Abstract
We introduce a new algorithm to compute the difference between values of the \(\log \Gamma\)-function in close points, where \(\Gamma\) denotes Euler’s gamma function. As a consequence, we obtain a way of computing the Dirichlet-multinomial log-likelihood function which is more accurate, has a better computational complexity and a wider range of application than the previously known ones.
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Notes
In the literature the digamma and polygamma functions are respectively denoted as \(\psi (u)\), \(\psi ^{(\ell )}(u)\) but in this context \(\psi\) has a different meaning.
See the previous footnote.
In fact, Pari/GP does not have a direct implementation of \(\digamma ^{(\ell )}\), \(\ell \in \mathbb {N}^*\), but these functions can be obtained using the Hurwitz zeta function \(\zeta (s, u)\), \(s>1, u>0\), since \(\digamma ^{(\ell )} (u) = (-1)^{\ell -1} (\ell !)\zeta (\ell +1, u)\). So it is more precise to say that in Pari/GP the implementation of \(\zeta (2, u)\) seems to be not optimised, at least for large values of u.
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Acknowledgements
The authors would like to thank Marco Vianello (Università di Padova) for having suggested one of the references. We would also like to thank the anonymous referees for their suggestions.
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Languasco, A., Migliardi, M. On the fast computation of the Dirichlet-multinomial log-likelihood function. Comput Stat 38, 1995–2013 (2023). https://doi.org/10.1007/s00180-022-01311-7
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DOI: https://doi.org/10.1007/s00180-022-01311-7