Logarithmic Binomial Theorem

For all integers $n$ and $\vert x\vert<a$,

$$\lambda_n^{(t)}(x+a)=\sum_{k=0}^\infty \left\lfloor{\matrix{n\cr k\cr}}\right\rceil \lambda_{n-k}^{(t)}(a)x^k,$$

where $\lambda_n^{(t)}$ is the Harmonic Logarithm and $\left\lfloor{\matrix{n\cr k\cr}}\right\rceil$ is a Roman Coefficient. For $t=0$, the logarithmic binomial theorem reduces to the classical Binomial Theorem for Positive $n$, since $\lambda^{(0)}_{n-k}(a)=a^{n-k}$ for $n\geq k$, $\lambda^{(0)}_{n-k}(a)=0$ for $n<k$, and $\left\lfloor{\matrix{n\cr k\cr}}\right\rceil ={n\choose k}$ when $n\geq k\geq 0$.

Similarly, taking $t=1$ and $n<0$ gives the Negative Binomial Series. Roman (1992) gives expressions obtained for the case $t=1$ and $n\geq 0$ which are not obtainable from the Binomial Theorem.

See also Harmonic Logarithm, Roman Coefficient

References

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