Sums of Powers of Sine and Generalized Bernoulli Polynomials

Technical Summary: Sums of Powers of Sine and Generalized Bernoulli Polynomials

Problem Statement
The paper addresses the derivation of explicit closed-form formulas for the sums of negative integer powers of the sine function evaluated at specific rational multiples of $\pi$. Specifically, the author investigates the quantity:
$$S(s,n) = \sum_{j=1}^{2n-2} \frac{1}{\sin^s \left( \frac{(2j-1)\pi}{2n} \right)}$$
The primary objective is to express these sums in terms of Generalized Bernoulli polynomials, Euler polynomials, and Bernoulli numbers. A secondary goal is to utilize these summation formulas to derive integral representations for the Riemann zeta function, particularly at odd integer arguments, and to establish connections between discrete sums and the zeta function values.

Methodology
The paper employs a combination of generating functions, combinatorial identities, and matrix algebra to derive its results.

1. Generating Functions and Polynomials: The core machinery relies on the generating functions for Generalized Bernoulli polynomials $B_n^{(\alpha)}(x)$ and Euler polynomials $E_n(x)$. The author utilizes known properties of these polynomials, including their derivatives, symmetry relations (e.g., $B_n^{(\alpha)}(\alpha-x) = (-1)^n B_n^{(\alpha)}(x)$), and relationships to Stirling numbers of the first and second kind.
2. Matrix Formulation: A significant portion of the methodology involves defining a set of $(2n-2) \times (2n-2)$ matrices, denoted $M_n$. These matrices relate powers of the sine function to linear combinations of sine functions with different arguments. Specifically, the paper establishes that:
   $$\frac{1}{\sin^r \left( \frac{(2j-1)\pi}{2n} \right)} = 2^r \sum_{k=1}^{2n-2} M_n(j,k) \sin \left( \frac{(2k-1)\pi}{2n} \right)$$
   The author analyzes the properties of these matrices, noting that for odd powers $r$, the matrices commute and are determined entirely by their first row.
3. Inductive Proofs: The derivation of the general formulas for $S(2m+1, n)$ (negative odd powers) and $S(2k, n)$ (negative even powers) is conducted via induction. The author demonstrates how multiplying by $\sin(\pi/2n)$ or $1/\sin(\pi/2n)$ transforms the coefficients of the matrix rows, allowing for the transition from power $r$ to $r \pm 1$.
4. Limit Processes: To connect the discrete sums to the Riemann zeta function, the paper utilizes the limit definition:
   $$\zeta(m) = \lim_{n \to \infty} \left( \frac{2^m \pi^m}{2^m - 1} \right) \frac{1}{2^{mn-m}} S(m,n)$$
   By analyzing the asymptotic behavior of the derived formulas as $n \to \infty$, the author isolates the leading terms to produce integral representations.

Key Contributions and Results

• Formulas for Odd Powers: The paper derives a general formula for $S(2m+1, n)$ expressed as a sum involving Euler polynomials and Generalized Bernoulli polynomials.

  • Theorem 4.1 presents the explicit formula:
    $$S(2m+1,n) = \frac{(-1)^m 2^{2mn}}{(2m)!} \sum_{j=1}^{2n-2} \sum_{k=0}^{m-1} \left[ \frac{2(n-1)(-2k-1)}{2^{2k+1}} \binom{2m-1}{2k} B_{2k}^{(2m)}(m) \right] \left( 2n-1 E_{2m-2k}\left(\frac{j}{2n-1}\right) - m E_{2m-2k-1}\left(\frac{j}{2n-1}\right) \right) \frac{1}{\sin \left( \frac{(2j-1)\pi}{2n} \right)}$$
  • This result generalizes specific cases (e.g., $S(3,n), S(5,n)$) previously listed in the introduction.

• Formulas for Even Powers: The paper provides a closed-form expression for $S(2k, n)$ involving Bernoulli numbers and the Riemann zeta function.

  • Theorem 5.1 states:
    $$S(2k,n) = \sum_{j=1}^{k} \frac{(-1)^{j+k} 2^{2j(n-2)+2k}(2^{2j}-1)}{\pi^{2j}(2(k-j))!} B_{2(k-j)}^{(2k)}(k) \zeta(2j)$$
  • This formula allows for the computation of $S(2k,n)$ using a finite sum of zeta values, significantly reducing computational complexity compared to direct summation.

• Integral Representation of $\zeta(2j+1)$:

  • Corollary 4.2 derives a new integral representation for the Riemann zeta function at odd integers:
    $$\zeta(2j+1) = \frac{(-1)^j 2^{2j} \pi^{2j+1}}{(2j)!(2^{2j+1}-1)} \int_0^{1/2} E_{2j}(x) \csc(\pi x) \, dx$$
  • This is obtained by taking the limit of the discrete sum formula for $S(2m+1, n)$ as $n \to \infty$, effectively converting the Riemann sum into an integral.

• Matrix Properties: The paper details the structure of the matrices $M_n$ for both odd and even powers. It proves that for odd powers, the matrices commute and share permutation structures, whereas for even powers, the matrices do not commute, though the sum of elements in specific columns vanishes.

Significance and Claims
The paper claims to produce explicit formulas for sums of powers of sine that were previously only known for low-order cases or required complex recursive computation. By linking these sums to Generalized Bernoulli and Euler polynomials, the author provides a unified algebraic framework for these trigonometric sums.

The primary significance highlighted by the author is the derivation of the integral representation for $\zeta(2j+1)$. While the Riemann zeta function at even integers is well-understood via Bernoulli numbers, the values at odd integers remain less tractable. The paper presents this integral form as a "nice" representation derived directly from the discrete summation properties of the sine function. Additionally, the paper notes the computational efficiency of the derived formulas for even powers, contrasting the $O(n)$ operations required for direct summation with the constant number of operations required when using the closed-form expression involving zeta values.

The work is presented as a theoretical exploration of the relationships between trigonometric sums, special polynomials, and the Riemann zeta function, relying on established references for foundational proofs while extending the specific formulas for general $n$ and $s$.