relation to classical theta functions

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… в–єHe 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. … в–єThis is closely connected with his interests in classical dynamical “chaos,” an area where he coauthored a book, Chaos in atomic physics with Reinhold Blümel. … в–є

20 Theta Functions

в–є

22 Jacobian Elliptic Functions

…

3: 20.1 Special Notation

… в–єThe main functions treated in this chapter are the theta functions

\theta_{j}\left(z\middle|\tau\right)=\theta_{j}\left(z,q\right)

where

j=1,2,3,4

and

q=e^{i\pi\tau}

. …Sometimes the theta functions are called the Jacobian or classical theta functions to distinguish them from generalizations; compare Chapter 21. в–єPrimes on the

\theta

symbols indicate derivatives with respect to the argument of the

\theta

function. … в–єThis notation simplifies the relationship of the theta functions to Jacobian elliptic functions (§22.2); see Neville (1951). в–єMcKean and Moll’s notation:

\vartheta_{j}(z|\tau)=\theta_{j}\left(\pi z\middle|\tau\right)

j=1,2,3,4

. …

4: 18.5 Explicit Representations

… в–є

Chebyshev

… в–є

§18.5(iii) Finite Power Series, the Hypergeometric Function, and Generalized Hypergeometric Functions

… в–є

Laguerre

… в–є

Hermite

… в–є …

5: 18.15 Asymptotic Approximations

§18.15 Asymptotic Approximations

… в–єas

n\to\infty

, uniformly with respect to

\theta\in[\delta,\pi-\delta]

. … в–єWith

\mu=\sqrt{2n+1}

the expansions in Chapter 12 are for the parabolic cylinder function

U\left(-\tfrac{1}{2}\mu^{2},\mu t\sqrt{2}\right)

, which is related to the Hermite polynomials via … в–єThe asymptotic behavior of the classical OP’s as

x\to\pm\infty

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. … в–єSee also Dunster (1999), Atia et al. (2014) and Temme (2015, Chapter 32).

6: Bibliography M

… в–є

A. I. Markushevich (1992) Introduction to the Classical Theory of Abelian Functions. American Mathematical Society, Providence, RI.

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Notes:: Translated from the 1979 Russian original by G. Bluher
External Links:: ISBN 0-8218-4542-X, MathReview, ZentralBlatt
Referenced by:: §21.8.
Encodings:: BibTeX

… в–є

F. Matta and A. Reichel (1971) Uniform computation of the error function and other related functions. Math. Comp. 25 (114), pp. 339–344.

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External Links:: FullText, MathReview, ZentralBlatt
Referenced by:: §7.22(i).
Encodings:: BibTeX

… в–є

G. J. Miel (1981) Evaluation of complex logarithms and related functions. SIAM J. Numer. Anal. 18 (4), pp. 744–750.

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External Links:: ISSN 0036-1429, Document, MathReview (M. Brannigan), ZentralBlatt
Referenced by:: §4.45(ii).
Encodings:: BibTeX

… в–є

S. C. Milne (1985c) A new symmetry related to

\mathit{SU}(n)

for classical basic hypergeometric series. Adv. in Math. 57 (1), pp. 71–90.

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External Links:: ISSN 0001-8708, Document, MathReview (George E. Andrews), ZentralBlatt
Referenced by:: §17.15.
Encodings:: BibTeX

… в–є

D. Mumford (1984) Tata Lectures on Theta. II. Birkhäuser Boston Inc., Boston, MA.

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Notes:: Jacobian theta functions and differential equations
External Links:: ISBN 0-8176-3110-0, MathReview (M. Kh. Gizatullin), ZentralBlatt
Referenced by:: §21.7(i), §21.7(ii), §21.7(iii), Ch.21.
Encodings:: BibTeX

…

7: 15.9 Relations to Other Functions

§15.9 Relations to Other Functions

в–є

§15.9(i) Orthogonal Polynomials

… в–є

Jacobi

… в–є

Legendre

… в–є

Meixner

…

8: Bibliography R

… в–є

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.

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External Links:: MathReview (Wadim Zudilin), ZentralBlatt
Referenced by:: §18.30(viii).
Encodings:: BibTeX

… в–є

H. E. Rauch and A. Lebowitz (1973) Elliptic Functions, Theta Functions, and Riemann Surfaces. The Williams & Wilkins Co., Baltimore, MD.

