Computing Generating Functions for Harmonic Numbers

This article delves into the complex world of generating functions for the squared first-order harmonic numbers. We explore a method to compute these functions using generating series and integrals involving polylogarithms. Let's break down the process step-by-step.

Introduction to Harmonic Numbers and Generating Functions

Harmonic numbers, denoted by H_n, are the sums of reciprocals of the first n natural numbers. They can be represented using the generating function:

H_n -[x^n] logleft(frac{1-x}{1-x}right)

Using this representation, we can find generating functions for related series. For instance, we are interested in the generating function for the squared first-order harmonic numbers, which can be expressed as:

H_2x sum_{n geq 1} H_n^2 x^n

We derive this and similar expressions by utilizing properties of polylogarithms.

Derivation of Generating Functions

The generating function for squared harmonic numbers can be written as:

H_2x frac{log^2(1-x)}{1-x} frac{operatorname{Li}_2(x)}{1-x}

For a more complex series, we define:

H_{2j}x sum_{n geq 1} frac{H_n^2}{n^j} x^n

By integrating twice, we can find expressions for specific values of j:

First Integration

For j1, the first integration yields:

H_{21}x operatorname{Li}_3(x) - operatorname{Li}_2(x) log(1-x) - frac{1}{3} log^3(1-x)

Second Integration

For j2, the second integration yields:

H_{22}x frac{operatorname{Li}_2(x)^2}{2} - 2 operatorname{Li}_4(1-x) 2 operatorname{Li}_4(x) - 2 operatorname{Li}_2(1-x) log^2(1-x) - 4 operatorname{Li}_3(1-x) log(1-x) H_{22}x - frac{1}{3} operatorname{Li}_2(1-x) log^3(1-x) - frac{pi^4}{45}

The last expression can be further simplified to:

H_{22}x frac{operatorname{Li}_2(x)^2}{2} - 2 operatorname{Li}_4(1-x) 2 operatorname{Li}_4(x) - 2 operatorname{Li}_2(1-x) log^2(1-x) - 4 operatorname{Li}_3(1-x) log(1-x) - frac{1}{3} operatorname{Li}_2(1-x) log^3(1-x) - frac{pi^4}{45}

Evaluation of the Desired Series

The desired series is given by:

H_{23}1/2

This is computed through an integral:

H_{23}x int_0^x frac{H_{22}t}{t} dt

The final integral to evaluate is:

int left(frac{operatorname{Li}_2(e^u)^2}{2} operatorname{Li}_4(e^u) - 2 operatorname{Li}_4(1-e^u) - 2 operatorname{Li}_2(1-e^u) log^2(1-e^u) - 4 operatorname{Li}_3(1-e^u) log(1-e^u) - frac{1}{3} u log^3(1-e^u) - frac{pi^4}{45}right) du bigg|_{u rightarrow log x}

Conclusion

While evaluating all the terms in the last integral is complex, the method outlined provides a comprehensive approach to computing generating functions for squared harmonic numbers. This process involves intricate manipulations of polylogarithms and integrals, which are fundamental tools in advanced analysis.

Keywords: harmonic numbers, generating functions, polylogarithms