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1: 20.11 Generalizations and Analogs
§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)). …
2: 11.8 Analogs to Kelvin Functions
§11.8 Analogs to Kelvin Functions
3: 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. …
4: 28.27 Addition Theorems
They are analogous to the addition theorems for Bessel functions (§10.23(ii)) and modified Bessel functions (§10.44(ii)). …
5: Sidebar 5.SB1: Gamma & Digamma Phase Plots
This pattern is analogous to one that would be seen in fluid flow generated by a semi-infinite line of vortices. … The fluid flow analogy in this case involves a line of vortices of alternating sign of circulation, resulting in a near cancellation of flow far from the real axis.
6: 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. …
7: 26.22 Software
  • GAP (website). A system for computational discrete algebra.

  • 8: 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.)
    9: 17.17 Physical Applications
    They were given this name because they play a role in quantum physics analogous to the role of Lie groups and special functions in classical mechanics. …
    10: 25.17 Physical Applications
    Analogies exist between the distribution of the zeros of ζ ( s ) on the critical line and of semiclassical quantum eigenvalues. …