Dougall 7F6(1) sum
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11: 26.9 Integer Partitions: Restricted Number and Part Size
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►The conjugate to the example in Figure 26.9.1 is .
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►It is also equal to the number of lattice paths from to that have exactly vertices , , , above and to the left of the lattice path.
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►It is also assumed everywhere that .
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►Also, when
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►where the inner sum is taken over all positive divisors of that are less than or equal to .
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12: 4.25 Continued Fractions
13: 4.39 Continued Fractions
14: 28.16 Asymptotic Expansions for Large
15: 24.2 Definitions and Generating Functions
16: Bibliography Q
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On two problems concerning means.
J. Hangzhou Inst. Elec. Engrg. 17, pp. 1–7 (Chinese).
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Asymptotic expansion of the Krawtchouk polynomials and their zeros.
Comput. Methods Funct. Theory 4 (1), pp. 189–226.
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“Best possible” upper and lower bounds for the zeros of the Bessel function
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Trans. Amer. Math. Soc. 351 (7), pp. 2833–2859.
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17: 3.4 Differentiation
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►The Lagrange -point formula is
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►where and .
►For the values of and used in the formulas below
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►With the choice (which is crucial when is large because of numerical cancellation) the integrand equals at the dominant points , and in combination with the factor in front of the integral sign this gives a rough approximation to .
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18: 26.6 Other Lattice Path Numbers
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is the number of paths from to that are composed of directed line segments of the form , , or .
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is the number of lattice paths from to that stay on or above the line and are composed of directed line segments of the form , , or .
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is the number of lattice paths from to that stay on or above the line , are composed of directed line segments of the form or , and for which there are exactly occurrences at which a segment of the form is followed by a segment of the form .
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is the number of paths from to that stay on or above the diagonal and are composed of directed line segments of the form , , or .
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26.6.6
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19: 7.17 Inverse Error Functions
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►With ,
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7.17.2
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►where and the other coefficients follow from the recursion
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7.17.2_5
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7.17.3
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20: 26.2 Basic Definitions
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►A partition of a nonnegative integer
is an unordered collection of positive integers whose sum is .
As an example, is a partition of 13.
…For the actual partitions () for see Table 26.4.1.
►The integers whose sum is are referred to as the parts in the partition.
The example has six parts, three of which equal 1.
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