# zbMATH — the first resource for mathematics

Discrete Mittag-Leffler functions in linear fractional difference equations. (English) Zbl 1220.39010
Summary: We investigate some initial value problems in discrete fractional calculus. We introduce a linear difference equation of fractional order along with suitable initial conditions of fractional type and prove the existence and uniqueness of the solution. Then the structure of the solutions space is discussed, and, in a particular case, an explicit form of the general solution involving discrete analogues of Mittag-Leffler functions is presented. All our observations are performed on a special time scale which unifies and generalizes ordinary difference calculus and $$q$$-difference calculus. Some of our results are new also in these particular discrete settings.

##### MSC:
 39A13 Difference equations, scaling ($$q$$-differences) 39A06 Linear difference equations 26A33 Fractional derivatives and integrals 33E12 Mittag-Leffler functions and generalizations
Full Text:
##### References:
 [1] R. P. Agarwal, “Certain fractional q-integrals and q-derivatives,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 66, pp. 365-370, 1969. · Zbl 0179.16901 [2] J. B. Diaz and T. J. Osler, “Differences of fractional order,” Mathematics of Computation, vol. 28, pp. 185-202, 1974. · Zbl 0282.26007 [3] H. L. Gray and N. F. Zhang, “On a new definition of the fractional difference,” Mathematics of Computation, vol. 50, no. 182, pp. 513-529, 1988. · Zbl 0648.39002 [4] K. S. Miller and B. Ross, “Fractional difference calculus,” in Univalent Functions, Fractional Calculus, and Their Applications (Koriyama, 1988), Ellis Horwood Series: Mathematics and Its Applications, pp. 139-152, Horwood, Chichester, UK, 1989. [5] F. M. Atici and P. W. Eloe, “A transform method in discrete fractional calculus,” International Journal of Difference Equations, vol. 2, no. 2, pp. 165-176, 2007. [6] F. M. Atici and P. W. Eloe, “Fractional q-calculus on a time scale,” Journal of Nonlinear Mathematical Physics, vol. 14, no. 3, pp. 341-352, 2007. · Zbl 1157.81315 [7] J. and L. Nechvátal, “On (q, h)-analogue of fractional calculus,” Journal of Nonlinear Mathematical Physics, vol. 17, no. 1, pp. 51-68, 2010. · Zbl 1189.26006 [8] M. Bohner and A. Peterson, Dynamic equations on time scales, Birkhauser, Boston, Mass, USA, 2001. · Zbl 0993.39010 [9] M. Bohner and A. Peterso, Eds., Advances in Dynamic Equations on Time Scales, Birkhauser, Boston, Mass, USA, 2003. · Zbl 1025.34001 [10] G. A. Anastassiou, “Foundations of nabla fractional calculus on time scales and inequalities,” Computers & Mathematics with Applications, vol. 59, no. 12, pp. 3750-3762, 2010. · Zbl 1198.26033 [11] G. E. Andrews, R. Askey, and R. Roy, Special Functions, vol. 71 of Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, UK, 1999. · Zbl 0920.33001 [12] F. M. Atici and P. W. Eloe, “Initial value problems in discrete fractional calculus,” Proceedings of the American Mathematical Society, vol. 137, no. 3, pp. 981-989, 2009. · Zbl 1166.39005 [13] F. M. Atici and P. W. Eloe, “Discrete fractional calculus with the nabla operator,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 2009, no. 2, p. 12, 2009. · Zbl 1189.39004 [14] Z. S. I. Mansour, “Linear sequential q-difference equations of fractional order,” Fractional Calculus & Applied Analysis, vol. 12, no. 2, pp. 159-178, 2009. · Zbl 1176.26005 [15] I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999. · Zbl 0924.34008 [16] A. Nagai, “On a certain fractional q-difference and its eigen function,” Journal of Nonlinear Mathematical Physics, vol. 10, supplement 2, pp. 133-142, 2003. · Zbl 1362.35251 [17] J. and T. Kisela, “Note on a discretization of a linear fractional differential equation,” Mathematica Bohemica, vol. 135, no. 2, pp. 179-188, 2010. · Zbl 1224.39003 [18] A. A. Kilbas, M. Saigo, and R. K. Saxena, “Solution of Volterra integrodifferential equations with generalized Mittag-Leffler function in the kernels,” Journal of Integral Equations and Applications, vol. 14, no. 4, pp. 377-396, 2002. · Zbl 1041.45011 [19] A. A. Kilbas, M. Saigo, and R. K. Saxena, “Generalized Mittag-Leffler function and generalized fractional calculus operators,” Integral Transforms and Special Functions, vol. 15, no. 1, pp. 31-49, 2004. · Zbl 1047.33011 [20] F. M. Atici and P. W. Eloe, “Linear systems of fractional nabla difference equations,” Rocky Mountain Journal of Mathematics, vol. 41, pp. 353-370, 2011. · Zbl 1218.39003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.