Geodesic
Koenigs function and fractional iterates of probability generating functions
Sbornik. Mathematics, Tome 193 (2002) no. 7, pp. 1009-1025 Cet article a éte moissonné depuis la source Math-Net.Ru

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The Koenigs function arises as the limit of an appropriately normalized sequence of iterates of holomorphic functions. On the other hand it is a solution of a certain functional equation and can be used for the definition of iterates of the original function. A description of the class of Koenigs functions corresponding to probability generating functions embeddable in a one-parameter group of fractional iterates is provided. The results obtained can be regarded as a test for the embeddability of a Galton–Watson process in a homogeneous Markov branching process. 
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V. V. Goryainov. Koenigs function and fractional iterates of probability generating functions. Sbornik. Mathematics, Tome 193 (2002) no. 7, pp. 1009-1025. http://geodesic.mathdoc.fr/item/SM_2002_193_7_a2/

[1] Valiron Zh., Analiticheskie funktsii, GITTL, M., 1957

[2] Harris T. E., “Some mathematical models for branching processes”, 2-d Berk. Symp., 1951, 305–328 | MR | Zbl

[3] Kharris T., Teoriya vetvyaschikhsya sluchainykh protsessov, Mir, M., 1966

[4] Karlin S., McGregor J., “Embeddability of discrete time simple branching processes into continuous time processes”, Trans. Amer. Math. Soc., 132 (1968), 115–136 | DOI | MR | Zbl

[5] Karlin S., McGregor J., “Embedding iterates of analytic functions with two fixed points into continuous semigroups”, Trans. Amer. Math. Soc., 132 (1968), 137–145 | DOI | MR | Zbl

[6] Athreya K. B., Ney P. E., Branching processes, Springer-Verlag, Berlin, 1972 | MR | Zbl

[7] Goryainov V. V., “Drobnoe iterirovanie veroyatnostnykh proizvodyaschikh funktsii i vlozhenie diskretnykh vetvyaschikhsya protsessov v nepreryvnye”, Matem. sb., 184:5 (1993), 55–74 | MR | Zbl

[8] Goryainov V. V., “Semigroups of probability generating functions, and infinitely splittable random variables”, Theory of Random Processes, 1(17) (1995), 2–9

[9] Seneta E., “Functional equations and the Galton–Watson processes”, Adv. in Appl. Probab., 1 (1969), 1–42 | DOI | MR | Zbl

[10] Kuczma M., Functional equations in a single variable, PWN–Polish Scientific Publishers, Warszawa, 1968 | MR | Zbl

[11] Königs G., “Recherches sur les intégrales des certaines equations fonctionelles”, Ann. Ecole Norm. Sup. (3), 1 (1884), 3–41 | MR | Zbl

[12] Schröder E., “Über itierte Funktionen”, Math. Ann., 3 (1871), 296–322 | DOI

[13] Baker I. N., “Fractional iteration near a fixpoint of multtiplier 1”, J. Austral. Math. Soc., 4 (1964), 143–148 | DOI | MR | Zbl

[14] Cowen C. C., “Iteration and the solution of functional equations for functions analytic in the unit disk”, Trans. Amer. Math. Soc., 265 (1981), 69–95 | DOI | MR | Zbl

[15] Goryainov V. V., “Odnoparametricheskie polugruppy analiticheskikh funktsii i kompozitsionnyi analog bezgranichnoi delimosti”, Trudy IPMM NAN Ukrainy, 5 (2000), 44–57 | MR | Zbl

[16] Denjoy A., “Sur l'iteration des fonctions analytiques”, C. R. Acad. Sci. Paris Sér. A, 182 (1926), 255–257 | Zbl

[17] Wolff J., “Sur l'iteration des fonctions”, C. R. Acad. Sci. Paris Sér. A, 182 (1926), 42–43 ; 200–201 | Zbl | Zbl

[18] Baker I. N., Pommerenke Ch., “On the iteration of analytic functions in a halfplane, II”, J. London Math. Soc. (2), 20 (1979), 255–258 | DOI | MR | Zbl

[19] Pommerenke Ch., “On asymptotic iteration of analytic functions in the disk”, Analysis (Munich), 1 (1981), 45–61 | MR | Zbl

[20] Pommerenke Ch., Univalent functions, Vandenhoeck and Ruprecht, Gottingen, 1975 | MR | Zbl