On power series, Bell polynomials, Hardy-Ramanujan-Rademacher problem and its statistical applications. (English) Zbl 0826.30002

A formula is derived for finding the common term of a power series which is an arbitrary complex power of a polynomial. This formula provides expressions for direct evaluation of the number of partitions of a nonnegative integer and the partitions themselves, which are useful in evaluating probabilities of distributions of order \(k\), derivatives of composite functions, and Bernoulli, Euler and Bell polynomials. Alternative expressions for the truncated Bell polynomials and some statistical applications are also given.


30B10 Power series (including lacunary series) in one complex variable
05A17 Combinatorial aspects of partitions of integers
11B68 Bernoulli and Euler numbers and polynomials
62G05 Nonparametric estimation
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[1] G. E. Andrews: The theory of partitions. Encyclopedia of Mathematics and Its Applications (Rota, Vol. 2, G.-C. Addison-Wesley, Reading 1976. · Zbl 0371.10001
[2] A. Cayley: Collected mathematical papers No. 3, 9, 11, 13. Cambridge 1889-1897.
[3] Ch. A. Charalambides: On the generalized discrete distributions and the Bell polynomials. Sankya 39B (1977), 36-44. · Zbl 0412.62010
[4] Ch. A. Charalambides: On discrete distributions of order k. Ann. Inst. Statist. Math. 38A (1986), 557-568. · Zbl 0611.60016
[5] P. C. Consul, L. R. Shenton: Use of Lagrange expansion for generating discrete generalized probability distributions. SIAM. J. Appl. Math. 23 (1972), 2, 239-272. · Zbl 0226.60017
[6] P. C. Consul, L. R. Shenton: Some interesting properties of Lagrange distributions. Commun. Statist. 2 (1973), 263-272. · Zbl 0267.60010
[7] H. Gupta: Tables of Partitions. Madras 1939. · Zbl 0023.10901
[8] K. Hirano: Some properties of the distribution of order k. Fibonacci Numbers and their Applications (A. N. Philippou and A. F. Horodam, Reidel, Dordrecht 1986, pp. 43-53.
[9] H. Hirano S. Aki N. Kashiwagi, H. Kiboki: On Ling’s binomial and negative binomial distributions of order k. Statist. Probab. Lett. 11 (1991), 503-509. · Zbl 0728.60017
[10] N. L. Johnson, S. Kotz: Discrete distributions 1969-1980. Internat. Statist. Rev. 50 (1982), 71-101. · Zbl 0497.62020
[11] K. D. Ling: On binomial distributions of order k. Statist. Probab. Lett. 6 (1988), 247-250. · Zbl 0638.60024
[12] P. A. Mac Mahon: Combinatorial Analysis. Part 2. London 1916.
[13] A. N. Philippou: The Poisson and compound Poisson distribution of order k and some of their properties. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. 130 (1983), 175-180. · Zbl 0529.60010
[14] A. N. Philippou C. Georghiou, G. N. Philippou: A generalized geometric distribution and some of its properties. Statist. Probab. Lett. 1 (1983), 171-175. · Zbl 0517.60010
[15] A. N. Philippou: The negative binomial distribution of order k and some of its properties. Biometrical J. 36 (1984), 789-794. · Zbl 0566.60014
[16] A. N. Philippou: On multiparameter distribution of order k. Ann. Inst. Statist. Math. 40 (1988), 3, 467-475. · Zbl 0679.62012
[17] J. Riordan: An Introduction to Combinatorial Analysis. John Wiley & Sons, Inc. 1958. · Zbl 0078.00805
[18] J. Riordan: Combinatorial Identities. John Wiley & Sons, Inc. 1968. · Zbl 0194.00502
[19] I. Schur: On Faber polynomials. Amer. J. Math. 69 (1945), 33-41. · Zbl 0060.20403
[20] V. G. Voinov R. Neymann, M. S. Nikulin: The Lagrange’s method of evaluation of quantiles and noncentrality parameters of probability distributions. Theory Probab. Appl. 31 (1986), 1, 185-187.
[21] V. G. Voinov, M. S. Nikulin: Unbiased Estimators and their Applications. Nauka, Moscow 1989. ( · Zbl 0665.62025
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