Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 17 February 2020 doi:10.20944/preprints202002.0245.v1 Peer-reviewed version available at Mathematics 2020, 8, 407; doi:10.3390/math8030407 Article Limiting values and functional and difference equations † N. -L. Wang 1, P. Agarwal 2,* and S. Kanemitsu 3 1 College of Applied Mathematics and Computer Science, ShangLuo University, Shangluo, 726000 Shaanxi, P.R. China; E-mail:
[email protected] 2 Anand International College of Engineering, Near Kanota, Agra Road, Jaipur-303012, Rajasthan, India 3 Faculty of Engrg Kyushu Inst. Tech. 1-1Sensuicho Tobata Kitakyushu 804-8555, Japan;
[email protected] * Correspondence:
[email protected] † Dedicated to Professor Dr. Yumiko Hironaka with great respect and friendship Version February 17, 2020 submitted to Journal Not Specified Abstract: Boundary behavior of a given important function or its limit values are essential in the whole spectrum of mathematics and science. We consider some tractable cases of limit values in which either a difference of two ingredients or a difference equation is used coupled with the relevant functional equations to give rise to unexpected results. This involves the expression for the Laurent coefficients including the residue, the Kronecker limit formulas and higher order coefficients as well as the difference formed to cancel the inaccessible part, typically the Clausen functions. We also state Abelian results which yield asymptotic formulas for weighted summatory function from that for the original summatory function. Keywords: limit values; modular relation; Lerch zeta-function; Hurwitz zeta-function; Laurent coefficients MSC: 11F03; 01A55; 40A30; 42A16 1. Introduction There have appeared enormous amount of papers on the Laurent coefficients of a large class of zeta-, L- and special functions.