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1: Alexander I. Bobenko
 Eitner), published by Springer in 2000, and Discrete Differential Geometry: Integrable Structure (with Y. …He is also coeditor of Discrete Integrable Geometry and Physics (with R.  Seiler), published by Oxford University Press in 1999, and Discrete Differential Geometry (with P. …
2: Vadim B. Kuznetsov
Kuznetsov published papers on special functions and orthogonal polynomials, the quantum scattering method, integrable discrete many-body systems, separation of variables, Bäcklund transformation techniques, and integrability in classical and quantum mechanics. …
3: 26.22 Software
  • GAP (website). A system for computational discrete algebra.

  • 4: 18.27 q -Hahn Class
    §18.27(vii) Discrete q -Hermite I and II Polynomials
    Discrete q -Hermite I
    Discrete q -Hermite II
    18.27.24 = - ( h ~ n ( c q ; q ) h ~ m ( c q ; q ) + h ~ n ( - c q ; q ) h ~ m ( - c q ; q ) ) q ( - c 2 q 2 ; q 2 ) = 2 ( q 2 , - c 2 q , - c - 2 q ; q 2 ) ( q , - c 2 , - c - 2 q 2 ; q 2 ) ( q ; q ) n q n 2 δ n , m , c > 0 .
    (For discrete q -Hermite II polynomials the measure is not uniquely determined.)
    5: 20.11 Generalizations and Analogs
    It is a discrete analog of theta functions. If both m , n are positive, then G ( m , n ) allows inversion of its arguments as a modular transformation (compare (23.15.3) and (23.15.4)): …This is the discrete analog of the Poisson identity (§1.8(iv)). …
    6: 18.1 Notation
  • Discrete q -Hermite I: h n ( x ; q ) .

  • Discrete q -Hermite II: h ~ n ( x ; q ) .

  • 7: 24.19 Methods of Computation
  • A method related to “Stickelberger codes” is applied in Buhler et al. (2001); in particular, it allows for an efficient search for the irregular pairs ( 2 n , p ) . Discrete Fourier transforms are used in the computations. See also Crandall (1996, pp. 120–124).

  • 8: Bibliography F
  • P. Flajolet and A. Odlyzko (1990) Singularity analysis of generating functions. SIAM J. Discrete Math. 3 (2), pp. 216–240.
  • A. S. Fokas, B. Grammaticos, and A. Ramani (1993) From continuous to discrete Painlevé equations. J. Math. Anal. Appl. 180 (2), pp. 342–360.
  • A. S. Fokas, A. R. Its, and A. V. Kitaev (1991) Discrete Painlevé equations and their appearance in quantum gravity. Comm. Math. Phys. 142 (2), pp. 313–344.
  • A. S. Fokas, A. R. Its, and X. Zhou (1992) Continuous and Discrete Painlevé Equations. In Painlevé Transcendents: Their Asymptotics and Physical Applications, D. Levi and P. Winternitz (Eds.), NATO Adv. Sci. Inst. Ser. B Phys., Vol. 278, pp. 33–47.
  • 9: 3.11 Approximation Techniques
    Now suppose that X k = 0 when k , that is, the functions ϕ k ( x ) are orthogonal with respect to weighted summation on the discrete set x 1 , x 2 , , x J . …
    Example. The Discrete Fourier Transform
    is called a discrete Fourier transform pair. … The direct computation of the discrete Fourier transform (3.11.38), that is, of …
    10: 31.4 Solutions Analytic at Two Singularities: Heun Functions
    For an infinite set of discrete values q m , m = 0 , 1 , 2 , , of the accessory parameter q , the function H ( a , q ; α , β , γ , δ ; z ) is analytic at z = 1 , and hence also throughout the disk | z | < a . …