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Fractional sum and fractional difference on non-uniform lattices and analogue of Euler and Cauchy beta formulas. (English) Zbl 07439146

Summary: As is well known, the definitions of fractional sum and fractional difference of \(f (z)\) on non-uniform lattices \(x(z) = c_1z^2 + c_2z + c_3\) or \(x(z) = c_1q^z + c_2q^{- z} + c_3\) are more difficult and complicated. In this article, for the first time we propose the definitions of the fractional sum and fractional difference on non-uniform lattices by two different ways. The analogue of Euler’s Beta formula, Cauchy’ Beta formula on non-uniform lattices are established, and some fundamental theorems of fractional calculas, the solution of the generalized Abel equation on non-uniform lattices are obtained etc.

MSC:

39A13 Difference equations, scaling (\(q\)-differences)
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
33D45 Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)
26A33 Fractional derivatives and integrals
34K37 Functional-differential equations with fractional derivatives
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[1] Agarwal, R. P., Certain fractional q-integral and q-derivative, Proc Camb Phil Soc, 66, 365-370 (1969) · Zbl 0179.16901
[2] Al-Salam, W. A., Some fractional q-integral and q-derivatives, Proc Edinb Math Soc v2, 15, 135-140 (1966) · Zbl 0171.10301
[3] Anastassiou, G. A., Nabla discrete fractional calculus and nalba inequalities, Mathematical and Computer Modelling, 51, 5-6, 562-571 (2010) · Zbl 1190.26001
[4] M H Annaby, Z S Mansour. q-Fractional Calculus and Equations, Springer, 2012.
[5] Andrews, G. E.; Askey, R.; Roy, R., Special functions (1999), Cambridge: Cambridge University Press, Cambridge
[6] R Askey, J A Wilson. Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Mem Amer Math Soc, 1985, 319. · Zbl 0572.33012
[7] Atakishiyev, N. M.; Suslov, S. K., Difference hypergeometric functions, 1-35 (1992), New York: Springer, New York · Zbl 0787.33003
[8] Atici, F. M.; Eloe, P. W., Discrete fractional calculus with the nable operator, Electronic Journal of Qualitative Theory of Differential Equations, Spec Ed I, 2009, 3, 1-12 (2009) · Zbl 1189.39004
[9] Baoguo, J.; Erbe, L.; Peterson, A., Two monotonicity results for nabla and delta fractional differences, Arch Math (Basel), 104, 589-597 (2015) · Zbl 1327.39011
[10] Cheng, J. F., Theory of Fractional Difference Equations (2011), Xiamen: Xiamen University Press, Xiamen
[11] Jia, L. K.; Cheng, J. F.; Feng, Z. S., A q-analogue of Rummer’s equation, Electron J Differential Equations, 2017, 31, 1-20 (2017)
[12] Cheng, J. F.; Dai, W. Z., Higer-order fractional Green and Gauss formulas, J Math Anal Appl, 462, 1, 157-171 (2018) · Zbl 1390.26008
[13] Cheng, J. F.; Jia, L. K., Hypergeometric Type Difference Equations on Nonuniform Lattices: Rodrigues Type Representation for the Second Kind Solution, Acta Mathematics Scientia, 39A, 4, 875-893 (2019) · Zbl 1449.33011
[14] Cheng, J. F.; Jia, L. K., Generalizations of Rodrigues type formulas for hypergeometric difference equations on nonuniform lattices, Journal of Difference Equations and Applications, 26, 4, 435-457 (2020) · Zbl 1453.33014
[15] Cheng, J. F., On the Complex Difference Equation of Hypergeometric Type on Non-uniform Lattices, Acta Mathematica Sinica, English Series, 36, 5, 487-511 (2020) · Zbl 1440.39007
[16] Cheng, J. F., Hypergeometric Equations and Discrete Fractionl Calculus on Non-uniform Lattices (2021), Beijing: Science Press, Beijing
[17] Diaz, J. B.; Osler, T. J., Differences of fractional order, Math Comp, 28, 125, 185-202 (1974) · Zbl 0282.26007
[18] Ferreira, R. A C.; Torres, D. F M., Fractional h—differences arising from the calculus of variations, Appl Anal Discrete Math, 5, 110-121 (2011) · Zbl 1289.39007
[19] Goodrich, C.; Peterson, A. C., Discrete Fractional Calculus (2015), Switzerland: Springer International Publishing, Switzerland Springer, Switzerland · Zbl 1350.39001
[20] Gray, H. L.; Zhang, N. F., On a new definition of the fractional difference, Mathematics of Computation, 50, 182, 513-529 (1988)
[21] Ismail, M. E H.; Zhang, R., Diagonalization of certain integral operators, Advance in Math Soc, 109, 1, 1-33 (1994) · Zbl 0838.33012
[22] Nikiforov, A. F.; Suslov, S. K.; Uvarov, V. B., Classical orthogonal polynomials of a discrete variable (1991), Berlin: SpringerVerlag, Berlin · Zbl 0743.33001
[23] Nikiforov, A. F.; Uvarov, V. B., Special functions of mathematical physics: A unified introduction with applications (1988), Basel: Birkhauser Verlag, Basel · Zbl 0624.33001
[24] Rahman, M.; Suslov, S. K., The Pearson equation and the Beta Integrals, SIAM J Math Anal, 25, 2, 646-693 (1994) · Zbl 0808.33002
[25] Suslov, S. K., On the theory of difference analogues of special functions of hypergeometric type, Russian Math Surveys, 44, 2, 227-278 (1989) · Zbl 0685.33013
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