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Product Code:  SURV/287 
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Softcover ISBN:  9781470478407 
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Product Code:  SURV/287.B 
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Softcover ISBN:  9781470478407 
Product Code:  SURV/287 
List Price:  $135.00 
MAA Member Price:  $121.50 
AMS Member Price:  $108.00 
eBook ISBN:  9781470478926 
Product Code:  SURV/287.E 
List Price:  $125.00 
MAA Member Price:  $112.50 
AMS Member Price:  $100.00 
Softcover ISBN:  9781470478407 
eBook ISBN:  9781470478926 
Product Code:  SURV/287.B 
List Price:  $260.00 $197.50 
MAA Member Price:  $234.00 $177.75 
AMS Member Price:  $208.00 $158.00 

Book DetailsMathematical Surveys and MonographsVolume: 287; 2024; Estimated: 680 ppMSC: Primary 39; Secondary 33; 34; 05; 12
The roots of the modern theories of differential and \(q\)difference equations go back in part to an article by George D. Birkhoff, published in 1913, dealing with the three “sister theories” of differential, difference and \(q\)difference equations. This book is about \(q\)difference equations and focuses on techniques inspired by differential equations, in line with Birkhoff's work, as revived over the last three decades. It follows the approach of the Ramis school, mixing algebraic and analytic methods. While it uses some \(q\)calculus and is illustrated by \(q\)special functions, these are not its main subjects.
After a gentle historical introduction with emphasis on mathematics and a thorough study of basic problems such as elementary \(q\)functions, elementary \(q\)calculus, and low order equations, a detailed algebraic and analytic study of scalar equations is followed by the usual process of transforming them into systems and back again. The structural algebraic and analytic properties of systems are then described using \(q\)difference modules (Newton polygon, filtration by the slopes). The final chapters deal with Fuchsian and irregular equations and systems, including their resolution, classification, RiemannHilbert correspondence, and Galois theory. Nine appendices complete the book and aim to help the reader by providing some fundamental yet not universally taught facts.
There are 535 exercises of various styles and levels of difficulty. The main prerequisites are general algebra and analysis as taught in the first three years of university. The book will be of interest to expert and nonexpert researchers as well as graduate students in mathematics and physics.
ReadershipGraduate students and researchers interested in \(q\)difference equations.

Table of Contents

Introduction

Prelude

Elementary special and $q$special functions

Basic notions and tools

Equations of low order, elementary approach

Resolution of (general) scalar equations and factorisation of $q$difference operators

Further analytic properties of solutions: Index theorems, growth

Equations and systems

Systems and modules

Further algebraic properties of $q$difference modules

Newton polygons and slope filtrations

Fuchsian $q$difference equations and systems: Local study

Fuchsian $q$difference equations and systems: Global study

Galois theory of Fuchsian systems

Irregular equations

Irregular systems

Some classical special functions

Riemann surfaces and vector bundles

Classical hypergeometric functions

Basic index theory

Cochain complexes

Base change and tensor products (and some more facts from linear algebra)

Tannaka duality (without schemes)

Čech cohomology of abelian sheaves

Čech cohomology of nonabelian sheaves

Bibliography

Index of terms

Index of notations

Index of names


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The roots of the modern theories of differential and \(q\)difference equations go back in part to an article by George D. Birkhoff, published in 1913, dealing with the three “sister theories” of differential, difference and \(q\)difference equations. This book is about \(q\)difference equations and focuses on techniques inspired by differential equations, in line with Birkhoff's work, as revived over the last three decades. It follows the approach of the Ramis school, mixing algebraic and analytic methods. While it uses some \(q\)calculus and is illustrated by \(q\)special functions, these are not its main subjects.
After a gentle historical introduction with emphasis on mathematics and a thorough study of basic problems such as elementary \(q\)functions, elementary \(q\)calculus, and low order equations, a detailed algebraic and analytic study of scalar equations is followed by the usual process of transforming them into systems and back again. The structural algebraic and analytic properties of systems are then described using \(q\)difference modules (Newton polygon, filtration by the slopes). The final chapters deal with Fuchsian and irregular equations and systems, including their resolution, classification, RiemannHilbert correspondence, and Galois theory. Nine appendices complete the book and aim to help the reader by providing some fundamental yet not universally taught facts.
There are 535 exercises of various styles and levels of difficulty. The main prerequisites are general algebra and analysis as taught in the first three years of university. The book will be of interest to expert and nonexpert researchers as well as graduate students in mathematics and physics.
Graduate students and researchers interested in \(q\)difference equations.

Introduction

Prelude

Elementary special and $q$special functions

Basic notions and tools

Equations of low order, elementary approach

Resolution of (general) scalar equations and factorisation of $q$difference operators

Further analytic properties of solutions: Index theorems, growth

Equations and systems

Systems and modules

Further algebraic properties of $q$difference modules

Newton polygons and slope filtrations

Fuchsian $q$difference equations and systems: Local study

Fuchsian $q$difference equations and systems: Global study

Galois theory of Fuchsian systems

Irregular equations

Irregular systems

Some classical special functions

Riemann surfaces and vector bundles

Classical hypergeometric functions

Basic index theory

Cochain complexes

Base change and tensor products (and some more facts from linear algebra)

Tannaka duality (without schemes)

Čech cohomology of abelian sheaves

Čech cohomology of nonabelian sheaves

Bibliography

Index of terms

Index of notations

Index of names