Harmonic Logarithm

For all Integers $n$ and Nonnegative Integers $t$, the harmonic logarithms $\lambda_n^{(t)}(x)$ of order $t$ and degree $n$ are defined as the unique functions satisfying

1. $\lambda_0^{(t)}(x)=(\ln x)^t$,

2. $\lambda_n^{(t)}(x)$ has no constant term except $\lambda_0^{(0)}(x)=1$,

3. ${d\over dx}\lambda_n^{(t)}(x)=\left\lfloor{n}\right\rceil \lambda_{n-1}^{(t)}(x)$,

where the ``Roman Symbol'' $\left\lfloor{n}\right\rceil$ is defined by

$$\left\lfloor{n}\right\rceil \equiv \cases{ n & for $n\not=0$\cr 1 & for $n=0$\cr}$$ (1)

(Roman 1992). This gives the special cases

$$\lambda_n^{(0)}(x) = \left\{\begin{array}{ll} x^n & \mbox{for $n\geq 0$}\\ 0 & \mbox{for $n<0$}\end{array}\right.$$ (2)

$$\lambda_n^{(1)}(x) = \left\{\begin{array}{ll} x^n(\ln x-H_n) & \mbox{for $n\geq 0$}\\ x^n & \mbox{for $n<0$,}\end{array}\right.$$ (3)

where $H_n$ is a Harmonic Number

$$H_n\equiv \sum_{k=1}^n {1\over k}.$$ (4)

The harmonic logarithm has the Integral

$$\int \lambda_n^{(1)}(x)\,dx = {1\over\left\lfloor{n+1}\right\rceil }\lambda_{n+1}^{(1)}(x).$$ (5)

The harmonic logarithm can be written

$$\lambda_n^{(t)}(x)=\left\lfloor{n}\right\rceil ! \tilde D^{-n}(\ln x)^t,$$ (6)

where $\tilde D$ is the Differential Operator, (so $\tilde D^{-n}$ is the $n$th Integral). Rearranging gives

$$\tilde D^k \lambda_n^{(t)}(x)=\left\lfloor{\left\lfloor{n}\r... ...loor{n-k}\right\rceil }\right\rceil !\,\lambda_{n-k}^{(t)}(x).$$ (7)

This formulation gives an analog of the Binomial Theorem called the Logarithmic Binomial Formula. Another expression for the harmonic logarithm is

$$\lambda_n^{(t)}(x)=x^n \sum_{j=0}^t (-1)^j(t)_jc_n^{(j)}(\ln x)^{t-j},$$ (8)

where $(t)_j=t(t-1)\cdots(t-j+1)$ is a Pochhammer Symbol and $c_n^{(j)}$ is a two-index Harmonic Number (Roman 1992).

See also Logarithm, Roman Factorial

References

Loeb, D. and Rota, G.-C. ``Formal Power Series of Logarithmic Type.'' Advances Math. 75, 1-118, 1989.

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