Table Of ContentElliptic
Poly:nornials
Library of Congress Cataloging-in-Publication Data
Lomont, JohnS., 1924-
Elliptic polynomials I J.S. Lomont, John Brillhart.
p.cm.
Includes bibliographical references and indexes.
ISBN 1-58488-210-7 (alk. paper)
1. Elliptic functions. 2. Polynomials. I. Brillhart, John, 1930-D. Title.
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Contents
Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
Tables ............................................................... ix
Preface .............................................................. xi
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv
1. Binomial Sequences of Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
(a) The functions F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
(b) The functions Fo- The sequences {Gm(z)} and {Hm(z)} ..... 8
2. The Binomial Sequences Generated from /-1, f E Fo ............ 25
3. The FUnctions F1 - Elliptic Polynomials of the First Kind . . . . . . . 33
4. The Moment Polynomials- Pn(x,y), Qn(x,y), and .Rn(x,y) ..... 43
5. The FUnctions :F11 - Elliptic Polynomials of Second Kind . . . . . . . 71
6. Inner Products, Integrals, and Moments- Favard's Theorem ..... 89
7. The FUnctions F2- Orthogonal Sequences {Gm(z)}, {Hm(z)} ... 97
8. The FUnctions in Classes I and II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
(a) Class I Functions-The elliptic functions sn (t, k) ........... 111
(b) Class II Functions- The polynomials f*(t, k) ............... 126
9. The Tangent Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
10. Class III Functions- The On(x) Polynomials .................... 145
11. Coefficients of the 0 (x) Polynomials ............................ 189
11
12. The >. (x) Polynomials ......................................... 197
11
13. The Orthogonal Sequences {Am(z)} and {Bm(z)} .............. 211
14: The Weight FUnctions for {Am(z)} and {Bm(z)} ............... 225
15. Miscellaneous R.esults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
16. Uniqueness and Completion R.esults ............................ 247
17. Polynomial Inequalities . .. . . . . . . . . . . .. . .. . . . . . . . . . .. . . . . . . . . . . . . 251
18. Some Concluding Questions . . . . . . . . . . . . . . . .. . . . . . .. .. . . . . . . . . . . 269
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
Name Index .................................................... 277
Subject Index .................................................. 281
Symbol Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
Tables
Table 1.1 ln(x), 0 ~ n ~ 11, I E :Fo ................................. 9
Table 1.2 Gm(z), 0 ~ m ~ 4, IE :Fo ............................... 18
Table 1.3 Hm(z), 0 ~ m ~ 4, IE :Fo ............................... 18
Table 2.1 iln, 1 $ n $ 5, I E :F . • . . . . . . • . • . . . . . • • . • . . • . . . . . • . • . • . . • 26
Table 2.2 ii2n+t. 0 ~ n ~ 4, IE :Fo ................................ 28
Table 2.3 /n(x), 0 ~ n ~ 5, I E :Fo ................................. 29
Table 3.1 Familiar Functicms in the Six Classes ................... 42
Table 4.1 Pn(X, y), 0 ~ n ~ 11 ..................................... 46
Table 4.2 Qn(x,y), 0 ~ n ~ 11 ..................................... 46
Table 4.3 Rn(x,y), 0 ~ n ~ 11 ..................................... 47
Table 5.1 a~n>, 0 ~ n ~ 13 ........................................ 86
Table 5.2 f3i2"), 0 ~ n ~ 13 .. .. .. .. .. .. .. . .. .. . .. .. .. .. .. .. .. .. . .. . 87
Table 7.1 J.'G(n), 0 ~ n ~ 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
Table 8.1 f>2n(k), 0 ~ n ~ 9 ....................................... 112
Table 8.2 6n(x,y), 0 ~ n ~ 9 ...................................... 114
Table 8.3 Am(z, k), 0 ~ m ~ 5 .................................... 118
Table 8.4 Bm(z, k), 0 ~ m ~ 5 .................................... 118
Table 8.5 -C2n+l(k), 0 ~ n ~ 11 ................................. 122
Table 8.6 D2n+2(k), 0 ~ n ~ 11 ................................... 123
Table 8.7 A~(z, k), 0 ~ m ~ 5 .................................... 127
Table 8.8 B~(z,k), 0 ~ m ~ 5 .................................... 127
Table 8.9 A~(z, ~), 0 ~ m ~ 5 .................................. 130
Table 8.10 B:.(z, ~), 0 ~ m ~ 5 ................................. 130
Table 10.1 6n(x), 0 ~ n ~ 11 ...................................... 148
cljn>,
Table 11.1 0 ~ n ~ 8, 0 ~ j ~ n .. .. .. .. .. .. .. .. .. .. .. .. .. .. . 191
m
Table 11.2 {r(m), 1 $ r $ 6, $ m $ 6 ......................... 192
Table 12.1 ~n(x), 0 ~ n ~ 11 ..................................... 199
Table 12.2 ~~n), 0 ~ n ~ 8, 0 ~ j ~ n .. .. .. . .. .. .. .. .. .. .. .. .. .. .. 202
Table 13.1 Am(z), 0 ~ m ~ 9 ..................................... 212
Table 13.2 Bm(z), 0 ~ m ~ 9 ..................................... 215
