On summation formulas due to Plana, Lindelöf and Abel, and related Gauss-Christoffel rules. III
This paper is the third and last one in a series of publications concerned with several summation formulas [for part II see ibid. 37, No. 4, 804-832 (1997; Zbl 0893.65001)]. The main emphasis is now on Planas summation formula, which is \[ \sum^\infty_{k= m} f(k)= {f(m)\over 2}+ \int^\infty_m f(\sigma) d\sigma- \int^\infty_{-\infty} g^P_m(\omega) w^P(\omega) d\omega, \] \[ \lim_{n\to\infty} T(m:n)f= -\int^\infty_{-\infty} g^P_m(\omega) w^P(\omega) d\omega. \] Here, \[ g^P_\sigma(\omega)= {f(\sigma+ i\omega)- f(\sigma- i\omega)\over 2i\omega},\quad w^P(\omega)= {|\omega|\over e^{| 2\pi\omega|}- 1} \] and \(f\) has to satisfy several technical assumptions. Moreover, some related density functions and summation formulas are investigated. Throughout the whole paper, a lot of numerical examples illustrate the author's considerations.
scientific article; zbMATH DE number 1275242