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relation to classical theta functions

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1: 21.2 Definitions
§21.2(iii) Relation to Classical Theta Functions
2: William P. Reinhardt
He has recently carried out research on non-linear dynamics of Bose–Einstein condensates that served to motivate his interest in elliptic functions. Older work on the scattering theory of the atomic Coulomb problem led to the discovery of new classes of orthogonal polynomials relating to the spectral theory of Schrödinger operators, and new uses of old ones: this work was strongly motivated by his original ownership of a 1964 hard copy printing of the original AMS 55 NBS Handbook of Mathematical Functions. … This is closely connected with his interests in classical dynamical “chaos,” an area where he coauthored a book, Chaos in atomic physics with Reinhold Blümel. …
  • 3: 20.1 Special Notation
    The main functions treated in this chapter are the theta functions θ j ( z | τ ) = θ j ( z , q ) where j = 1 , 2 , 3 , 4 and q = e i π τ . …Sometimes the theta functions are called the Jacobian or classical theta functions to distinguish them from generalizations; compare Chapter 21. Primes on the θ symbols indicate derivatives with respect to the argument of the θ function. … This notation simplifies the relationship of the theta functions to Jacobian elliptic functions22.2); see Neville (1951). McKean and Moll’s notation: ϑ j ( z | τ ) = θ j ( π z | τ ) , j = 1 , 2 , 3 , 4 . …
    4: 18.5 Explicit Representations
    Chebyshev
    §18.5(iii) Finite Power Series, the Hypergeometric Function, and Generalized Hypergeometric Functions
    Laguerre
    Hermite
    5: 18.15 Asymptotic Approximations
    §18.15 Asymptotic Approximations
    as n , uniformly with respect to θ [ δ , π δ ] . … With μ = 2 n + 1 the expansions in Chapter 12 are for the parabolic cylinder function U ( 1 2 μ 2 , μ t 2 ) , which is related to the Hermite polynomials via … The asymptotic behavior of the classical OP’s as x ± with the degree and parameters fixed is evident from their explicit polynomial forms; see, for example, (18.2.7) and the last two columns of Table 18.3.1. … See also Dunster (1999), Atia et al. (2014) and Temme (2015, Chapter 32).
    6: Bibliography M
  • A. I. Markushevich (1992) Introduction to the Classical Theory of Abelian Functions. American Mathematical Society, Providence, RI.
  • F. Matta and A. Reichel (1971) Uniform computation of the error function and other related functions. Math. Comp. 25 (114), pp. 339–344.
  • G. J. Miel (1981) Evaluation of complex logarithms and related functions. SIAM J. Numer. Anal. 18 (4), pp. 744–750.
  • S. C. Milne (1985c) A new symmetry related to 𝑆𝑈 ( n ) for classical basic hypergeometric series. Adv. in Math. 57 (1), pp. 71–90.
  • D. Mumford (1984) Tata Lectures on Theta. II. Birkhäuser Boston Inc., Boston, MA.
  • 7: 15.9 Relations to Other Functions
    §15.9 Relations to Other Functions
    §15.9(i) Orthogonal Polynomials
    Jacobi
    Legendre
    Meixner
    8: Bibliography R
  • M. Rahman (2001) The Associated Classical Orthogonal Polynomials. In Special Functions 2000: Current Perspective and Future Directions (Tempe, AZ), NATO Sci. Ser. II Math. Phys. Chem., Vol. 30, pp. 255–279.
  • H. E. Rauch and A. Lebowitz (1973) Elliptic Functions, Theta Functions, and Riemann Surfaces. The Williams & Wilkins Co., Baltimore, MD.
  • W. P. Reinhardt (2021a) Erratum to:Relationships between the zeros, weights, and weight functions of orthogonal polynomials: Derivative rule approach to Stieltjes and spectral imaging. Computing in Science and Engineering 23 (4), pp. 91.
  • R. Roy (2017) Elliptic and modular functions from Gauss to Dedekind to Hecke. Cambridge University Press, Cambridge.
  • J. Rushchitsky and S. Rushchitska (2000) On Simple Waves with Profiles in the form of some Special Functions—Chebyshev-Hermite, Mathieu, Whittaker—in Two-phase Media. In Differential Operators and Related Topics, Vol. I (Odessa, 1997), Operator Theory: Advances and Applications, Vol. 117, pp. 313–322.
  • 9: Bibliography K
  • A. Khare, A. Lakshminarayan, and U. Sukhatme (2003) Cyclic identities for Jacobi elliptic and related functions. J. Math. Phys. 44 (4), pp. 1822–1841.
  • D. A. Kofke (2004) Comment on “The incomplete beta function law for parallel tempering sampling of classical canonical systems” [J. Chem. Phys. 120, 4119 (2004)]. J. Chem. Phys. 121 (2), pp. 1167.
  • S. Koizumi (1976) Theta relations and projective normality of Abelian varieties. Amer. J. Math. 98 (4), pp. 865–889.
  • T. H. Koornwinder (1975c) Two-variable Analogues of the Classical Orthogonal Polynomials. In Theory and Application of Special Functions, R. A. Askey (Ed.), pp. 435–495.
  • T. H. Koornwinder (2007b) The structure relation for Askey-Wilson polynomials. J. Comput. Appl. Math. 207 (2), pp. 214–226.
  • 10: Bibliography
  • V. S. Adamchik and H. M. Srivastava (1998) Some series of the zeta and related functions. Analysis (Munich) 18 (2), pp. 131–144.
  • C. Adiga, B. C. Berndt, S. Bhargava, and G. N. Watson (1985) Chapter 16 of Ramanujan’s second notebook: Theta-functions and q -series. Mem. Amer. Math. Soc. 53 (315), pp. v+85.
  • N. I. Akhiezer (2021) The classical moment problem and some related questions in analysis. Classics in Applied Mathematics, Vol. 82, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.
  • T. M. Apostol and T. H. Vu (1984) Dirichlet series related to the Riemann zeta function. J. Number Theory 19 (1), pp. 85–102.
  • F. M. Arscott (1964a) Integral equations and relations for Lamé functions. Quart. J. Math. Oxford Ser. (2) 15, pp. 103–115.