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generalized hypergeometric differential equation

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11: Bibliography B
  • W. N. Bailey (1928) Products of generalized hypergeometric series. Proc. London Math. Soc. (2) 28 (2), pp. 242–254.
  • W. N. Bailey (1929) Transformations of generalized hypergeometric series. Proc. London Math. Soc. (2) 29 (2), pp. 495–502.
  • W. N. Bailey (1964) Generalized Hypergeometric Series. Stechert-Hafner, Inc., New York.
  • W. Bühring (1992) Generalized hypergeometric functions at unit argument. Proc. Amer. Math. Soc. 114 (1), pp. 145–153.
  • J. C. Butcher (1987) The Numerical Analysis of Ordinary Differential Equations. Runge-Kutta and General Linear Methods. John Wiley & Sons Ltd., Chichester.
  • 12: Bibliography O
  • A. B. Olde Daalhuis and F. W. J. Olver (1995a) Hyperasymptotic solutions of second-order linear differential equations. I. Methods Appl. Anal. 2 (2), pp. 173–197.
  • A. B. Olde Daalhuis and F. W. J. Olver (1995b) On the calculation of Stokes multipliers for linear differential equations of the second order. Methods Appl. Anal. 2 (3), pp. 348–367.
  • F. W. J. Olver (1977c) Second-order differential equations with fractional transition points. Trans. Amer. Math. Soc. 226, pp. 227–241.
  • F. W. J. Olver (1993a) Exponentially-improved asymptotic solutions of ordinary differential equations I: The confluent hypergeometric function. SIAM J. Math. Anal. 24 (3), pp. 756–767.
  • P. J. Olver (1993b) Applications of Lie Groups to Differential Equations. 2nd edition, Graduate Texts in Mathematics, Vol. 107, Springer-Verlag, New York.
  • 13: 15.17 Mathematical Applications
    §15.17(i) Differential Equations
    The logarithmic derivatives of some hypergeometric functions for which quadratic transformations exist (§15.8(iii)) are solutions of Painlevé equations. … … Harmonic analysis can be developed for the Jacobi transform either as a generalization of the Fourier-cosine transform (§1.14(ii)) or as a specialization of a group Fourier transform. … These monodromy groups are finite iff the solutions of Riemann’s differential equation are all algebraic. …
    14: Bibliography N
  • J. Negro, L. M. Nieto, and O. Rosas-Ortiz (2000) Confluent hypergeometric equations and related solvable potentials in quantum mechanics. J. Math. Phys. 41 (12), pp. 7964–7996.
  • J. J. Nestor (1984) Uniform Asymptotic Approximations of Solutions of Second-order Linear Differential Equations, with a Coalescing Simple Turning Point and Simple Pole. Ph.D. Thesis, University of Maryland, College Park, MD.
  • V. A. Noonburg (1995) A separating surface for the Painlevé differential equation x ′′ = x 2 - t . J. Math. Anal. Appl. 193 (3), pp. 817–831.
  • L. N. Nosova and S. A. Tumarkin (1965) Tables of Generalized Airy Functions for the Asymptotic Solution of the Differential Equations ϵ ( p y ) + ( q + ϵ r ) y = f . Pergamon Press, Oxford.
  • V. Yu. Novokshënov (1985) The asymptotic behavior of the general real solution of the third Painlevé equation. Dokl. Akad. Nauk SSSR 283 (5), pp. 1161–1165 (Russian).
  • 15: 35.7 Gaussian Hypergeometric Function of Matrix Argument
    §35.7 Gaussian Hypergeometric Function of Matrix Argument
    §35.7(iii) Partial Differential Equations
    Subject to the conditions (a)–(c), the function f ( T ) = F 1 2 ( a , b ; c ; T ) is the unique solution of each partial differential equationSystems of partial differential equations for the F 1 0 (defined in §35.8) and F 1 1 functions of matrix argument can be obtained by applying (35.8.9) and (35.8.10) to (35.7.9). …
    16: Bibliography M
  • A. P. Magnus (1995) Painlevé-type differential equations for the recurrence coefficients of semi-classical orthogonal polynomials. J. Comput. Appl. Math. 57 (1-2), pp. 215–237.
  • R. S. Maier (2005) On reducing the Heun equation to the hypergeometric equation. J. Differential Equations 213 (1), pp. 171–203.
  • A. R. Miller (1997) A class of generalized hypergeometric summations. J. Comput. Appl. Math. 87 (1), pp. 79–85.
  • A. R. Miller (2003) On a Kummer-type transformation for the generalized hypergeometric function F 2 2 . J. Comput. Appl. Math. 157 (2), pp. 507–509.
  • J. C. P. Miller (1950) On the choice of standard solutions for a homogeneous linear differential equation of the second order. Quart. J. Mech. Appl. Math. 3 (2), pp. 225–235.
  • 17: 32.10 Special Function Solutions
    For certain combinations of the parameters, P II P VI  have particular solutions expressible in terms of the solution of a Riccati differential equation, which can be solved in terms of special functions defined in other chapters. …
    §32.10(iv) Fourth Painlevé Equation
    §32.10(vi) Sixth Painlevé Equation
    where the fundamental periods 2 ϕ 1 and 2 ϕ 2 are linearly independent functions satisfying the hypergeometric equation
    18: 19.16 Definitions
    §19.16(ii) R - a ( b ; z )
    All elliptic integrals of the form (19.2.3) and many multiple integrals, including (19.23.6) and (19.23.6_5), are special cases of a multivariate hypergeometric function … For generalizations and further information, especially representation of the R -function as a Dirichlet average, see Carlson (1977b).
    §19.16(iii) Various Cases of R - a ( b ; z )
    19: 2.6 Distributional Methods
    Although divergent, these integrals may be interpreted in a generalized sense. … To each function in this equation, we shall assign a tempered distribution (i. … Corresponding results for the generalized Stieltjes transformThese equations again hold only in the sense of distributions. … For rigorous derivations of these results and also order estimates for δ n ( x ) , see Wong (1979) and Wong (1989, Chapter 6).
    20: Errata
  • Section 1.13

    In Equation (1.13.4), the determinant form of the two-argument Wronskian

    1.13.4 𝒲 { w 1 ( z ) , w 2 ( z ) } = det [ w 1 ( z ) w 2 ( z ) w 1 ( z ) w 2 ( z ) ] = w 1 ( z ) w 2 ( z ) - w 2 ( z ) w 1 ( z )

    was added as an equality. In ¶Wronskian (in §1.13(i)), immediately below Equation (1.13.4), a sentence was added indicating that in general the n -argument Wronskian is given by 𝒲 { w 1 ( z ) , , w n ( z ) } = det [ w k ( j - 1 ) ( z ) ] , where 1 j , k n . Immediately below Equation (1.13.4), a sentence was added giving the definition of the n -argument Wronskian. It is explained just above (1.13.5) that this equation is often referred to as Abel’s identity. Immediately below Equation (1.13.5), a sentence was added explaining how it generalizes for n th-order differential equations. A reference to Ince (1926, §5.2) was added.

  • Equations (10.15.1), (10.38.1)

    These equations have been generalized to include the additional cases of J - ν ( z ) / ν , I - ν ( z ) / ν , respectively.

  • Equations (13.2.9), (13.2.10)

    There were clarifications made in the conditions on the parameter a in U ( a , b , z ) of those equations.

  • Equation (14.2.7)

    The Wronskian was generalized to include both associated Legendre and Ferrers functions.

  • Equation (10.13.4)

    has been generalized to cover an additional case.