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Stephen C. . Ed(S): Milne - Infinite Families of Exact Sums of Squares Formulas, Jacobi Elliptic Functions, Continued Fractions, and Schur Functions - 9781402004919 - V9781402004919
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Infinite Families of Exact Sums of Squares Formulas, Jacobi Elliptic Functions, Continued Fractions, and Schur Functions

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Description for Infinite Families of Exact Sums of Squares Formulas, Jacobi Elliptic Functions, Continued Fractions, and Schur Functions Hardback. Jacobi's elliptic function approach dates from his epic Fundamenta Nova of 1829. This title employs his combinatorial/elliptic function methods to derive many infinite families of explicit exact formulas involving either squares or triangular numbers, two of which generalize Jacobi's (1829) 4 and 8 squares identities to 4n2 or 4n(n+1) squares. Editor(s): Milne, Stephen C. Series: Developments in Mathematics. Num Pages: 143 pages, biography. BIC Classification: PBH. Category: (P) Professional & Vocational; (UP) Postgraduate, Research & Scholarly. Dimension: 164 x 236 x 12. Weight in Grams: 398.

The problem of representing an integer as a sum of squares of integers is one of the oldest and most significant in mathematics. It goes back at least 2000 years to Diophantus, and continues more recently with the works of Fermat, Euler, Lagrange, Jacobi, Glaisher, Ramanujan, Hardy, Mordell, Andrews, and others. Jacobi's elliptic function approach dates from his epic Fundamenta Nova of 1829. Here, the author employs his combinatorial/elliptic function methods to derive many infinite families of explicit exact formulas involving either squares or triangular numbers, two of which generalize Jacobi's (1829) 4 and 8 squares identities to 4n2 or ... Read more

The author derives his formulas by utilizing combinatorics to combine a variety of methods and observations from the theory of Jacobi elliptic functions, continued fractions, Hankel or Turanian determinants, Lie algebras, Schur functions, and multiple basic hypergeometric series related to the classical groups. His results (in Theorem 5.19) generalize to separate infinite families each of the 21 of Jacobi's explicitly stated degree 2, 4, 6, 8 Lambert series expansions of classical theta functions in sections 40-42 of the Fundamental Nova. The author also uses a special case of his methods to give a derivation proof of the two Kac and Wakimoto (1994) conjectured identities concerning representations of a positive integer by sums of 4n2 or 4n(n+1) triangular numbers, respectively. These conjectures arose in the study of Lie algebras and have also recently been proved by Zagierusing modular forms. George Andrews says in a preface of this book, `This impressive work will undoubtedly spur others both in elliptic functions and in modular forms to build on these wonderful discoveries.'

Audience: This research monograph on sums of squares is distinguished by its diversity of methods and extensive bibliography. It contains both detailed proofs and numerous explicit examples of the theory. This readable work will appeal to both students and researchers in number theory, combinatorics, special functions, classical analysis, approximation theory, and mathematical physics.

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Product Details

Format
Hardback
Publication date
2002
Publisher
Kluwer Academic Publishers United States
Number of pages
143
Condition
New
Series
Developments in Mathematics
Number of Pages
143
Place of Publication
New York, NY, United States
ISBN
9781402004919
SKU
V9781402004919
Shipping Time
Usually ships in 15 to 20 working days
Ref
99-15

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