Distribution of roots of Mittag-Leffler functions

A. Yu. Popov ; A. M. Sedletskii

Contemporary Mathematics. Fundamental Directions, Theory of functions, Tome 40 (2011), pp. 3-171.

Bibliographie

[1] Bari N. K., Trigonometricheskie ryady, Fizmatlit, M., 1961 | MR

[2] Beitmen G., Erdeii A., Vysshie transtsendentnye funktsii. Ellipticheskie i avtomorfnye funktsii. Funktsii Lame i Mate, Nauka, M., 1967 | MR

[3] Bekkenbakh E., Bellman R., Neravenstva, Mir, M., 1965 | MR

[4] Dzhrbashyan M. M., Integralnye preobrazovaniya i predstavleniya funktsii v kompleksnoi oblasti, Nauka, M., 1966 | Zbl

[5] Evgrafov M. E., Asimptoticheskie otsenki i tselye funktsii, Nauka, M., 1979 | MR | Zbl

[6] Levin B. Ya., “Interpolyatsiya tselymi funktsiyami eksponentsialnogo tipa”, Matem. fizika i funkts. anal., 1, FTINT AN USSR, 1969, 136–146 | MR

[7] Levin B. Ya., Tselye funktsii, Kurs lektsii, MGU, M., 1971

[8] Markushevich A. I., Teoriya analiticheskikh funktsii, v. 1, Nauka, M., 1967 | Zbl

[9] Markushevich A. I., Kratkii kurs teorii analiticheskikh funktsii, Nauka, M., 1978 | MR

[10] Nakhushev A. M., “Zadacha Shturma–Liuvillya dlya obyknovennogo differentsialnogo uravneniya vtorogo poryadka s drobnymi proizvolnymi v mladshikh chlenakh”, DAN CCCP, 234:2 (1977), 308–311 | MR | Zbl

[11] Olver F., Vvedenie v asimptoticheskie metody i spetsialnye funktsii, Nauka, M., 1978 | MR

[12] Polia G., Sege G., Zadachi i teoremy iz analiza, Ch. 1, Nauka, M., 1978

[13] Popov A. Yu., “O spektralnykh znacheniyakh odnoi kraevoi zadachi i nulyakh funktsii Mittag-Lefflera”, Diff. uravn., 38:5 (2002), 611–621 | MR | Zbl

[14] Popov A. Yu., “O kolichestve veschestvennykh sobstvennykh znachenii odnoi kraevoi zadachi dlya uravneniya vtorogo poryadka s drobnoi proizvodnoi”, Fund. i prikl. mat., 12:6 (2006), 137–155 | MR | Zbl

[15] Prudnikov A. P., Brychkov Yu. A., Marichev O. I., Integraly i ryady, Fizmatlit, M., 2003

[16] Pskhu A. V., “O veschestvennykh nulyakh funktsii tipa Mittag-Lefflera”, Mat. zametki, 77:4 (2005), 592–599 | MR | Zbl

[17] Sedletskii A. M., “Bazisy iz eksponent v prostranstvakh $E_p$ na vypuklykh mnogougolnikakh”, Izv. AN SSSR. Ser. matem., 42:5 (1978), 1101–1119 | MR | Zbl

[18] Sedletskii A. M., “Asimptoticheskie formuly dlya nulei funktsii tipa Mittag-Lefflera”, Anal. Math., 20 (1994), 117–132 | DOI | MR

[19] Sedletskii A. M., “O nulyakh funktsii tipa Mittag-Lefflera”, Mat. zametki, 68:5 (2000), 710–724 | MR | Zbl

[20] Sedletskii A. M., “Neasimptoticheskie svoistva nulei funktsii tipa Mittag-Lefflera”, Mat. zametki, 75:3 (2004), 405–420 | MR | Zbl

[21] Sedletskii A. M., “O nulyakh preobrazovanii Laplasa”, Mat. zametki, 76:6 (2004), 883–892 | MR | Zbl

[22] Sedletskii A. M., Klassy analiticheskikh preobrazovanii Fure i eksponentsialnye approksimatsii, Fizmatlit, M., 2005

[23] Sedletskii A. M., “Asimptotika nulei vyrozhdennoi gipergeometricheskoi funktsii”, Mat. zametki, 82:2 (2007), 262–271 | MR | Zbl

