Ramanujan%20partition%20identity
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1: 26.10 Integer Partitions: Other Restrictions
§26.10 Integer Partitions: Other Restrictions
►§26.10(i) Definitions
… ►§26.10(iv) Identities
►Equations (26.10.13) and (26.10.14) are the Rogers–Ramanujan identities. … ►2: 27.20 Methods of Computation: Other Number-Theoretic Functions
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►The recursion formulas (27.14.6) and (27.14.7) can be used to calculate the partition function for .
…To compute a particular value it is better to use the Hardy–Ramanujan–Rademacher series (27.14.9).
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►A recursion formula obtained by differentiating (27.14.18) can be used to calculate Ramanujan’s function , and the values can be checked by the congruence (27.14.20).
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3: 27.14 Unrestricted Partitions
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§27.14(v) Divisibility Properties
►Ramanujan (1921) gives identities that imply divisibility properties of the partition function. For example, the Ramanujan identity …Ramanujan also found that and for all . … ►4: Bibliography R
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On the partition function p(n).
Proc. London Math. Soc. (2) 43 (4), pp. 241–254.
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Congruence properties of partitions.
Math. Z. 9 (1-2), pp. 147–153.
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Some properties of Bernoulli’s numbers (J. Indian Math. Soc. 3 (1911), 219–234.).
In Collected Papers,
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Collected Papers of Srinivasa Ramanujan.
Chelsea Publishing Co., New York.
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On the definition and properties of generalized - symbols.
J. Math. Phys. 20 (12), pp. 2398–2415.
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5: 20.11 Generalizations and Analogs
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§20.11(ii) Ramanujan’s Theta Function and -Series
►Ramanujan’s theta function is defined by … ►§20.11(iii) Ramanujan’s Change of Base
… ►These results are called Ramanujan’s changes of base. …6: 26.2 Basic Definitions
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Partition
… ►As an example, , , is a partition of . … ►The total number of partitions of is denoted by . …For the actual partitions () for see Table 26.4.1. ►The integers whose sum is are referred to as the parts in the partition. …7: Bibliography
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Exact linearization of a Painlevé transcendent.
Phys. Rev. Lett. 38 (20), pp. 1103–1106.
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Chapter of Ramanujan’s second notebook: Theta-functions and -series.
Mem. Amer. Math. Soc. 53 (315), pp. v+85.
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Gauss, Landen, Ramanujan, the arithmetic-geometric mean, ellipses, , and the Ladies Diary.
Amer. Math. Monthly 95 (7), pp. 585–608.
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Ramanujan Revisited.
Academic Press Inc., Boston, MA.
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Multiple series Rogers-Ramanujan type identities.
Pacific J. Math. 114 (2), pp. 267–283.
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8: 26.9 Integer Partitions: Restricted Number and Part Size
§26.9 Integer Partitions: Restricted Number and Part Size
►§26.9(i) Definitions
… ►Unrestricted partitions are covered in §27.14. … ►§26.9(ii) Generating Functions
… ►§26.9(iii) Recurrence Relations
…9: 20 Theta Functions
Chapter 20 Theta Functions
…10: Bibliography L
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Algorithm 917: complex double-precision evaluation of the Wright function.
ACM Trans. Math. Software 38 (3), pp. Art. 20, 17.
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An asymptotic estimate for the Bernoulli and Euler numbers.
Canad. Math. Bull. 20 (1), pp. 109–111.
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Ramanujan’s function
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Duke Math. J. 10 (3), pp. 483–492.
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The vanishing of Ramanujan’s function
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Duke Math. J. 14 (2), pp. 429–433.
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A Lie theoretic interpretation and proof of the Rogers-Ramanujan identities.
Adv. in Math. 45 (1), pp. 21–72.
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