# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
On the fractional difference equations of order $(2, q)$. (English) Zbl 1227.39007
Summary: This paper presents a kind of new definition of fractional difference, fractional summation, and fractional difference equations and gives methods for explicitly solving fractional difference equations of order $(2, q)$.

##### MSC:
 39A20 Generalized difference equations
##### Keywords:
fractional summation; fractional difference equations
Full Text:
##### References:
 [1] J. K. Hale, Ordinary Differential Equations, John Wiley & Sons, New York, NY, USA, 1969. · Zbl 0211.12301 · doi:10.1007/BF00281436 [2] P. Hartman, Ordinary Differential Equations, Birkhäuser, Boston, Mass, USA, 1982. · Zbl 0506.43005 [3] R. P. Agarwal, Difference Equations and Inequalities, Marcel Dekker, New York, NY, USA, 1992. · Zbl 0977.68567 [4] S. G. Samko, A. A. Kilbas, and O. I. Marichev, Integrals and Derivatives of Fractional Order and Some of their Applications, Nauka I Tekhnika, Minsk, Belarus, 1987. · Zbl 0697.26004 [5] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, NY, USA, 1993. · Zbl 0943.82582 · doi:10.1007/BF01048101 [6] K. S. Miller, “Derivatives of noninteger order,” Mathematics Magazine, vol. 68, no. 3, pp. 183-192, 1995. · Zbl 0837.26006 · doi:10.2307/2691413 [7] F. Mainardi and R. Gorenflo, “On Mittag-Leffler-type functions in fractional evolution processes,” Journal of Computational and Applied Mathematics, vol. 118, no. 1-2, pp. 283-299, 2000. · Zbl 0970.45005 · doi:10.1016/S0377-0427(00)00294-6 [8] K. Diethelm and N. J. Ford, “Multi-order fractional differential equations and their numerical solution,” Applied Mathematics and Computation, vol. 154, no. 3, pp. 621-640, 2004. · Zbl 1060.65070 · doi:10.1016/S0096-3003(03)00739-2 [9] V. Daftardar-Gejji and A. Babakhani, “Analysis of a system of fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 293, no. 2, pp. 511-522, 2004. · Zbl 1058.34002 · doi:10.1016/j.jmaa.2004.01.013 [10] R. W. Ibrahim and S. Momani, “On the existence and uniqueness of solutions of a class of fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 334, no. 1, pp. 1-10, 2007. · Zbl 1123.34302 · doi:10.1016/j.jmaa.2006.12.036 [11] V. Lakshmikantham and A. S. Vatsala, “Basic theory of fractional differential equations,” Nonlinear Analysis, vol. 69, no. 8, pp. 2677-2682, 2008. · Zbl 1161.34001 · doi:10.1016/j.na.2007.08.042 [12] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, Calif, USA, 1999. · Zbl 0924.34008 [13] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, The Netherlands, 2006. · Zbl 1092.45003 [14] J. Cheng, Theory of Fractional Difference Equations, Xiamen University Press, Xiamen, 2011, (in Chinese).