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1: Bibliography
  • M. J. Ablowitz and H. Segur (1977) Exact linearization of a Painlevé transcendent. Phys. Rev. Lett. 38 (20), pp. 1103–1106.
  • A. Adelberg (1992) On the degrees of irreducible factors of higher order Bernoulli polynomials. Acta Arith. 62 (4), pp. 329–342.
  • A. Adelberg (1996) Congruences of p -adic integer order Bernoulli numbers. J. Number Theory 59 (2), pp. 374–388.
  • F. A. Alhargan (2000) Algorithm 804: Subroutines for the computation of Mathieu functions of integer orders. ACM Trans. Math. Software 26 (3), pp. 408–414.
  • R. W. B. Ardill and K. J. M. Moriarty (1978) Spherical Bessel functions j n and y n of integer order and real argument. Comput. Phys. Comm. 14 (3-4), pp. 261–265.
  • 2: 18.39 Applications in the Physical Sciences
    The nature of, and notations and common vocabulary for, the eigenvalues and eigenfunctions of self-adjoint second order differential operators is overviewed in §1.18. … The fundamental quantum Schrödinger operator, also called the Hamiltonian, , is a second order differential operator of the form … ϵ 0 is referred to as the ground state, all others, n = 1 , 2 , , in order of increasing energy being excited states. … Namely the k th eigenfunction, listed in order of increasing eigenvalues, starting at k = 0 , has precisely k nodes, as real zeros of wave-functions away from boundaries are often referred to. … Derivations of (18.39.42) appear in Bethe and Salpeter (1957, pp. 12–20), and Pauling and Wilson (1985, Chapter V and Appendix VII), where the derivations are based on (18.39.36), and is also the notation of Piela (2014, §4.7), typifying the common use of the associated Coulomb–Laguerre polynomials in theoretical quantum chemistry. …
    3: Bibliography C
  • R. Chelluri, L. B. Richmond, and N. M. Temme (2000) Asymptotic estimates for generalized Stirling numbers. Analysis (Munich) 20 (1), pp. 1–13.
  • H. S. Cohl (2010) Derivatives with respect to the degree and order of associated Legendre functions for | z | > 1 using modified Bessel functions. Integral Transforms Spec. Funct. 21 (7-8), pp. 581–588.
  • M. Colman, A. Cuyt, and J. Van Deun (2011) Validated computation of certain hypergeometric functions. ACM Trans. Math. Software 38 (2), pp. Art. 11, 20.
  • M. D. Cooper, R. H. Jeppesen, and M. B. Johnson (1979) Coulomb effects in the Klein-Gordon equation for pions. Phys. Rev. C 20 (2), pp. 696–704.
  • C. M. Cosgrove (2006) Chazy’s second-degree Painlevé equations. J. Phys. A 39 (39), pp. 11955–11971.
  • 4: 19.36 Methods of Computation
    For R F the polynomial of degree 7, for example, is … For both R D and R J the factor ( r / 4 ) 1 / 6 in Carlson (1995, (2.18)) is changed to ( r / 5 ) 1 / 8 when the following polynomial of degree 7 (the same for both) is used instead of its first seven terms: …Polynomials of still higher degree can be obtained from (19.19.5) and (19.19.7). … If the iteration of (19.36.6) and (19.36.12) is stopped when c s < r t s ( M and T being approximated by a s and t s , and the infinite series being truncated), then the relative error in R F and R G is less than r if we neglect terms of order r 2 . … For computation of Legendre’s integral of the third kind, see Abramowitz and Stegun (1964, §§17.7 and 17.8, Examples 15, 17, 19, and 20). …