A partition formula for the integer coefficients of the theta function nome
Authors:
Helaman Rolfe Pratt Ferguson, Dale E. Nielsen and Grant Cook
Journal:
Math. Comp. 29 (1975), 851855
MSC:
Primary 33A25
MathSciNet review:
0367322
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Abstract 
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Abstract: In elliptic function theory, the nome q can be given as a power series in with integer coefficients, . Heretofore, the first 14 coefficients were calculated with considerable difficulty. In this paper, an explicit and general formula involving partitions is given for all the . A table of the first 59 of these integers is given. The table is of numbertheoretical interest as well as useful for calculating complete and incomplete elliptic integrals.
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E. T. WHITTAKER & G. N. WATSON, A Course of Modem Analysis, 4th ed., Cambridge Univ. Press, New York, 1927, Sect. 21.8, pp. 485486.
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Eugene
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M. MilneThomson, Jacobian elliptic function tables. A guide to
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𝑑𝑛𝑢, 𝑍(𝑢), Dover Publications
Inc., New York, N. Y., 1950. MR 0048143
(13,987g)
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Einar
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Higher Mathematics, Ginn and Co., Boston, Mass.New YorkToronto, Ont.,
1962. MR
0201608 (34 #1490)
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I. S. GRADŠTEĬN & I. M. RYŽIK, Tables of Integrals, Series and Products, 4th ed., Fizmatgiz, Moscow, 1963; English transl., Academic Press, New York, 1965, pp. 921925. MR 28 #5198; 33 #5952.
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A. WEIERSTRASS, "Zur Theorie der Elliptischen Funktionen," Werke, v. 2, 1895, pp. 275276.
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L. M. MILNETHOMSON, "Ten decimal table of the nome q," J. London Math. Soc., v. 5, 1930, pp. 148149.
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A.
N. Lowan, G.
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𝑞series associated with Jacobian elliptic functions, Bull. Amer. Math. Soc. 48 (1942), 737–738. MR 0007119
(4,90e), http://dx.doi.org/10.1090/S000299041942077706
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C. E. VAN ORSTRAND, "Reversion of power series," Philos. Mag., v. 19, 1910, pp. 366376.
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James
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H. RADEMACHER, "On the partition function," Proc. London Math. Soc., v. 43, 1937, pp. 241254.
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We thank the BYU Scientific Computation Center and their resurrected IBM 7030 (an old STRETCH) for doing the extensive number crunching of which Table II is a distillation.
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Louis
M. Rauch, Some general inversion formulae for analytic
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 [1]
 E. T. WHITTAKER & G. N. WATSON, A Course of Modem Analysis, 4th ed., Cambridge Univ. Press, New York, 1927, Sect. 21.8, pp. 485486.
 [2]
 E. JAHNKE & F. EMDE, Tables of Functions, with Formulae and Curves, 4th ed., Dover, New York, 1945 (note that on page 74 the fifth coefficient is misprinted, it should be 1707 instead of 1701 this has been corrected in some later editions). MR 7, 485. MR 0015900 (7:485b)
 [3]
 L. M. MILNETHOMSON, Jacobian Elliptic Functions Tables. A Guide to Practical Computation with Elliptic Functions and Integrals Together with Tables of sn u, cn u, dn u, , Dover, New York, 1950. MR 13, 987. MR 0048143 (13:987g)
 [4]
 E. HILLE, Analytic Function Theory. Vol. II, Introductions to Higher Math., Ginn, Boston, Mass., 1962, p. 160. MR 34 #1490. MR 0201608 (34:1490)
 [5]
 I. S. GRADŠTEĬN & I. M. RYŽIK, Tables of Integrals, Series and Products, 4th ed., Fizmatgiz, Moscow, 1963; English transl., Academic Press, New York, 1965, pp. 921925. MR 28 #5198; 33 #5952.
 [6]
 A. WEIERSTRASS, "Zur Theorie der Elliptischen Funktionen," Werke, v. 2, 1895, pp. 275276.
 [7]
 L. M. MILNETHOMSON, "Ten decimal table of the nome q," J. London Math. Soc., v. 5, 1930, pp. 148149.
 [8]
 A. N. LOWAN, G. BLANCH & W. HORENSTEIN, "On the inversion of the qseries associated with Jacobian elliptic functions," Bull. Amer. Math. Soc., v. 48, 1942, pp. 737738. MR 4, 90. MR 0007119 (4:90e)
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 L. B. W. JOLLEY, Summation of Series, 2nd rev. ed., Dover Books on Advanced Math., Dover, New York, 1961, pp. 3031. MR 24 #B511. MR 0134458 (24:B511)
 [10]
 C. E. VAN ORSTRAND, "Reversion of power series," Philos. Mag., v. 19, 1910, pp. 366376.
 [11]
 M. McMAHON, "On the general term in the reversion of series," Bull. Amer. Math. Soc., v. 3, 1894, pp. 170172. MR 1557325
 [12]
 H. RADEMACHER, "On the partition function," Proc. London Math. Soc., v. 43, 1937, pp. 241254.
 [13]
 We thank the BYU Scientific Computation Center and their resurrected IBM 7030 (an old STRETCH) for doing the extensive number crunching of which Table II is a distillation.
 [14]
 LOUIS M. RAUCH, "Some general inversion formulae for analytic functions," Duke Math. J., v. 18, 1951, pp. 131146. MR 12, 813. MR 0041211 (12:813c)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718197503673228
PII:
S 00255718(1975)03673228
Keywords:
Elliptic integrals,
theta functions,
nome,
partitions,
reversion
Article copyright:
© Copyright 1975 American Mathematical Society