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External Links:: MathReview (H. H. Martens), ZentralBlatt
Referenced by:: §20.11(iv), §20.12(ii).
Encodings:: BibTeX

… в–є

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.

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External Links:: ISSN 1521-9615, Document, Link
Referenced by:: §18.39(iv), §18.40(ii).
Encodings:: BibTeX

… в–є

R. Roy (2017) Elliptic and modular functions from Gauss to Dedekind to Hecke. Cambridge University Press, Cambridge.

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External Links:: ISBN 978-1-107-15938-9, Document, Link, MathReview (Paul M. Jenkins), ZentralBlatt
Referenced by:: Profile Ranjan Roy.
Encodings:: BibTeX

… в–є

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.

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External Links:: ISBN 3-7643-6287-1, MathReview Entry, ZentralBlatt
Referenced by:: 5th item.
Encodings:: BibTeX

…

9: Bibliography K

… в–є

A. Khare, A. Lakshminarayan, and U. Sukhatme (2003) Cyclic identities for Jacobi elliptic and related functions. J. Math. Phys. 44 (4), pp. 1822–1841.

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External Links:: ISSN 0022-2488, Document, MathReview (Zhi Guo Liu), ZentralBlatt
Referenced by:: §22.9(iv), §22.9(v).
Encodings:: BibTeX

… в–є

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.

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External Links:: Document
Referenced by:: §8.24(ii).
Encodings:: BibTeX

в–є

S. Koizumi (1976) Theta relations and projective normality of Abelian varieties. Amer. J. Math. 98 (4), pp. 865–889.

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External Links:: FullText, MathReview (T. Oda), ZentralBlatt
Referenced by:: §21.6(ii).
Encodings:: BibTeX

… в–є

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.

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External Links:: MathReview (R. A. Askey), ZentralBlatt
Referenced by:: §18.37(ii).
Encodings:: BibTeX

… в–є

T. H. Koornwinder (2007b) The structure relation for Askey-Wilson polynomials. J. Comput. Appl. Math. 207 (2), pp. 214–226.

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External Links:: ISSN 0377-0427, Document, Link, MathReview Entry
Referenced by:: (18.9.18_5).
Encodings:: BibTeX

…

10: Bibliography

… в–є

V. S. Adamchik and H. M. Srivastava (1998) Some series of the zeta and related functions. Analysis (Munich) 18 (2), pp. 131–144.

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External Links:: ISSN 0174-4747, MathReview (Jack Quine), ZentralBlatt
Referenced by:: (25.11.37), (25.11.38), (25.11.39), (25.11.40), §25.11, §25.11(xi), (25.8.1), (25.8.4), §25.8.
Encodings:: BibTeX

… в–є

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.

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External Links:: ISSN 0065-9266, MathReview (R. A. Askey), ZentralBlatt
Referenced by:: §20.11(ii).
Encodings:: BibTeX

… в–є

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.

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Notes:: Reprint of the 1965 edition [ 0184042], Translated by N. Kemmer, With a foreword by H. J. Landau
External Links:: ISBN 978-1-611976-38-0, MathReview Entry
Referenced by:: §18.40(ii).
Encodings:: BibTeX

… в–є

T. M. Apostol and T. H. Vu (1984) Dirichlet series related to the Riemann zeta function. J. Number Theory 19 (1), pp. 85–102.

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External Links:: ISSN 0022-314X, Document, MathReview (Kai Man Tsang), ZentralBlatt
Referenced by:: (25.16.10), (25.16.11), (25.16.12), (25.16.14), (25.16.15), (25.16.5), (25.16.6), (25.16.7), (25.16.8), (25.16.9), §25.16(ii), §25.16(ii), §25.16.
Encodings:: BibTeX

… в–є

F. M. Arscott (1964a) Integral equations and relations for Lamé functions. Quart. J. Math. Oxford Ser. (2) 15, pp. 103–115.

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External Links:: Document, MathReview (I. Marx), ZentralBlatt
Referenced by:: §29.8.
Encodings:: BibTeX

…

relation to classical theta functions

1—10 of 12 matching pages

1: 21.2 Definitions

§21.2(iii) Relation to Classical Theta Functions

2: William P. Reinhardt

3: 20.1 Special Notation

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

6: Bibliography M

7: 15.9 Relations to Other Functions

§15.9 Relations to Other Functions

§15.9(i) Orthogonal Polynomials

Jacobi

Legendre

Meixner

8: Bibliography R

9: Bibliography K

10: Bibliography