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1: Bibliography S
  • R. Spigler, M. Vianello, and F. Locatelli (1999) Liouville-Green-Olver approximations for complex difference equations. J. Approx. Theory 96 (2), pp. 301–322.
  • R. Spigler and M. Vianello (1992) Liouville-Green approximations for a class of linear oscillatory difference equations of the second order. J. Comput. Appl. Math. 41 (1-2), pp. 105–116.
  • R. Spigler and M. Vianello (1997) A Survey on the Liouville-Green (WKB) Approximation for Linear Difference Equations of the Second Order. In Advances in Difference Equations (Veszprém, 1995), S. Elaydi, I. Győri, and G. Ladas (Eds.), pp. 567–577.
  • A. N. Stokes (1980) A stable quotient-difference algorithm. Math. Comp. 34 (150), pp. 515–519.
  • B. I. Suleĭmanov (1987) The relation between asymptotic properties of the second Painlevé equation in different directions towards infinity. Differ. Uravn. 23 (5), pp. 834–842 (Russian).
  • 2: Bibliography W
  • Z. Wang and R. Wong (2003) Asymptotic expansions for second-order linear difference equations with a turning point. Numer. Math. 94 (1), pp. 147–194.
  • Z. Wang and R. Wong (2005) Linear difference equations with transition points. Math. Comp. 74 (250), pp. 629–653.
  • R. S. Ward (1987) The Nahm equations, finite-gap potentials and Lamé functions. J. Phys. A 20 (10), pp. 2679–2683.
  • R. Wong and H. Li (1992a) Asymptotic expansions for second-order linear difference equations. II. Stud. Appl. Math. 87 (4), pp. 289–324.
  • R. Wong and H. Li (1992b) Asymptotic expansions for second-order linear difference equations. J. Comput. Appl. Math. 41 (1-2), pp. 65–94.
  • 3: 36.5 Stokes Sets
    The Stokes set takes different forms for z = 0 , z < 0 , and z > 0 . … where u satisfies the equationHere u is the root of the equationwhere u is the root of the equationwhere u is the positive root of the equation
    4: 28.8 Asymptotic Expansions for Large q
    28.8.1 a m ( h 2 ) b m + 1 ( h 2 ) } 2 h 2 + 2 s h 1 8 ( s 2 + 1 ) 1 2 7 h ( s 3 + 3 s ) 1 2 12 h 2 ( 5 s 4 + 34 s 2 + 9 ) 1 2 17 h 3 ( 33 s 5 + 410 s 3 + 405 s ) 1 2 20 h 4 ( 63 s 6 + 1260 s 4 + 2943 s 2 + 486 ) 1 2 25 h 5 ( 527 s 7 + 15617 s 5 + 69001 s 3 + 41607 s ) + .
    5: Bibliography V
  • G. Vedeler (1950) A Mathieu equation for ships rolling among waves. I, II. Norske Vid. Selsk. Forh., Trondheim 22 (25–26), pp. 113–123.
  • H. Volkmer (1998) On the growth of convergence radii for the eigenvalues of the Mathieu equation. Math. Nachr. 192, pp. 239–253.
  • H. Volkmer (2004a) Error estimates for Rayleigh-Ritz approximations of eigenvalues and eigenfunctions of the Mathieu and spheroidal wave equation. Constr. Approx. 20 (1), pp. 39–54.
  • H. Volkmer (2004b) Four remarks on eigenvalues of Lamé’s equation. Anal. Appl. (Singap.) 2 (2), pp. 161–175.
  • A. P. Vorob’ev (1965) On the rational solutions of the second Painlevé equation. Differ. Uravn. 1 (1), pp. 79–81 (Russian).
  • 6: 28.35 Tables
    §28.35 Tables
  • Ince (1932) includes eigenvalues a n , b n , and Fourier coefficients for n = 0 or 1 ( 1 ) 6 , q = 0 ( 1 ) 10 ( 2 ) 20 ( 4 ) 40 ; 7D. Also ce n ( x , q ) , se n ( x , q ) for q = 0 ( 1 ) 10 , x = 1 ( 1 ) 90 , corresponding to the eigenvalues in the tables; 5D. Notation: a n = 𝑏𝑒 n 2 q , b n = 𝑏𝑜 n 2 q .

  • Kirkpatrick (1960) contains tables of the modified functions Ce n ( x , q ) , Se n + 1 ( x , q ) for n = 0 ( 1 ) 5 , q = 1 ( 1 ) 20 , x = 0.1 ( .1 ) 1 ; 4D or 5D.

