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11: 34.1 Special Notation
{ j 11 j 12 j 13 j 21 j 22 j 23 j 31 j 32 j 33 } .
12: 1.3 Determinants, Linear Operators, and Spectral Expansions
1.3.2 det [ a j k ] = | a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 | = a 11 | a 22 a 23 a 32 a 33 | a 12 | a 21 a 23 a 31 a 33 | + a 13 | a 21 a 22 a 31 a 32 | = a 11 a 22 a 33 a 11 a 23 a 32 a 12 a 21 a 33 + a 12 a 23 a 31 + a 13 a 21 a 32 a 13 a 22 a 31 .
13: 30.9 Asymptotic Approximations and Expansions
2 14 β 3 = 33 q 5 1594 q 3 5621 q + 128 m 2 ( 37 q 3 + 167 q ) 2048 m 4 q .
2 9 c 3 = 33 q 5 114 q 3 37 q + 2 m 2 ( 23 q 3 + 25 q ) 13 m 4 q .
14: 28.11 Expansions in Series of Mathieu Functions
See Meixner and Schäfke (1954, §2.28), and for expansions in the case of the exceptional values of q see Meixner et al. (1980, p. 33). …
15: 29.7 Asymptotic Expansions
29.7.7 τ 3 = p 2 14 ( ( 1 + k 2 ) 4 ( 33 p 4 + 410 p 2 + 405 ) 24 k 2 ( 1 + k 2 ) 2 ( 7 p 4 + 90 p 2 + 95 ) + 16 k 4 ( 9 p 4 + 130 p 2 + 173 ) ) ,
16: Bibliography R
  • M. Rahman (1981) A non-negative representation of the linearization coefficients of the product of Jacobi polynomials. Canad. J. Math. 33 (4), pp. 915–928.
  • W. Reinhardt (1982) Complex Coordinates in the Theory of Atomic and Molecular Structure and Dynamics. Annual Review of Physical Chemistry 33, pp. 223–255.
  • S. O. Rice (1954) Diffraction of plane radio waves by a parabolic cylinder. Calculation of shadows behind hills. Bell System Tech. J. 33, pp. 417–504.
  • 17: 23.21 Physical Applications
    18: 28.26 Asymptotic Approximations for Large q
    28.26.4 Fc m ( z , h ) 1 + s 8 h cosh 2 z + 1 2 11 h 2 ( s 4 + 86 s 2 + 105 cosh 4 z s 4 + 22 s 2 + 57 cosh 2 z ) + 1 2 14 h 3 ( s 5 + 14 s 3 + 33 s cosh 2 z 2 s 5 + 124 s 3 + 1122 s cosh 4 z + 3 s 5 + 290 s 3 + 1627 s cosh 6 z ) + ,
    19: 18.39 Applications in the Physical Sciences
    The spectrum is mixed, as in §1.18(viii), the positive energy, non- L 2 , scattering states are the subject of Chapter 33. … Namely for fixed l the infinite set labeled by p describe only the L 2 bound states for that single l , omitting the continuum briefly mentioned below, and which is the subject of Chapter 33, and so an unusual example of the mixed spectra of §1.18(viii). … This is also the normalization and notation of Chapter 33 for Z = 1 , and the notation of Weinberg (2013, Chapter 2). … As the scattering eigenfunctions of Chapter 33, are not OP’s, their further discussion is deferred to §18.39(iv), where discretized representations of these scattering states are introduced, Laguerre and Pollaczek OP’s then playing a key role. … The Coulomb–Pollaczek polynomials provide alternate representations of the positive energy Coulomb wave functions of Chapter 33. …
    20: 26.6 Other Lattice Path Numbers
    Table 26.6.1: Delannoy numbers D ( m , n ) .
    m n
    9 1 19 181 1159 5641 22363 75517 2 24143 5 98417 14 62563 33 17445
    10 1 21 221 1561 8361 36365 1 34245 4 33905 12 56465 33 17445 80 97453