Extensions of Euler-Type Sums and Ramanujan-Type Sums

Abstract

We define a new kind of classical digamma function, and establish some of its fundamental identities. Then we apply the formulas obtained, and extend tools developed by Flajolet and Salvy to study more general Euler-type sums. The main results of Flajolet and Salvy's paper (Expo. Math. 7(1) (1998), 15-35) are immediate corollaries of the main results in this paper. Furthermore, we provide some parameterized extensions of Ramanujan-type identities that involve hyperbolic series. Some interesting new consequences and illustrative examples are considered.

Comparison of Contents Design in Mathematical Inquiry in High School Textbooks

“…We remark that all the degree bounds in Theorem 1.1 and Theorem 1.2 are optimal. We record six examples to illustrate Theorem 1.1 and Theorem 1.2 (for the first four examples, see [16,Examples 4.2 and 4.5] and [25,Examples 6.4 and 8.13]).…”