## Logarithmic Binomial Theorem

For all integers and ,

where is the Harmonic Logarithm and is a Roman Coefficient. For , the logarithmic binomial theorem reduces to the classical Binomial Theorem for Positive , since for , for , and when .

Similarly, taking and gives the Negative Binomial Series. Roman (1992) gives expressions obtained for the case and which are not obtainable from the Binomial Theorem.

References

Roman, S. The Logarithmic Binomial Formula.'' Amer. Math. Monthly 99, 641-648, 1992.