On Fractional Orthonormal Polynomials of a Discrete Variable

Discrete Dynamics in Nature and Society

Volumn 2015, Issue , 2015, Pages

  Area, I ^a   Djida, J D ^b   Losada, J ^c   Nieto, Juan J ^c,d  

^a UNIVERSITY OF VIGO   (Spain)

^b African Institute for Mathematical Sciences (AIMS) Limbe Crystal Gardens   (Cameroon)

^c UNIVERSITY OF SANTIAGO DE COMPOSTELA   (Spain)

^d KING ABDULAZIZ UNIVERSITY   (Saudi Arabia)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84937775579     PISSN: 10260226     EISSN: 1607887X     Source Type: Journal    
DOI: 10.1155/2015/141325     Document Type: Article

References (35)

1. B. Ross, "Fractional calculus," Mathematics Magazine, vol. 50, no. 3, pp. 115-122, 1977.
2. G. W. Leibniz, "Letter from Hanover, Germany; to G.F.A. L'Hôpital," in Mathematische Schriften 1849, vol. 2, pp. 301-302, Olms, Hidesheim, Germany, 1695, reprinted in 1962.
3. C. Li, F. Zeng, and F. Liu, "Spectral approximations to the fractional integral and derivative," Fractional Calculus and Applied Analysis, vol. 15, no. 3, pp. 383-406, 2012.
4. A. M. Legendre, "Sur l'attraction des sphéroïdes homogénes," Mémoires de Mathématique et de Physique, Présentés à l'Académie Royale des Sciences, par Divers Savans, et lus dans ses Assemblées, vol. 10, pp. 411-434, 1785.
5. NIST Digital Library of Mathematical Functions, Release 1.0.8, 2014, http://dlmf.nist.gov/.
6. M. E. H. Ismail, Classical and QuantumOrthogonal Polynomials in One Variable, Cambridge University Press, Cambridge, UK, 2005.
7. R. Koekoek, P. A. Lesky, and R. F. Swarttouw, Hypergeometric Orthogonal Polynomials and Their q-Analogues, Springer, Berlin, Germany, 2010.
8. A. F. Nikiforov, S. K. Suslov, and V. B. Uvarov, Classical Orthogonal Polynomials of a Discrete Variable, Springer, Berlin, Germany, 1991.
9. F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, Eds., NIST Handbook of Mathematical Functions, Cambridge University Press, New York, NY, USA, 2010.
10. M. R. Rapaić, T. B. Šekara, and V. Govedarica, "A novel class of fractionally orthogonal quasi-polynomials and new fractional quadrature formulas," Applied Mathematics and Computation, vol. 245, pp. 206-219, 2014.
11. I. Area, J. Losada, and A. Manintchap, "On some fractional Pearson equations," Fractional Caculus and Applied Analysis, vol. 18, no. 5, 2015.
12. P. L. Chebyshev, "Sur l'interpolation des valeurs équidistantes," Mémoires de l'Académie Impériale des Sciences de Saint Pétersbourg, vol. 25, no. 5, pp. 219-242, 1875, Oeuvres, vol. 2, 219-242.
13. A. B. J. Kuijlaars and E. A. Rakhmanov, "Zero distributions for discrete orthogonal polynomials," Journal of Computational and Applied Mathematics, vol. 99, no. 1-2, pp. 255-274, 1998.
14. I. Area, D. K. Dimitrov, E. Godoy, and V. Paschoa, "Approximate calculation of sums I: bounds for the zeros of Gram polynomials," SIAMJournal on Numerical Analysis, vol. 52, no. 4, pp. 1867-1886, 2014.
15. A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions, vol. 2, Robert E Krieger Publishing, Melbourne, Fla, USA, 1981.
16. G. Szegö, Orthogonal Polynomials, American Mathematical Society, Providence, RI, USA, 4th edition, 1975.
17. J. P. Gram, "Ueber die Entwickelung reeller Functionen in Reihen mittelst der Methode der kleinsten Quadrate," Journal für die reine und angewandte Mathematik, vol. 1883, no. 94, pp. 41-73, 2009.
18. J. Baik, T. Kriecherbauer, K. T.-R. McLaughlin, and P. D. Miller, Discrete Orthogonal Polynomials, vol. 164 of Annals of Mathematics Studies, Princeton University Press, Princeton, NJ, USA, 2007.
19. G. Dahlquist and Å. Björck, Numerical Methods in Scientific Computing, vol. 1, Society for Industrial and AppliedMathematics (SIAM), Philadelphia, Pa, USA, 2008.
20. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204, Elsevier Science, Amsterdam, The Netherlands, 2006.
21. K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, NY, USA, 1993.
22. K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, NY, USA, 1974.
23. S. Samko, A. A. Kilbas, and O. Marichev, Fractional Integrals and Derivatives, Taylor & Francis, 1993.
24. F. M. Atici, A. Cabada, and J. B. Ferreiro, "First order difference equations with maxima and nonlinear functional boundary value conditions," Journal of Difference Equations and Applications, vol. 12, no. 6, pp. 565-576, 2006.
25. F. M. Atici and P.W. Eloe, "A transformmethod in discrete fractional calculus," International Journal of Difference Equations, vol. 2, no. 2, pp. 165-176, 2007.
26. J. Cermak and T. Kisela, "Note on a discretization of a linear fractional differential equation," Mathematica Bohemica, vol. 135, no. 2, pp. 179-188, 2010.
27. J. Čermák, T. Kisela, and L. Nechvátal, "Discrete Mittag-Leffler functions in linear fractional difference equations," Abstract and Applied Analysis, vol. 2011, Article ID 565067, 21 pages, 2011.
28. K. S. Miller and B. Ross, "Fractional difference calculus," in Univalent Functions, Fractional Calculus, and Their Applications (Koriyama, 1988), Ellis Horwood Series inMathematics and Its Applications, pp. 139-152, Horwood, Chichester, UK, 1989.
29. M. D. Ortigueira, "Fractional central differences and derivatives," Journal of Vibration and Control, vol. 14, no. 9-10, pp. 1255-1266, 2008.
30. J. Tariboon, S. K. Ntouyas, and P. Agarwal, "New concepts of fractional quantumcalculus and applications to impulsive fractional α-difference equations,"Advances inDifference Equations, vol. 2015, article 18, 2015.
31. I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, 1999.
32. T. Abdeljawad, D. Baleanu, F. Jarad, and R. P. Agarwal, "Fractional sums and differences with binomial coefficients," Discrete Dynamics in Nature and Society, vol. 2013, Article ID 104173, 6 pages, 2013.
33. J. Diblík, M. Fećkan, and M. Posipíšil, "Nonexistence of periodic solutions and S-asymptotically periodic solutions in fractional difference equations," Applied Mathematics and Computation, vol. 257, pp. 230-240, 2015.
34. T. S. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach Science Publishers, New York, NY, USA, 1978.
35. Wolfram Research Inc, Mathematica, version 10.0.2.0, Wolfram Research Inc, Champaign, Ill, USA, 2010.