Contributions To The Theory Of Zeta-functions: The Modular Relation Supremacy: The Modular Relation Supremacy

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:

• Prelude
• Grocery of Special Functions
• Unprocessed Modular Relations
• Fourier-Bessel Expansion H^1;1_1;1↔ H^2;0_0;2
• The Ewald Expansion or the Incomplete Gamma Series
• The Riesz Sums
• The General Modular Relation
• The Hecke Type Zeta-functions
• The Product of Zeta-functions
• Miscellany

Readership: 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 studied
• Given 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 disciplines
• The 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