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11: 26.9 Integer Partitions: Restricted Number and Part Size
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denotes the number of partitions of into at most parts.
See Table 26.9.1.
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►It follows that also equals the number of partitions of into parts that are less than or equal to .
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is the number of partitions of into at most parts, each less than or equal to .
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12: 34.8 Approximations for Large Parameters
§34.8 Approximations for Large Parameters
►For large values of the parameters in the , , and symbols, different asymptotic forms are obtained depending on which parameters are large. … ►
34.8.1
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►Uniform approximations in terms of Airy functions for the and symbols are given in Schulten and Gordon (1975b).
For approximations for the , , and symbols with error bounds see Flude (1998), Chen et al. (1999), and Watson (1999): these references also cite earlier work.
13: 22.9 Cyclic Identities
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►These identities are cyclic in the sense that each of the indices in the first product of, for example, the form are simultaneously permuted in the cyclic order: ; .
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§22.9(iii) Typical Identities of Rank 3
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22.9.22
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22.9.23
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14: 11 Struve and Related Functions
Chapter 11 Struve and Related Functions
…15: 34.7 Basic Properties: Symbol
16: 24.2 Definitions and Generating Functions
17: 26.12 Plane Partitions
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►As an example, there are six plane partitions of 3:
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►The notation denotes the sum over all plane partitions contained in , and denotes the number of elements in .
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►where is the sum of the squares of the divisors of .
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26.12.26
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18: 34.9 Graphical Method
§34.9 Graphical Method
… ►For specific examples of the graphical method of representing sums involving the , and symbols, see Varshalovich et al. (1988, Chapters 11, 12) and Lehman and O’Connell (1973, §3.3).19: Staff
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Richard B. Paris, University of Abertay, Chaps. 8, 11
Nico M. Temme, Centrum Wiskunde Informatica, Chaps. 3, 6, 7, 12
Amparo Gil, Universidad de Cantabria, for Chap. 3
Javier Segura, Universidad de Cantabria, for Chap. 3
Nico M. Temme, Centrum Wiskunde & Informatica (CWI), for Chaps. 3, 6, 7, 12
20: 23.5 Special Lattices
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►Then and the parallelogram with vertices at , , , is a rectangle.
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►Also, and have opposite signs unless , in which event both are zero.
►As functions of , and are decreasing and is increasing.
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►The parallelogram , , , is a square, and
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►The parallelogram , , , , is a rhombus: see Figure 23.5.1.
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