In this paper we give an extension of the results of the generalized waitingtime problem given by El-Desouky and Hussen (1990). An urn contains mtypes of balls of unequal numbers, and balls are drawn with replacement untilfirst duplication. In the case of finite memory of order k, letni be the number of typei, i = 1, 2, . . ., m. The probability of successpi = ni / N, i = 1, 2, . . ., m,where ni is a positive integer andN = ∑i=1mni.Let Ym,k be the number of drawings requireduntil first duplication. We obtain some new expressions of the probabilityfunction, in terms of Stirling numbers, symmetric polynomials, and generalizedharmonic numbers. Moreover, some special cases are investigated. Finally, someimportant new combinatorial identities are obtained.
"On a generalization of a waiting time problem and some combinatorial identities." J. Appl. Probab. 52 (4) 981 - 989, December 2015. https://doi.org/10.1239/jap/1450802747