A two variable Riemann zeta-function
This paper is essentially a pendant to two earlier papers of the author [J. Lond. Math. Soc., II. Ser. 20, 101--108 (1979; Zbl 0401.12017); ibid. 11, 256 (1975; Zbl 0335.10017)]. The numbers \(A^*_{m,n}\) may be defined by \[ A^*_{m,n}=\frac{1}{n+1}\sum^{n+1}_{j=1}(-1)^ j \left( \begin{matrix} n+1\\ j\end{matrix} \right) B_{m+j} B_{n+1-j}\quad (m,n\geq 0) \] where \(B_ n\) is the \(n\)th Bernoulli number (in the notation with \(B_{2m+1}=0\) for \(m\geq 1)\). The author suggests that the \(A^*_{m,n}\) are a genuinely two variable analogue of the Bernoulli numbers. In this direction he proves the following results. Define the ''poly-Dirichlet series'' \(Y(w,z)\), for \(\text{Re}\,w>2\), \(\text{Re}\,z>2\), by \[ Y(w,z) = (z-1)\sum^{\infty}_{m\leq n, m=1}\sum^{\infty}_{n=1}m^{-w+1} n^{- z}-(w-1)\sum^{\infty}_{m>n, m=1}\sum^{\infty}_{n=1}m^{-w} n^{-z+1}. \] Then \(Y(w,z)\) is analytically continuable to an entire function of \((w,z)\in\mathbb C\times\mathbb C\) and satisfies the equation \(Y(w,z)+Y(z,w)=Z(z+w-1)\) where \(Z(s)=(s-1)\zeta (s)\), \(\zeta\) the Riemann zeta-function. Furthermore \[ Y(1-m,1-n) = (-1)^{m+1} A^*_{m,n}+(-1)^ m B_ m \delta_{n0} \] where \(\delta_{nm}\) is the Kronecker delta. Various relationships and identities for the \(A^*_{m,n}\) are also exhibited. At the paper's conclusion, the author raises the question whether there exists a ``poly-Dirichlet series'' entire function \(X\) of two complex variables satisfying \(X(z,w)=X(w,z)\) and \(X(1-m,1-n)=A^*_{m,n}\).
two variable analogue of the Bernoulli numbers
scientific article; zbMATH DE number 4010612