Generalized hypergeometric, digamma and trigamma distributions

  • Masaaki Sibuya


In this paper we introduce and study new probability distributions named “digamma” and “trigamma” defined on the set of all positive integers. They are obtained as limits of the zero-truncated Type B3 generalized hypergeometric distributions (inverse Pólya-Eggenberger or negative binomial beta distributions), and also by compounding the logarithmic series distributions.

The family of digamma distributions has the logarithmic series as a limit and the trigamma as another limit. The trigamma distributions are very close to the zeta (Zipf) distributions. Thus, our new distributions are useful as substitutes of the logarithmic series when the observed frequency data have such a long tail that cannot be fitted by the latter distributions.

In the beginning sections we summarize properties of the Type B3 generalized hypergeometric distributions. It is emphasized that the distributions are obtained by compounding a Poisson distribution by “gamma product-ratio” distributions.


Hypergeometric Distribution Factorial Moment Logarithmic Series Entropy Distance Logarithmic Series Distribution 


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Copyright information

© The Institute of Statistical Mathematics, Tokyo 1979

Authors and Affiliations

  • Masaaki Sibuya
    • 1
  1. 1.IBM JapanJapan

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