Contents
Preface vii
Chapter 1. Symmetri c function s 1
1.1. Alphabet s 1
1.2. Partition s 1
1.3. Generatin g Function s o f Symmetri c Function s 5
1.4. Matri x Generatin g Function s 7
1.5. Cauch y Formul a 13
1.6. Scala r Produc t 14
1.7. Differentia l Calculu s 16
1.8. Operator s o n Isobaric Determinant s 18
1.9. Pier i Formula s 2 2
Exercises 2 4
Chapter 2 . Symmetri c Function s a s Operator s an d A-Ring s 3 1
2.1. Algebrai c Operation s o n Alphabet s 3
2.2. Lambd a Operation s 3 2
2.3. Interpretin g Polynomial s an d g-serie s 3 3
2.4. Lagrang e Inversio n 3 5
2.5. Som e relations betwee n multiple s o f alphabets 3 6
Exercises 3 8
Chapter 3 . Euclidea n Divisio n 4 7
3.1. Euclid' s Algorith m 4 7
3.2. Remainder s a s Schu r function s 5 0
3.3. Companio n Matri x 5
3.4. Bezout' s Matri x 5 2
3.5. Sylvester' s Summation s 5 4
3.6. Stur m Sequenc e 5 5
3.7. Wrohski' s Algorith m 5 7
3.8. Divisio n an d Continue d Fraction s 5 7
Exercises 5 9
Chapter 4 . Reciproca l Difference s an d Continue d Fraction s 6 3
4.1. Euler' s Recursion s 6 3
4.2. Continue d Fractio n Expressio n o f a Formal Serie s 6 4
4.3. Interpolatio n o f a Function b y a Continue d Fractio n 6 6
4.4. Relatio n betwee n Stieltje s an d Wronsk i Continue d Fraction s 6 9
4.5. Jacobi' s Tridiagona l Matri x 7 0
4.6. Motzki n Path s 7
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