Behaviour of the first-order q-difference equation
Since the need to investigate many aspects of q-difference equations cannot be ruled out, this article aims to explore response of the mechanism modelled by linear and nonlinear q-difference equations. Therefore, analysis of an important bundle of nonlinear q-difference equations, in particular the q-Bernoulli difference equation, has been developed. In this context, capturing the behaviour of the q-Bernoulli difference equation as well as linear q-difference equations are considered to be a significant contribution here. Illustrative examples related to the difference equations are also presented.
[1] Finkelstein, R.J. and Marcus, E. (1995). Transformation theory of the q-oscillator. Journal of Mathematical Physics, 36(6), 2652–2672.
[2] Finkelstein, R.J. (1996). The q-coulomb problem. Journal of Mathematical Physics, 37(6), 2628–2636
[3] Floreanini, R. and Vinet, L. (1993). Automorphisms of the q-oscillator algebra and basic orthogonal polynomials. Physics Letters A, 180(6), 393–401.
[4] Floreanini, R. and Vinet, L. (1994). Symmetries of the q-difference heat equation. Letters in Mathematical Physics, 32(1), 37–44.
[5] Floreanini, R. and Vinet, L. (1995). Quantum symmetries of q-difference equations. Journal of Mathematical Physics, 36(6), pp. 3134– 3156.
[6] Freund, P.G. and Zabrodin, A.V. (1995). The spectral problem for the q-knizhnikzamolodchikov equation and continuous q-jacobi polynomials. Communications in Mathematical Physics, 173(1), 17–42.
[7] Han, G.N. and Zeng, J. (1999). On a qsequence that generalizes the median genocchi numbers. Ann. Sci. Math. Qu´ebec, 23(1), 63–72, 1999.
[8] Jaulent, M. and Miodek, I. (1976). Nonlinear evolution equations associated with ‘enegrydependent schr¨odinger potentials’. Letters in Mathematical Physics, 1(3), 243–250.
[9] Jackson, F. (1903). A basic-sine and cosine with symbolical solutions of certain differential equations. Proceedings of the Edinburgh Mathematical Society, 22, 28–39.
[10] Jackson, F. (1910). q-difference equations. American Journal of Mathematics, 32(4), 305–314.
[11] Carmichael, R.D. (1912). The general theory of linear q-difference equations. American Journal of Mathematics, 34(2), 147–168.
[12] Mason, T.E. (1915). On properties of the solutions of linear q-difference equations with entire function coefficients. American Journal of Mathematics, 37(4), 439–444.
[13] Adams, C.R. (1928). On the linear ordinary q-difference equation. Annals of Mathematics, 195–205.
[14] Trjitzinsky, W.J. (1933). Analytic theory of linear q-difference equations. Acta mathematica, 61(1), 1–38.
[15] Andrews, G.E. (1986). q-Series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics and Computer Algebra: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra. No. 66, American Mathematical Society.
[16] Ernst, T. (2000). The history of q-calculus and a new method. Citeseer.
[17] Ernst, T. (2012). A comprehensive treatment of q-calculus. Springer Science & Business Media.
[18] Gasper, G., Rahman, M. and George, G. (2004). Basic hypergeometric series, vol. 96. Cambridge university press.
[19] Kac, V. and Cheung, P. (2001). Quantum calculus. Springer Science & Business Media.
[20] Annaby, M. and Mansour, Z. (2008). qtaylor and interpolation series for jackson qdifference operators. Journal of Mathematical Analysis and Applications, 344(1), 472–483.
[21] Bangerezako, G. (2004). Variational qcalculus. Journal of Mathematical Analysis and Applications, 289(2), 650–665.
[22] Ahmad, B. and Ntouyas, S.K. (2011). Boundary value problems for q-difference inclusions. Abstract and Applied Analysis, vol. 2011, Hindawi.
[23] Dobrogowska, A. and Odzijewicz, A. (2006). Second order q-difference equations solvable by factorization method. Journal of Computational and Applied Mathematics, 193(1), 319–346.
[24] Gasper, G. and Rahman, M. (2007). Some systems of multivariable orthogonal q-racah polynomials. The Ramanujan Journal, 13(1- 3), 389–405.
[25] Ismail, M.E. and Simeonov, P. (2009) qdifference operators for orthogonal polynomials. Journal of Computational and Applied Mathematics, 233(3), 749–761.
[26] El-Shahed, M. and Hassan, H. (2010). Positive solutions of q-difference equation. Proceedings of the American Mathematical Society, 138(5), 1733–1738.
[27] Exton, H. (1983). q-Hypergeometric functions and applications. Horwood.
[28] Gould, H.W. (1961). The q-series generalization of a formula of sparre andersen. Mathematica Scandinavica, 9(1a), 90–94.