On the zeros of Lerch's transcendental function with real parameters (Q753989)

The power series \(f_{\kappa,\lambda}(z):=\sum^{\infty}_{n=0}(n+\lambda)^{\kappa}z^ n\) for \(| z| <1\) together with its analytic extension into \({\mathbb{C}}\setminus [1,\infty)\) is considered for \(\kappa >0\), \(0\leq \lambda <1\). If \(k<\kappa \leq k+1\) holds for some \(k\in {\mathbb{N}}_ 0\), then \(f_{\kappa,\lambda}\) has exactly \(k+1\) zeros in \({\mathbb{C}}\setminus [1,\infty)\), which are simple, real and \(\leq 0\) and may be denoted by \(z_{\kappa,k}(\lambda)<z_{\kappa,k- 1}(\lambda)<...<z_{\kappa,1}(\lambda)<z_{\kappa,0}(\lambda)\leq 0\). The authors derive the following asymptotic estimate for \(\kappa\to \infty\) and for \(\nu\) in an interval depending on \(\kappa\) : \[ (1)\quad z_{\kappa,\nu}(\lambda)=-\exp (-\pi \cot an(\frac{2(\nu +\lambda)+1}{\kappa +1}\frac{\pi}{2}))+r_{\kappa}, \] where an explicit upper bound for \(| r_{\kappa -1}|\) \((=O(c^{\kappa /2}/\kappa)\) for some \(0<c<1)\) is given which is too complicated to be staed here. For \(\kappa =m\in {\mathbb{N}}\), \(f_{m,\lambda}(z)=P_{m,\lambda}(z)/(1-z)^{m+1}\) holds. In this case upper and lower estimates for the main root \(\max \{z_{m,\nu}(\lambda):\) \(0\leq \nu \leq m\), \(z_{m,\nu}(\lambda)\leq -1\}\) for the Euler-Frobenius polynomial \(P_{m,\lambda}(z)\) are obtained from (1). Furthermore, the authors derive monotonicity properties of \(z_{\kappa,\nu}(\lambda)\) as functions of \(\kappa\) and \(\lambda\).