Journal of Applied Probability

On a generalization of a waiting time problem and some combinatorial identities

B. S. El-Desouky, F. A. Shiha, and A. M. Magar

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In this paper we give an extension of the results of the generalized waiting time problem given by El-Desouky and Hussen (1990). An urn contains m types of balls of unequal numbers, and balls are drawn with replacement until first duplication. In the case of finite memory of order k, let ni be the number of type i, i = 1, 2, . . ., m. The probability of success pi = ni / N, i = 1, 2, . . ., m, where ni is a positive integer and N = ∑i=1mni. Let Ym,k be the number of drawings required until first duplication. We obtain some new expressions of the probability function, in terms of Stirling numbers, symmetric polynomials, and generalized harmonic numbers. Moreover, some special cases are investigated. Finally, some important new combinatorial identities are obtained.

Article information

J. Appl. Probab., Volume 52, Number 4 (2015), 981-989.

First available in Project Euclid: 22 December 2015

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Zentralblatt MATH identifier

Primary: 60C05: Combinatorial probability 05A10: Factorials, binomial coefficients, combinatorial functions [See also 11B65, 33Cxx]
Secondary: 05A19: Combinatorial identities, bijective combinatorics 11B73: Bell and Stirling numbers 11C08: Polynomials [See also 13F20]

Stirling number generating function waiting time symmetric polynomial harmonic number


El-Desouky, B. S.; Shiha, F. A.; Magar, A. M. On a generalization of a waiting time problem and some combinatorial identities. J. Appl. Probab. 52 (2015), no. 4, 981--989. doi:10.1239/jap/1450802747.

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