  • National Bureau of Standards (1967) includes the eigenvalues a n ( q ) , b n ( q ) for n = 0 ( 1 ) 3 with q = 0 ( .2 ) 20 ( .5 ) 37 ( 1 ) 100 , and n = 4 ( 1 ) 15 with q = 0 ( 2 ) 100 ; Fourier coefficients for ce n ( x , q ) and se n ( x , q ) for n = 0 ( 1 ) 15 , n = 1 ( 1 ) 15 , respectively, and various values of q in the interval [ 0 , 100 ] ; joining factors g e , n ( q ) , f e , n ( q ) for n = 0 ( 1 ) 15 with q = 0 ( .5  to  10 ) 100 (but in a different notation). Also, eigenvalues for large values of q . Precision is generally 8D.

  • Zhang and Jin (1996, pp. 521–532) includes the eigenvalues a n ( q ) , b n + 1 ( q ) for n = 0 ( 1 ) 4 , q = 0 ( 1 ) 50 ; n = 0 ( 1 ) 20 ( a ’s) or 19 ( b ’s), q = 1 , 3 , 5 , 10 , 15 , 25 , 50 ( 50 ) 200 . Fourier coefficients for ce n ( x , 10 ) , se n + 1 ( x , 10 ) , n = 0 ( 1 ) 7 . Mathieu functions ce n ( x , 10 ) , se n + 1 ( x , 10 ) , and their first x -derivatives for n = 0 ( 1 ) 4 , x = 0 ( 5 ) 90 . Modified Mathieu functions Mc n ( j ) ( x , 10 ) , Ms n + 1 ( j ) ( x , 10 ) , and their first x -derivatives for n = 0 ( 1 ) 4 , j = 1 , 2 , x = 0 ( .2 ) 4 . Precision is mostly 9S.

  • 7: Bibliography O
  • K. Okamoto (1987a) Studies on the Painlevé equations. I. Sixth Painlevé equation P VI . Ann. Mat. Pura Appl. (4) 146, pp. 337–381.
  • A. B. Olde Daalhuis (2004a) Inverse factorial-series solutions of difference equations. Proc. Edinb. Math. Soc. (2) 47 (2), pp. 421–448.
  • J. Oliver (1977) An error analysis of the modified Clenshaw method for evaluating Chebyshev and Fourier series. J. Inst. Math. Appl. 20 (3), pp. 379–391.
  • F. W. J. Olver (1967a) Numerical solution of second-order linear difference equations. J. Res. Nat. Bur. Standards Sect. B 71B, pp. 111–129.
  • F. W. J. Olver (1967b) Bounds for the solutions of second-order linear difference equations. J. Res. Nat. Bur. Standards Sect. B 71B (4), pp. 161–166.
  • 8: 14.30 Spherical and Spheroidal Harmonics
    Sometimes Y l , m ( θ , ϕ ) is denoted by i l 𝔇 l m ( θ , ϕ ) ; also the definition of Y l , m ( θ , ϕ ) can differ from (14.30.1), for example, by inclusion of a factor ( 1 ) m . … As an example, Laplace’s equation 2 W = 0 in spherical coordinates (§1.5(ii)): … In the quantization of angular momentum the spherical harmonics Y l , m ( θ , ϕ ) are normalized solutions of the eigenvalue equations
    14.30.11_5 L z Y l , m = m Y l , m , m = l , 1 + 1 , , 0 , , l 1 , l ,
    14.30.13 L z = i ϕ ;
    9: Bibliography D
  • A. Decarreau, M.-Cl. Dumont-Lepage, P. Maroni, A. Robert, and A. Ronveaux (1978a) Formes canoniques des équations confluentes de l’équation de Heun. Ann. Soc. Sci. Bruxelles Sér. I 92 (1-2), pp. 53–78.
  • A. Decarreau, P. Maroni, and A. Robert (1978b) Sur les équations confluentes de l’équation de Heun. Ann. Soc. Sci. Bruxelles Sér. I 92 (3), pp. 151–189.
  • B. Döring (1966) Complex zeros of cylinder functions. Math. Comp. 20 (94), pp. 215–222.
  • C. F. Dunkl (1989) Differential-difference operators associated to reflection groups. Trans. Amer. Math. Soc. 311 (1), pp. 167–183.
  • T. M. Dunster (1989) Uniform asymptotic expansions for Whittaker’s confluent hypergeometric functions. SIAM J. Math. Anal. 20 (3), pp. 744–760.
  • 10: 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.
  • I. Cherednik (1995) Macdonald’s evaluation conjectures and difference Fourier transform. Invent. Math. 122 (1), pp. 119–145.
  • R. C. Y. Chin and G. W. Hedstrom (1978) A dispersion analysis for difference schemes: Tables of generalized Airy functions. Math. Comp. 32 (144), pp. 1163–1170.
  • 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.