Orders and types of the Wright and Mittag--Leffler functions
Trudy Instituta matematiki, Tome 17 (2009) no. 2, pp. 15-22.

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An entire function, with coefficients involving products and quotients of a finite number of gamma functions, is considered. The order and the type of such a function, known as the generalized Wright function, are evaluated. Applications are given to evaluate the orders and types of the generalized Mittag–Leffler functions will even and odd parameters and of the generalized hypergeometric function. Special cases involving in particular Wright, Mittag–Leffler and the confluent hypergeometric Kummer functions are presented.
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A. A. Kilbas; V. V. Lipnevich. Orders and types of the Wright and Mittag--Leffler functions. Trudy Instituta matematiki, Tome 17 (2009) no. 2, pp. 15-22. http://geodesic.mathdoc.fr/item/TIMB_2009_17_2_a1/

[1] Beitmen G., Erdeii A., Vysshie transtsendentnye funktsii, v. 1, Gipergeometricheskaya funktsiya Gaussa. Funktsii Lezhandra, M., 1973

[2] Kilbas A.A., Srivastava H.M., Trujillo J.J., Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006 | MR | Zbl

[3] Wright E.M., “The asymptotic expansion of the generalized hypergeometric function”, J. London Math. Soc., 10 (1935), 286–293 | DOI

[4] Wright E.M., “The asymptotic expansion of integral functions defined by Taylor series”, Philos. Trans. Roy. Soc. London. Ser. A, 238 (1940), 423–451 | DOI | MR

[5] Wright E.M., “The asymptotic expansion of the generalized hypergeometric function II”, Proc. London Math. Soc. Ser. 2, 46 (1940), 389–409 | DOI | MR

[6] Wright E.M., “On the coefficient of power series having exponential singularities”, J. London Math. Soc., 8 (1933), 71–79 | DOI | Zbl

[7] Beitmen G., Erdeii A., Vysshie transtsendentnye funktsii, v. 3, Ellipticheskie i avtomorfnye funktsii. Funktsii Lame i Mate, M., 1967

[8] Marichev O.I., Metod vychisleniya integralov ot spetsialnykh funktsii (teoriya i tablitsy formul), Minsk, 1978 | MR

[9] Kiryakova V.S., Generalized Fractional Calculus and Applications, Longman, Harlow, 1994 | MR | Zbl

[10] Wright E.M., “The asymptotic expansion of the generalized Bessel function”, Proc. London Math. Soc. Ser. 2, 38 (1934), 257–270 | DOI

[11] Wright E.M., “The generalized Bessel functions of order greater than one”, Quart. J. Math. Oxford. Ser. 2, 11 (1940), 36–48 | DOI | MR

[12] Fox C., “The asymptotic expansion of generalized hypergeometric function”, Proc. London Math. Soc. Ser. 2, 27 (1928), 389–400 | DOI | Zbl

[13] Kilbas A.A., Saigo M., Trujillo J.J., “On the generalized Wright function”, Frac. Calc. Appl. Anal., 5 (2002), 437–460 | MR | Zbl

[14] Kilbas A.A., “Fractional calculus of the generalized Wright function”, Frac. Calc. Appl. Anal., 8:2 (2005), 113–126 | MR | Zbl

[15] Kilbas A.A., Koroleva A.A., “Rasshirennye obobschennye funktsii Mittag–Lefflera kak $H$-funktsiya. Obobschennye funktsii Raita i ikh formuly differentsirovaniya”, Vestn. BGU. Ser. 1, 2005, no. 2, 53–60

[16] Kilbas A.A., Saigo M., H-Transforms. Theory and Applications, Boca Raton–London–New York–Washington, 2006 | MR

[17] Levin B.Ya., Raspredelenie kornei tselykh funktsii, M., 1956

[18] Al-Bassam M.A., Luchko Y.F., “On generalized fractional calculus and its application to the solution of integro-differential equation”, J. Frac. Calc., 7 (1995), 69–88 | MR | Zbl

[19] Dzhrbashyan M.M., Integralnye preobrazovaniya i predstavleniya funktsii v kompleksnoi oblasti, M., 1966

[20] Dzhrbashyan M.M., “Ob integralnykh preobrazovaniyakh, porozhdennykh obobschennoi funktsiei Mittag–Lefflera”, Izv. AN Arm. SSR. Ser. fiz.-mat. nauk, 13:3 (1960), 21–63 | MR | Zbl

[21] Kilbas A.A., Koroleva A.A., “Obobschennaya funktsiya Mittag–Lefflera i ee rasshirenie”, Tr. Instituta matematiki. Minsk, 13:1 (2005), 23–32 | Zbl

[22] Prabhakar T.R., “A singular integral equation with a generalized Mittag–Leffler function in the kernel”, Yokohama Math. J., 9 (1971), 7–15 | MR