physics A: Mathematical and General, 25, 5563, 1992.
[10] Şahin, A. On the Q analogue of Fibonacci and Lucas matrices and Fibonacci polynomials, Utilitas Mathematica, 100, 2016.
[11] Cigler, J., q-Fibonacci polynomials, Fibonacci Quarterly, 41, 31-40, 2003.
[12] Cigler, J., Einige q-Analoga der Lucas- und Fibonacci-Polynome, Sitzungsberichte Abt.II., 211, 3-20, 2002.
[13] Cigler, J., A new class of q-Fibonacci polynomials, The Electronic Journal of Combinatorics, 10, R19, 2003.
[14] Rogers, L. J., On a three-fold symmetry in the elements of Heine’s series, Proceedings of the London Mathematical Society, 24, 171–179, 1893.
[15] Rogers, L. J., On the expansion of some infinite products, Proceedings of the London Mathematical Society, 24, 337–352, 1893.
[16] Szegő, G., Ein Beitrag zur Theorie der Thetafunktionen, S. B. Preuss. Akad. Wiss. Phys.- Math. Kl, 242–252, 1926.
[17] Andrews, G., The Theory of Partitions, Encyclopedia of Mathematics and Its Applications, Addison-Wesley, Reading, Mass.-London-Amsterdam, 1976.
[18] Jagannathan, R., Sridhar, R., (p, q)-Rogers-Szegő Polynomial and the (p, q)-Oscillator, arXiv:1005.4309v1[math.QA], 2010.
[19] Lee, G.Y., Kim, J.S., The linear algebra of the k-Fibonacci matrix, Linear Algebra and Its Applications, 373, 75-87, 2003.
[20] Kılıç, E., Taşcı, D., The linear algebra of the Pell matrix, Boletin de la Sociedad Matematica Mexicana, 2(11), 163-174, 2005.
[21] Lee, G.Y., Kim, J.S., Lee, S.G., Factorizations and eigenvalues of Fibonacci and symmetric Fibonacci matrices, Fibonacci Quarterly, 40(3), 203-211, 2002.
[22] Fonseca, C.M.da.,Petronilho, J., Explicit inverses of some tridiagonal matrices, Linear Algebra and Its Applications, 325,7–21, 2001.
[23] Şahin, A., Ramirez, J.L., Determinantal and permanental representations of convolved Lucas polynomials, Applied Mathematics and Computation, 281, 314-322, 2016.
[24] Şahin, A. On the generalized Perrin and Cordonnier matrices, Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 66(1), 242-253, 2017.
[25] Fonseca, C.M. da., Unifying some Pell and Fibonacci identities, Applied Mathematics and Computation, 236, 41-42, 2014.
[26] Kaygısız, K., Şahin, A., Determinant and permanent of Hessenberg matrix and generalized Lucas polynomials, Bulletin of the Iranian Mathematical Society., 39(6), 1065-1078, 2013.
[27] Anđelić, M., Du, Z., C.M. da Fonseca, Kılıç, E., A matrix approach to some second-order difference equations with sign-alternating coefficients, Journal of Difference Equations and Applications, 26:2, 149-162, 2020.
[28] Cahill, N.D., D'Errico, J.R., Narayan, D.A., Narayan, J.Y., Fibonacci determinants, College Mathematics Journal, 33, 221-225, 2002.
[29] Gibson, P.M., An identity between permanents and determinants, The American Mathematical Monthly, 76, 270-271, 1969.