On summation formulas due to Plana, Lindelöf and Abel, and related Gauss-Christoffel rules. II (Q1378462)

This paper is the second one in a series of publications on certain summation formulas, listed in the title, and is a continuation of the author's paper [ibid. 37, No. 2, 256-295 (1997; Zbl 0878.65002)]. The main emphasis is on the following summation formula, due to Lindelöf: For a function \(f\), satisfying certain technical conditions, and \(m\in\mathbb{N}_0\), Lindelöf's summation formula reads: \[ \sum^\infty_{k=m} (-1)^kf(k)= (-1)^m \int^\infty_{-\infty} {f\bigl(m- {1\over 2} +i\omega \bigr)\over 2\cosh (\pi\omega)} d\omega. \] The weight function \[ w_L (\omega)= {1\over 2\cosh (\pi\omega)} \] is called Lindelöf density. The paper under review elaborates many interesting results connected with this summation formula, e.g., the three-term recurrence relation for the orthogonal polynomials for the Lindelöf density, weights and nodes for the connected Gauss-Lindelöf formula etc.. What the reviewer maybe impressed most is the large number of numerical results and reports on numerical tests that the author of the paper has examined.