A generating function for a class of generalized Bernoulli polynomials
The authors prove: Theorem 1: Let \(B_0(x)\) be a periodic function of period \(1\) with a continuous derivative in the open interval \((0,1)\). Let \(a_0=\int_0^1B_0(x)\,dx\). Let \(\{B_n\}_{n\geq 0} \) be the sequence defined by \[ B_n(x)=\int_0^xB_{n-1}(y)\,dy+\int_0^1(y-1)B_{n-1}(y)\,dy,\quad n\geq 1. \] For each \(x\in (0,1)\) we have \[ \sum_{k=0}^\infty B_k(x)t^k=\frac{te^{xt}}{e^t-1}\left[a_0-\int_0^1B_0'(1-y)\frac{e^{ty}-1}{t}\,dy\right]+\int_0^xe^{t(x-y)}B_0'(y)\,dy. \] Let \(a_j\) and \(b_j\) be the Fourier coefficients of \(B_o(x)\). Then \[ B_n(x)=\text{Re}\left[2\sum_{j=1}^\infty\frac{e^{2\pi ijx}}{(2\pi ij)^n}(a_j-a_0-ib_j)\right]. \] Theorem 2: Let \(\{\lambda_j\}_{j\geq 1}\) be a periodic sequence of period \(T\geq 2\). Let \(B_0(x)\), \(\{B_n\}_{n\geq 0} \), \(a_j\) and \(b_j\) be as in Theorem 1. Consider the Dirichlet series \[ f(s)=\sum_{j=1}^\infty\frac{a_j-a_0}{j^s}\lambda_j \] and \[ g(s)=\sum_{j=1}^\infty\frac{b_j}{j^s}\lambda_j. \] For each \(j\geq 1\) let \[ \alpha_j=\sum_{k=1}^T\lambda_k\cos\left(2\pi j\frac{k}{T}\right)\quad\text{and}\quad \beta_j=\sum_{k=1}^T\lambda_k\sin\left(2\pi j\frac{k}{T}\right). \] If \(\lambda_{T-j}=\lambda_j\) for each \(j\in\{1,\ldots,T-1\} \) then for all \(n\geq 0\) \[ f(2n)=\frac{(-1)^n}{2T}(2\pi)^{2n}\sum_{j=1}^T\alpha_jB_{2n}\left(\frac{j}{T}\right), \] \[ g(2n+1)=\frac{(-1)^{n+1}}{2T}(2\pi)^{2n+1}\sum_{j=1}^T\alpha_jB_{2n+1}\left(\frac{j}{T}\right). \] If \(\lambda_{T-j}=-\lambda_j \) for each \(j\in\{1,\ldots,T-1\} \) and \(\lambda_T=0\) then for all \(n\geq 0\) \[ f(2n+1)=\frac{(-1)^n}{2T}(2\pi)^{2n+1}\sum_{j=1}^T\beta_jB_{2n+1}\left(\frac{j}{T}\right), \] \[ g(2n)=\frac{(-1)^n}{2T}(2\pi)^{2n}\sum_{j=1}^T\beta_jB_{2n}\left(\frac{j}{T}\right). \] As application one obtains classical formulas of Leibniz, Newton and Euler.
scientific article; zbMATH DE number 5636950
scientific article