[24] Sedletskii A. M., “Asimptotika nulei funktsii tipa Mittag-Lefflera poryadka $1/2$”, Vestn. MGU Ser. matem. i mekh., 2007, no. 1, 22–28 | MR

[25] Sedletskii A. M., “Lokalizatsiya malykh nulei finitnykh sinus- i kosinus-preobrazovanii Fure”, Vestn. Mosk. un-ta. Ser. 1. Mat., mekh., 2009, no. 4, 35–41 | MR

[26] Seneta E., Pravilno menyayuschiesya funktsii, Nauka, M., 1985 | MR | Zbl

[27] Sleiter L., Vyrozhdennye gipergeometricheskie funktsii, VTs AN SSSR, M., 1966 | MR

[28] Tikhonov I. V., Obratnye, nelokalnye i kraevye zadachi dlya evolyutsionnykh uravnenii, Avtoref. disc. $\dots$ dokt. fiz.-matem. nauk, MGU, M., 2008

[29] Khardi G. G., Littlvud D. E., Polia G., Neravenstva, IL, M., 1948

[30] Khardi G. G., Rogozinskii V. V., Ryady Fure, Fizmatgiz, M., 1959

[31] Tsvetkov G. E., “O kornyakh funktsii Whittaker'a”, DAN SSSR, 32:1 (1941), 10–12

[32] Tsvetkov G. E., “O kompleksnykh kornyakh funktsii Whittaker'a”, DAN SSSR, 33:4 (1941), 290–291

[33] Shilov G. E., Gurevich B. L., Integral, mera i proizvodnaya, Nauka, M., 1967 | Zbl

[34] Yukhimenko A. A., “Ob odnom klasse funktsii tipa sinusa”, Mat. zametki, 83:6 (2008), 941–954 | MR | Zbl

[35] Yanke E., Emde F., Lesh F., Spetsialnye funktsii, Nauka, M., 1977

[36] Anh V. V., Leonenko N. N., “Spectral analysis of fractional kinetic equations with random data”, J. Statist. Phys., 104:5–6 (2001), 1349–1387 | DOI | MR | Zbl

[37] Beghin L., Orsinger E., “Fractional Poisson processes and related planar random motions”, Electron. J. Probab., 14:61 (2009), 1790–1827 | MR

[38] Mikusinski J. G., “Sur les fonctions $k_n(x)=\sum_{\nu=0}^\infty(-1)^\nu\frac{\varkappa^{n+k\nu}}{(n+k\nu)!}$ ($k=1,2,\dots$; $n=0,1,\dots,k-1$)”, Ann. Soc. Polon. Math., 21 (1948), 46–51 | MR | Zbl

[39] Mittag-Leffler M. G., “Sur la fonction nouvelle $E_{1/\alpha}(x)$”, C. R. Acad. Sci. Paris (2), 137 (1903), 554–558

[40] Ostrovskii I. V., Peresyolkova I. N., “Nonasymptotic results on distribution of zeros of the function $E_\rho(z;\mu)$”, Anal. Math., 23 (1997), 283–296 | DOI | MR | Zbl

[41] Polya G., “Über die Nullstellen gewisser ganzer Funktionen”, Math. Z., 2 (1918), 352–383 | DOI | MR | Zbl

[42] Polya G., “Bemerkung über die Mittag-Lefflerschen Funktionen $E_{1/\alpha}(x)$”, Tôhoku Math. J., 19 (1921), 241–248 | Zbl

[43] Sedletskii A. M., “On zeros of Laplace transforms of finite measure”, Integral Transforms Spec. Funct., 1 (1993), 51–59 | DOI | MR

[44] Sedletskii A. M., “Addition to Polya's theorem on zeros of Fourier sine-transforms”, Integral Transforms Spec. Funct., 9 (2000), 65–68 | DOI | MR | Zbl

[45] Wiman A., “Über die Nullstellen der Funktionalen $E_\alpha(z)$”, Acta Math., 29 (1905), 217–234 | DOI | MR | Zbl

[46] Wright E. M., “On the coefficient of power series having exponential singularities”, J. London Math. Soc., 8:29 (1933), 71–79 | DOI | Zbl

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