Overview

This volume provides a systematic survey of almost all the equivalent assertions to the functional equations — zeta symmetry — which zeta-functions satisfy, thus streamlining previously published results on zeta-functions. The equivalent relations are given in the form of modular relations in Fox H -function series, which at present include all that have been considered as candidates for ingredients of a series. The results are presented in a clear and simple manner for readers to readily apply without much knowledge of zeta-functions.This volume aims to keep a record of the 150-year-old heritage starting from Riemann on zeta-functions, which are ubiquitous in all mathematical sciences, wherever there is a notion of the norm. It provides almost all possible equivalent relations to the zeta-functions without requiring a reader's deep knowledge on their definitions. This can be an ideal reference book for those studying zeta-functions.Contents: PreludeGrocery of Special FunctionsUnprocessed Modular RelationsFourier-Bessel Expansion H 1;11;1↔ H 2;00;2The Ewald Expansion or the Incomplete Gamma SeriesThe Riesz SumsThe General Modular RelationThe Hecke Type Zeta-functionsThe Product of Zeta-functionsMiscellanyReadership: Graduate students and researchers in functional equations and special functions.Key Features: The book gives a thorough description of a.a. possible equivalent relations to the functional equation in terms of the H -function series for zeta-functions that have been studiedGiven a functional equation, the book immediately informs the reader of the direction to be taken to reach the ultimate aim concerning the zeta-function in both pure and applied disciplinesThe reader can forget about the difficult definition of zeta-functions themselves and can refer to the form of functional equation and check how multiple is the gamma factor

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