Geodesic
Classes of formulas and expansions of Taylor–Maclaurin type associated with differential operators of fractional order
Izvestiya. Mathematics, Tome 9 (1975) no. 1, pp. 63-112 
M. M. Dzhrbashyan; B. A. Saakyan. Classes of formulas and expansions of Taylor–Maclaurin type associated with differential operators of fractional order. Izvestiya. Mathematics, Tome 9 (1975) no. 1, pp. 63-112. http://geodesic.mathdoc.fr/item/IM2_1975_9_1_a4/
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     title = {Classes of formulas and expansions of {Taylor{\textendash}Maclaurin} type associated with differential operators of fractional order},
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Voir la notice de l'article provenant de la source Math-Net.Ru

This article contains new applications of functions of Mittag–Leffler type $E_\rho(z,\mu)=\sum_0^\infty\Gamma^{-1}\bigl(\mu+\frac k\rho\bigr)z^k$ in mathematical analysis. We construct an apparatus of formulas and expansions of Taylor–Maclaurin type in systems generated by a system of functions of the form $\bigl\{E_\rho\bigl(-\lambda_nx^{1/\rho},\frac1\rho\bigr)\bigr\}_0^\infty$ and an arbitrary sequence $\{\lambda_n\}_0^\infty$ ($0=\lambda_0\leqslant\lambda_n\leqslant\lambda_{n+1}$). In addition, in the article we introduce essentially new classes $\langle\rho,\lambda_n\rangle$ of absolutely monotone functions and establish theorems on their representations. Bibliography: 9 items.

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[2] Dzhrbashyan M. M., “Rasshirenie kvazianaliticheskikh klassov Danzhua–Karlemana”, Izv. AN Arm. SSR, Matematika, 3:4 (1968), 171–248 | Zbl

[3] Dzhrbashyan M. M., Nersesyan A. B., “Drobnye proizvodnye i zadacha Koshi dlya differentsialnykh uravnenii drobnogo poryadka”, Izv. AN Arm. SSR, Matematika, 3:1 (1968), 3–29 | Zbl

[4] Dzhrbashyan M. M., Nersesyan A. B., “O prilozhenii nekotorykh integro-differentsialnykh operatorov”, Dokl. AN SSSR, 121:2 (1958), 210–213 | MR | Zbl

[5] Bernshtein S. N., “Absolyutno monotonnye funktsii”, Sobr. soch., t. I, no. 35, 1952, 370–425

[6] Widder P. V., The Laplace transforms, Princeton, 1946 | Zbl

[7] Dzhrbashyan M. M., Nersesyan A. B., “O postroenii nekotorykh spetsialnykh biortogonalnykh sistem”, Izv. AN Arm. SSR, seriya fiz.-matem. nauk, 12:5 (1959), 17–42 | MR

[8] Nözlund N. E., Legons sur lis series d'interpolation, Paris, 1926

[9] Badalyan G. V., “Obobschenie ryada Teilora i nekotorye voprosy teorii analiticheskikh i kvazianaliticheskikh funktsii”, Izv. AN Arm. SSR, ser. fiz.-matem. nauk, 1954, no. 1, 3–33 | MR | Zbl