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1: 16.16 Transformations of Variables
§16.16(i) Reduction Formulas
See Erdélyi et al. (1953a, §5.10) for these and further reduction formulas. …
2: 20.7 Identities
§20.7(iv) Reduction Formulas for Products
§20.7(ix) Addendum to 20.7(iv) Reduction Formulas for Products
3: Bibliography W
  • P. L. Walker (2012) Reduction formulae for products of theta functions. J. Res. Nat. Inst. Standards and Technology 117, pp. 297–303.
  • 4: 15.11 Riemann’s Differential Equation
    §15.11(ii) Transformation Formulas
    The reduction of a general homogeneous linear differential equation of the second order with at most three regular singularities to the hypergeometric differential equation is given by …
    5: 19.29 Reduction of General Elliptic Integrals
    §19.29 Reduction of General Elliptic Integrals
    §19.29(i) Reduction Theorems
    §19.29(ii) Reduction to Basic Integrals
    The reduction of I ( m ) is carried out by a relation derived from partial fractions and by use of two recurrence relations. … Another method of reduction is given in Gray (2002). …
    6: Bibliography C
  • B. C. Carlson and J. FitzSimons (2000) Reduction theorems for elliptic integrands with the square root of two quadratic factors. J. Comput. Appl. Math. 118 (1-2), pp. 71–85.
  • B. C. Carlson (2002) Three improvements in reduction and computation of elliptic integrals. J. Res. Nat. Inst. Standards Tech. 107 (5), pp. 413–418.
  • B. C. Carlson (2006b) Table of integrals of squared Jacobian elliptic functions and reductions of related hypergeometric R -functions. Math. Comp. 75 (255), pp. 1309–1318.
  • P. A. Clarkson and M. D. Kruskal (1989) New similarity reductions of the Boussinesq equation. J. Math. Phys. 30 (10), pp. 2201–2213.
  • P. A. Clarkson (1991) Nonclassical Symmetry Reductions and Exact Solutions for Physically Significant Nonlinear Evolution Equations. In Nonlinear and Chaotic Phenomena in Plasmas, Solids and Fluids (Edmonton, AB, 1990), W. Rozmus and J. A. Tuszynski (Eds.), pp. 72–79.
  • 7: Bibliography N
  • D. Naylor (1984) On simplified asymptotic formulas for a class of Mathieu functions. SIAM J. Math. Anal. 15 (6), pp. 1205–1213.
  • D. Naylor (1987) On a simplified asymptotic formula for the Mathieu function of the third kind. SIAM J. Math. Anal. 18 (6), pp. 1616–1629.
  • W. J. Nellis and B. C. Carlson (1966) Reduction and evaluation of elliptic integrals. Math. Comp. 20 (94), pp. 223–231.
  • G. Nemes (2013a) An explicit formula for the coefficients in Laplace’s method. Constr. Approx. 38 (3), pp. 471–487.
  • G. Nemes (2013c) Generalization of Binet’s Gamma function formulas. Integral Transforms Spec. Funct. 24 (8), pp. 597–606.
  • 8: 31.7 Relations to Other Functions
    §31.7(i) Reductions to the Gauss Hypergeometric Function
    Other reductions of H to a F 1 2 , with at least one free parameter, exist iff the pair ( a , p ) takes one of a finite number of values, where q = α β p . Below are three such reductions with three and two parameters. … For additional reductions, see Maier (2005). Joyce (1994) gives a reduction in which the independent variable is transformed not polynomially or rationally, but algebraically. …
    9: Bibliography G
  • F. Gao and V. J. W. Guo (2013) Contiguous relations and summation and transformation formulae for basic hypergeometric series. J. Difference Equ. Appl. 19 (12), pp. 2029–2042.
  • G. Gasper (1975) Formulas of the Dirichlet-Mehler Type. In Fractional Calculus and its Applications, B. Ross (Ed.), Lecture Notes in Math., Vol. 457, pp. 207–215.
  • W. Gautschi (1983) How and how not to check Gaussian quadrature formulae. BIT 23 (2), pp. 209–216.
  • H. W. Gould (1972) Explicit formulas for Bernoulli numbers. Amer. Math. Monthly 79, pp. 44–51.
  • N. Gray (2002) Automatic reduction of elliptic integrals using Carlson’s relations. Math. Comp. 71 (237), pp. 311–318.