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Fractional calculus operators and their applications involving power functions and summation of series. (English) Zbl 0863.26008
Many earlier works on the subject of fractional calculus contain interesting accounts of the theory and applications of fractional calculus operators in a number of areas of mathematical analysis (such as ordinary and partial differential equations, integral equations, summation of series, etc.). The main object of the paper is to examine rather systematically (and extensively) some of the most recent contributions on the applications of fractional calculus operators involving power functions and in finding the sums of several interesting families of infinite series. Various other classes of infinite sums found in the mathematical literature by these (or other) means, and their validity or hitherto unnoticed connections with some known results, are also considered.

MSC:
26A33Fractional derivatives and integrals (real functions)
33B15Gamma, beta and polygamma functions
33C05Classical hypergeometric functions, ${}_2F_1$
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Full Text: EuDML
References:
[1] Kiryakova, V.: Generalized fractional calculus and applications. Pitman research notes in mathematics series 301 (1994)
[2] Miller, K. S.; Ross, B.: An introduction to the fractional calculus and fractional differential equations. (1993) · Zbl 0789.26002
[3] Nishimoto, K.: Fractional calculus. II, III, and IV (1984) · Zbl 0605.26006
[4] Nishimoto, K.: An essence of nishimoto’s fractional calculus (Calculus in the 21st century): integrations and differentiations of arbitrary order. (1991) · Zbl 0798.26007
[5] Oldham, K. B.; Spanier, J.: The fractional calculus: theory and applications of differentiation and integration to arbitrary order. (1974) · Zbl 0292.26011
[6] Samko, S. G.; Kilbas, A. A.; Marichev, O. I.: Fractional integrals and derivatives: theory and applications. (1993) · Zbl 0818.26003
[7] Srivastava, H. M.; Buschman, R. G.: Theory and applications of convolution integral equations. Mathematics and its applications 79 (1992) · Zbl 0755.45002
[8] Srivastava, H. M.; Owa, S.: Univalent functions, fractional calculus, and their applications. (1989) · Zbl 0683.00012
[9] Ross, B.; Samko, S. G.; Love, E. R.: Functions that have no. First-order derivative might have fractional derivatives of all orders less than 1. Real anal. Exchange 20, 140-157 (1994/1995) · Zbl 0820.26002
[10] Vyas, D. N.; Banerji, P. K.: Fractional integral formula of the function ({$\alpha$}z + {$\beta$})a. J. fractional calculus 2, 83-86 (1992) · Zbl 0806.30004
[11] Nishimoto, K.: Power functions in fractional calculus of nishimoto and that of lacroix and Riemann-Liouville. J. fractional calculus 2, 11-25 (1992) · Zbl 0790.26006
[12] Tu, S. -T.; Nishimoto, K.: On the fractional calculus of functions (cz - a)${\beta}$ and $log(cz - a)$. J. fractional calculus 5, 35-43 (1994) · Zbl 0827.26008
[13] Srivastava, H. M.; Nishimoto, K.: A note on a certain fractional integral formula. J. fractional calculus 3, 87-89 (1993) · Zbl 0848.26005
[14] Erdélyi, A.; Magnus, W.; Oberhettinger, F.; Tricomi, F. G.: Tables of integral transforms. (1954) · Zbl 0055.36401
[15] Ross, B.: A formula for the fractional integration and differentiation of (a + bx)c. J. fractional calculus 5, 87-89 (1994) · Zbl 0858.26007
[16] Srivastava, H. M.; Saigo, M.; Owa, S.: A class of distortion theorems involving certain operators of fractional calculus. J. math. Anal. appl. 131, 412-420 (1988) · Zbl 0628.30014
[17] Nishimoto, K.; Srivastava, H. M.: Certain classes of infinite series summable by means of fractional calculus. J. college engrg. Nihon univ. Ser. B 30, 97-106 (1989) · Zbl 0651.33002
[18] Srivastava, H. M.: A simple algorithm for the evaluation of a class of generalized hypergeometric series. Stud. appl. Math. 86, 79-86 (1992) · Zbl 0724.33002
[19] Al-Saqabi, B. N.; Kalla, S. L.; Srivastava, H. M.: A certain family of infinite series associated with digamma functions. J. math. Anal. appl. 159, 361-372 (1991) · Zbl 0726.33002
[20] De Durán, J. Aular; Kalla, S. L.; Srivastava, H. M.: Fractional calculus and the sums of certain families of infinite series. J. math. Anal. appl. 190, 738-754 (1995) · Zbl 0827.26006
[21] Tu, S. -T.; Chyan, D. -K.: A certain family of infinite series, differinte-grable functions and psi functions. Preceedings of the first international workshop on transform methods and special functions, 310-316 (1995)
[22] Galué, L.; Kalla, S. L.; Nishimoto, K.: Application of fractional calculus to infinite sums. J. fractional calculus 1, 17-21 (1992) · Zbl 0798.26009
[23] Galué, L.: Application of fractional calculus to infinite sums (II). J. fractional calculus 7, 61-67 (1995)
[24] Watanabe, Y.: Notes on the generalized derivative of Riemann-Liouville and its applications to Leibniz’s formula. II. Tôhoku math. J. 34, 28-41 (1931) · Zbl 0002.25304
[25] Osler, T. J.: Leibniz rule for fractional derivatives generalized and an application to infinite series. SIAM J. Appl. math. 18, 658-674 (1970) · Zbl 0201.44102
[26] Al-Zamel, A.; Kalla, S.: An application of generalized Leibniz rule to infinite sums. J. fractional calculus 7, 29-33 (1995) · Zbl 0842.26005
[27] Owa, S.: An application of the fractional derivative to hypergeometric series. Sugaku 38, 360-362 (1986) · Zbl 0643.33006
[28] Erdélyi, A.; Magnus, W.; Oberhettinger, F.; Tricomi, F. G.: Higher transcendental functions. (1953) · Zbl 0051.30303
[29] Samtani, R. K.; Bhatt, R. C.: A useful hypergeometric transformation. Ganita sandesh 8, 65-67 (1994) · Zbl 0831.33001
[30] Kummer, E. E.: Über die hypergeometrische reihe $1 {\alpha}{\cdot}{\beta} 1{\cdot}{\gamma}{\chi} + {\alpha}({\alpha}$+1)${\beta}({\beta}$+1) $1{\cdot}2{\cdot}{\gamma}({\gamma}$+1)${\chi}2 + {\alpha}({\alpha}$+1)({$\alpha$}+2)${\beta}({\beta}$+1)({$\beta$}+2) $1{\cdot}2{\cdot}3{\cdot}{\gamma}({\gamma}$+1)({$\gamma$}+2)${\chi}3 + \dots $. J. reine angew. Math. 15, 127-172 (1836) · Zbl 015.0533cj