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11: 21.7 Riemann Surfaces
These are Riemann surfaces that may be obtained from algebraic curves of the form
12: 32.10 Special Function Solutions
The solution (32.10.34) is an essentially transcendental function of both constants of integration since P VI  with α = β = γ = 0 and δ = 1 2 does not admit an algebraic first integral of the form P ( z , w , w , C ) = 0 , with C a constant. …
13: Bibliography T
  • F. Tu and Y. Yang (2013) Algebraic transformations of hypergeometric functions and automorphic forms on Shimura curves. Trans. Amer. Math. Soc. 365 (12), pp. 6697–6729.
  • 14: 2.6 Distributional Methods
    It is easily seen that K + forms a commutative, associative linear algebra. …
    15: 23.20 Mathematical Applications
    An algebraic curve that can be put either into the form
    16: 19.14 Reduction of General Elliptic Integrals
    Legendre (1825–1832) showed that every elliptic integral can be expressed in terms of the three integrals in (19.1.2) supplemented by algebraic, logarithmic, and trigonometric functions. …It then improves the classical method by first applying Hermite reduction to (19.2.3) to arrive at integrands without multiple poles and uses implicit full partial-fraction decomposition and implicit root finding to minimize computing with algebraic extensions. The choice among 21 transformations for final reduction to Legendre’s normal form depends on inequalities involving the limits of integration and the zeros of the cubic or quartic polynomial. …
    17: 18.38 Mathematical Applications
    Zhedanov Algebra
    See Zhedanov (1991), Granovskiĭ et al. (1992, §3), Koornwinder (2007a, §2) and Terwilliger (2011). Similar algebras can be associated with all families of OP’s in the q -Askey scheme and the Askey scheme. … Algebraic structures were built of which special representations involve Dunkl type operators. In the q -case this algebraic structure is called the double affine Hecke algebra (DAHA), introduced by Cherednik. …
    18: Errata
    The spectral theory of these operators, based on Sturm-Liouville and Liouville normal forms, distribution theory, is now discussed more completely, including linear algebra, matrices, matrices as linear operators, orthonormal expansions, Stieltjes integrals/measures, generating functions. …
    19: 3.6 Linear Difference Equations
    Let us assume the normalizing condition is of the form w 0 = λ , where λ is a constant, and then solve the following tridiagonal system of algebraic equations for the unknowns w 1 ( N ) , w 2 ( N ) , , w N 1 ( N ) ; see §3.2(ii). …
    20: Bruce R. Miller
    While developing the supporting theories, he discovered a passion for symbolic computation and computer algebra. …There, he carried out research in non-linear dynamics and celestial mechanics, developing a specialized computer algebra system for high-order Lie transformations. … He is the developer of the tools used to process the DLMF into both book and web forms. …