Asymptotic expansions of integrals and Nielsen's polylogarithms

1. Problem Statement

The authors investigate the asymptotic behavior of a specific family of integrals as the parameter $n \to \infty$. The integral is defined as:
$$I_n = \int_0^1 f(u)(1 + q \cdot u^n)^{w/n} \, du$$
where:

• $f(u)$ is an analytic function on $[0, 1]$ with finite absolute integrals.
• $w \in \mathbb{R} \setminus \{0\}$ is a real parameter.
• $q \in (-1, 1]$ is a parameter (with specific conditions for $q=-1$).

The primary goal is to derive a full asymptotic expansion of the form:
$$I_n \sim \sum_{p=0}^{\infty} \frac{a_p}{n^p}$$
and to characterize the coefficients $a_p$ in terms of special functions, specifically Nielsen's generalized polylogarithms, Multiple Zeta Values (MZVs), and ordinary Riemann zeta values.

The paper addresses three specific motivations:

1. Mathematical Problems: Solving specific integrals posed in the American Mathematical Monthly (e.g., involving $|\sin^n x - \cos^n x|$).
2. Number Theory: Understanding the reducibility of coefficients to ordinary zeta values versus alternating multiple zeta values.
3. Probability Theory: Analyzing the moments of norms of random vectors (e.g., $Z_n = \|(U, 1-U)\|_n$ where $U$ is uniform).

2. Methodology

The derivation proceeds through a multi-step analytical approach combining series expansion, moment analysis, and combinatorial identities.

A. Series Expansion and Stirling Numbers

The term $(1 + q u^n)^{w/n}$ is expanded using the exponential-logarithmic representation:
$$(1 + q u^n)^{w/n} = \exp\left(\frac{w}{n} \log(1 + q u^n)\right) = 1 + \sum_{k=1}^\infty \frac{1}{k!} \left(\frac{w}{n} \log(1 + q u^n)\right)^k$$
The authors then expand $\log^k(1 + q u^n)$ using unsigned Stirling numbers of the first kind ($\left[ \begin{smallmatrix} m \\ k \end{smallmatrix} \right]$):
$$\log^k(1 + qx) = k! \sum_{m=0}^\infty \frac{(-1)^{m-k}}{m!} \left[ \begin{smallmatrix} m \\ k \end{smallmatrix} \right] q^m x^m$$
This transforms the integral into a double series involving the moments of $f(u)$, defined as $\phi_r = \int_0^1 f(u) u^r du$.

B. Asymptotics of Moments (Watson's Lemma)

To handle the moments $\phi_r$ as $r \to \infty$, the authors apply Watson's Lemma for Laplace-type integrals. By substituting $u = e^{-t}$, the moment becomes:
$$\phi_r = \int_0^\infty e^{-rt} g(t) \, dt, \quad \text{where } g(t) = e^{-t}f(e^{-t})$$
This yields an asymptotic expansion for $\phi_r$ in terms of the derivatives of $f$ at $u=1$:
$$\phi_r \sim \sum_{\nu=0}^\infty \frac{\beta_\nu}{r^{\nu+1}}, \quad \beta_\nu = (-1)^\nu \sum_{\ell=0}^\nu \left\{ \begin{smallmatrix} \nu+1 \\ \ell+1 \end{smallmatrix} \right\} f^{(\ell)}(1)$$
where $\left\{ \begin{smallmatrix} n \\ k \end{smallmatrix} \right\}$ are Stirling numbers of the second kind.

C. Connection to Nielsen's Polylogarithms

Substituting the moment expansion back into the integral expansion, the coefficients $a_p$ are expressed as linear combinations of Nielsen's generalized polylogarithms $S_{m,p}(z)$:
$$S_{m,p}(z) = \frac{(-1)^{m+p-1}}{(m-1)!p!} \int_0^1 \frac{\log^{m-1}(t) \log^p(1-zt)}{t} dt$$
The key identity established is:
$$S_{m,p}(-q) = \sum_{j=p}^\infty \left[ \begin{smallmatrix} j \\ p \end{smallmatrix} \right] \frac{(-q)^j}{j! j^m}$$
This links the integral coefficients directly to $S_{m,p}(-q)$.

