The Pollaczek-Geiringer distribution is another name for the stuttering Poisson Distribution (SPD), a non-negative discrete compound probability distribution .
Historical Notes on the Pollaczek-Geiringer Distribution
The Pollaczek-Geiringer distribution makes a sparse entry in 
The reference  No. 9 refers to the obscure and seldom-reference book A Summary of Known Distribution Functions, published in 1945.
Hilda Geiringer was born in Vienna in 1893. Her papers betwen 1923 and 1934 appeared under the hyphenated name Pollaczek-Geiringer due to her (brief) marriage to the statistician Felix Pollaczek (1892-1981). One paper was on “The Poisson distribution and the development of arbitrary distributions” which stirred up debate:
“…namely, expansions of a discrete distribution with an infinite number of values in a series in successive derivatives of the Poisson distribution with respect to the parameter. These expansions were first proposed by the Swedish astronomer C. L. W. Charlier (1862-1934) in 1905.” 
 Zhang, H. et al. On nonnegative integer-valued Lévy processes and applications in probabilistic number theory and inventory policies. American Journal of Theoretical and Applied Statistics 2013; 2(5): 110-121. doi: 10.11648/j.ajtas.20130205.11
 Haight, F. (1958). Index to the Distributions of Mathematical Statistics. National Bureau of Standards Report.
 HALLER, B. Verteilungsfunktionen und ihre Auszeichnung durch Funktionalgleichungen. Mitteilungen der Vereinigung schweizerischer Versicherungsmathematiker . 45 Band, Heft 1, 21 April 1945, pp. 97 – 163. Translated by R, E. Kalaba, and published by the RAND Corporation under the title A Summary of Known Distribution Functions, T – 27, 7 January 1953.
 Siegmund-Schultze, R. Human Side of the Emancipation of Applied Mathematics at the University of Berlin During the 1920s. Historia Mathematica 20 (1993). 364-381.