×

zbMATH — the first resource for mathematics

On \((q, h)\)-analogue of fractional calculus. (English) Zbl 1189.26006
Summary: The paper discusses fractional integrals and derivatives appearing in the so-called \((q, h)\)-calculus which is reduced for \(h = 0\) to quantum calculus and for \(q = h = 1\) to difference calculus. We introduce delta as well as nabla version of these notions and present their basic properties. Furthermore, we give comparisons with the known results and discuss possible extensions to more general settings.

MSC:
26A33 Fractional derivatives and integrals
26E70 Real analysis on time scales or measure chains
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Agarwal R. P., Proc. Camb. Phil. Soc. 66 pp 365–
[2] Al-Salam W. A., Proc. Edin. Math. Soc. 15 pp 135– · Zbl 0171.10301
[3] Atici F. M., Int. J. Diff. Eq. 2 pp 165–
[4] Atici F. M., J. Nonlin. Math. Phys. 14 pp 341–
[5] Benaoum H. B., J. Phys. A. 32 pp 2037– · Zbl 1042.33501
[6] Bohner M., Dynamic Equations on Time Scales. An Introduction with Applications (2001) · Zbl 0978.39001
[7] Díaz J. B., Math. Comp. 28 pp 185– · Zbl 0282.26007
[8] Díaz R., J. Nonlin. Math. Phys. 12 pp 118– · Zbl 1075.33010
[9] Díaz R., Divulg. Mat. 15 pp 179–
[10] Granger C. W. J., J. Time Ser. Anal. 1 pp 15– · Zbl 0503.62079
[11] Grey H. L., Math. Comp. 50 pp 513–
[12] Jackson F. H., Quart. J. Pure Appl. Math. 41 pp 193–
[13] Moak D. S., Aeq. Math. 20 pp 278– · Zbl 0437.33002
[14] Miller K. S., An Introduction to the Fractional Calculus and Fractional Differential Equations (1993) · Zbl 0789.26002
[15] Nagai A., J. Nonlin. Math. Phys. 10 pp 133– · Zbl 1362.35251
[16] Parashar D., J. Geom. Phys. 48 pp 297– · Zbl 1029.05011
[17] Podlubny I., Fractional Differential Equations (1999) · Zbl 0924.34008
[18] Rosengren H., J. Geom. Phys. 32 pp 349– · Zbl 1043.33011
[19] Zdun M. C., Aeq. Math. 38 pp 163– · Zbl 0686.39009
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.