D. Appell Sequences and Symmetry

The paper introduces a class of functions $f(u)$ related to Appell sequences of polynomials (e.g., Bernoulli, Euler, Genocchi, Hermite). If $f(u) = b \cdot A(c \ln u)$ where $A(t)$ is the seed of an Appell sequence, the coefficients $\beta_\nu$ simplify to evaluations of the corresponding polynomials.

A critical contribution is the derivation of a symmetry condition on the coefficients $\beta_\nu$. The authors prove that if the coefficients satisfy a specific weighted binomial transform symmetry, the alternating multiple zeta values (arising from $q=1$) reduce to ordinary zeta values.

3. Key Contributions and Results

1. General Asymptotic Formula (Theorem 3)

The paper provides a complete asymptotic expansion for $I_n$:
$$I_n \sim \int_0^1 f(u) du + \sum_{p=2}^\infty \frac{a_p}{n^p}$$
where the coefficients are:
$$a_p = \sum_{\ell=1}^{p-1} (-w)^{p-\ell} \beta_{\ell-1} S_{\ell, p-\ell}(-q)$$

• For $q = -1$, $S_{\ell, p-\ell}(1)$ reduces to ordinary Multiple Zeta Values (specifically $\zeta(\ell+1, \{1\}^{p-\ell-1})$), which are known to be polynomials in ordinary Riemann zeta values $\zeta(k)$.
• For $q = 1$, the coefficients involve alternating multiple zeta values (colored MZVs).

2. Reduction to Ordinary Zeta Values (Theorem 5)

The authors establish a precise condition under which the $q=1$ case (typically alternating) reduces to ordinary zeta values. This occurs if the coefficients $\beta_\nu$ satisfy a symmetry condition related to a binomial transform:
$$\sum_{\ell=1}^\nu (-1)^{\ell-\nu} \binom{\nu-1}{\ell-1} w^\ell \beta_{p-\ell-1} = \sum_{\ell=1}^{p-\nu} (-1)^{\ell-p+\nu} \binom{p-\nu-1}{\ell-1} w^\ell \beta_{p-\ell-1}$$
This condition is equivalent to the generating polynomial of the transformed sequence being symmetric (or anti-symmetric) about $x=1/2$.

3. Application to Appell Families (Section 3.4)

The paper verifies that the symmetry condition holds for several classical families of polynomials under specific parameter constraints:

• Generalized Euler Polynomials: Satisfy the condition if $cd = w+2$.
• Generalized Genocchi Polynomials: Satisfy the condition if $cd = w+2$ and $d$ is even.
• Generalized Bernoulli Polynomials: Satisfy the condition if $cd = w+2$.
• Probabilist's Hermite Polynomials: Satisfy the condition if $w = -2$.

4. Probabilistic Applications (Section 4)

The theory is applied to concrete problems:

• Sine-Cosine Integrals: The integral $\int_0^{\pi/2} |\sin^n x - \cos^n x|^{1/n} dx$ is solved, providing explicit coefficients involving $\zeta(3), \zeta(5)$, etc.
• Norms of Random Vectors:
  • For $Z_n = \|(U, 1-U)\|_n$, the moments are expanded using generalized Euler polynomials.
  • For $Z_n = \|(e^{Y/2}, e^{-Y/2})\|_n$ with $Y \sim N(0,1)$, the inverse square moment ($w=-2$) is expanded using Hermite polynomials, yielding an expansion in ordinary zeta values.

4. Significance

• Unification: The paper unifies the study of diverse integrals (from analysis, number theory, and probability) under a single framework involving Nielsen's polylogarithms.
• Reduction Criterion: It provides a rigorous, verifiable criterion (the symmetry of the binomial transform of coefficients) for determining when complex alternating multiple zeta values collapse into the simpler ordinary zeta values. This resolves ambiguity in previous studies where such reductions were observed empirically but not theoretically justified.
• Explicit Coefficients: The authors provide explicit formulas for the asymptotic coefficients for a wide range of functions $f(u)$, including those related to Bernoulli, Euler, and Hermite polynomials, allowing for high-precision numerical approximations of these integrals.
• Probabilistic Insight: The work clarifies the asymptotic behavior of norms of random vectors, linking the convergence of moments to the structure of multiple zeta values.

In summary, Kuba and Levy provide a powerful analytical toolkit that bridges asymptotic analysis, special functions, and combinatorial number theory, offering both general theoretical results and concrete solutions to specific open problems.