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11: 31.11 Expansions in Series of Hypergeometric Functions
μ = γ + δ 2 .
12: Howard S. Cohl
Cohl has published papers in orthogonal polynomials and special functions, and is particularly interested in fundamental solutions of linear partial differential equations on Riemannian manifolds, associated Legendre functions, generalized and basic hypergeometric functions, eigenfunction expansions of fundamental solutions in separable coordinate systems for linear partial differential equations, orthogonal polynomial generating function and generalized expansions, and q -series. …
13: 31.10 Integral Equations and Representations
For suitable choices of the branches of the P -symbols in (31.10.9) and the contour C , we can obtain both integral equations satisfied by Heun functions, as well as the integral representations of a distinct solution of Heun’s equation in terms of a Heun function (polynomial, path-multiplicative solution). …
14: 31.8 Solutions via Quadratures
Here Ψ g , N ( λ , z ) is a polynomial of degree g in λ and of degree N = m 0 + m 1 + m 2 + m 3 in z , that is a solution of the third-order differential equation satisfied by a product of any two solutions of Heun’s equation. … When λ = 4 q approaches the ends of the gaps, the solution (31.8.2) becomes the corresponding Heun polynomial. …
15: Bibliography C
  • P. A. Clarkson (2005) Special polynomials associated with rational solutions of the fifth Painlevé equation. J. Comput. Appl. Math. 178 (1-2), pp. 111–129.
  • 16: 13.2 Definitions and Basic Properties
    Standard Solutions
    The first two standard solutions are: …
    §13.2(v) Numerically Satisfactory Solutions
    17: 31.9 Orthogonality
    For corresponding orthogonality relations for Heun functions (§31.4) and Heun polynomials31.5), see Lambe and Ward (1934), Erdélyi (1944), Sleeman (1966a), and Ronveaux (1995, Part A, pp. 59–64).
    §31.9(ii) Double Orthogonality
    Heun polynomials w j = 𝐻𝑝 n j , m j , j = 1 , 2 , satisfy …and the integration paths 1 , 2 are Pochhammer double-loop contours encircling distinct pairs of singularities { 0 , 1 } , { 0 , a } , { 1 , a } . … For bi-orthogonal relations for path-multiplicative solutions see Schmidt (1979, §2.2). …
    18: 3.8 Nonlinear Equations
    Solutions are called roots of the equation, or zeros of f . … and the solutions are called fixed points of ϕ . …
    §3.8(iv) Zeros of Polynomials
    The polynomial
    Example. Wilkinson’s Polynomial
    19: 29.19 Physical Applications
    §29.19 Physical Applications
    §29.19(ii) Lamé Polynomials
    Shail (1978) treats applications to solutions of elliptic crack and punch problems. Hargrave (1978) studies high frequency solutions of the delta wing equation. …
    20: 18.36 Miscellaneous Polynomials
    In §18.39(i) it is seen that the functions, w ( x ) H ^ n + 3 ( x ) , are solutions of a Schrödinger equation with a rational potential energy; and, in spite of first appearances, the Sturm oscillation theorem, Simon (2005c, Theorem 3.3, p. 35), is satisfied. …