relation to theta functions

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11: Bille C. Carlson

… ►He then went to Oxford as a Rhodes Scholar and completed a doctoral degree in physics. … ►Also, the homogeneity of the

R

-function has led to a new type of mean value for several variables, accompanied by various inequalities. … ►In Symmetry in c, d, n of Jacobian elliptic functions (2004) he found a previously hidden symmetry in relations between Jacobian elliptic functions, which can now take a form that remains valid when the letters c, d, and n are permuted. This invariance usually replaces sets of twelve equations by sets of three equations and applies also to the relation between the first symmetric elliptic integral and the Jacobian functions. In Permutation symmetry for theta functions (2011) he found an analogous hidden symmetry between theta functions. …

12: 15.9 Relations to Other Functions

§15.9 Relations to Other Functions

►

§15.9(i) Orthogonal Polynomials

… ►

Jacobi

… ►

Legendre

… ►

Meixner

…

13: 18.35 Pollaczek Polynomials

… ►

18.35.7

(1-z{\mathrm{e}}^{\mathrm{i}\theta})^{-\lambda+\mathrm{i}\tau_{a,b}(\theta)}(1% -z{\mathrm{e}}^{-\mathrm{i}\theta})^{-\lambda-\mathrm{i}\tau_{a,b}(\theta)}=% \sum_{n=0}^{\infty}P^{(\lambda)}_{n}\left(\cos\theta;a,b\right)z^{n},

|z|<1

0<\theta<\pi

ⓘ

Symbols:: $P^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b}\right)$ : Pollaczek polynomial, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $z$ : complex variable, $n$ : nonnegative integer and $\tau_{a,b}(\theta)$
Proved:: Ismail (2009, (5.4.8))(proved)
Permalink:: http://dlmf.nist.gov/18.35.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §18.35(iii), §18.35 and Ch.18

…

14: 10.21 Zeros

… ►The functions

\rho_{\nu}(t)

and

\sigma_{\nu}(t)

are related to the inverses of the phase functions

\theta_{\nu}\left(x\right)

and

\phi_{\nu}\left(x\right)

defined in §10.18(i): if

\nu\geq 0

, then …

15: 18.5 Explicit Representations

… ►

Chebyshev

… ►Related formula: … ►

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

… ►

Laguerre

… ►

Hermite

…

16: 14.19 Toroidal (or Ring) Functions

… ►This form of the differential equation arises when Laplace’s equation is transformed into toroidal coordinates

(\eta,\theta,\phi)

, which are related to Cartesian coordinates

(x,y,z)

by …

17: 27.11 Asymptotic Formulas: Partial Sums

… ►The behavior of a number-theoretic function

f(n)

for large

n

is often difficult to determine because the function values can fluctuate considerably as

n

increases. It is more fruitful to study partial sums and seek asymptotic formulas of the form …Dirichlet’s divisor problem (unsolved as of 2022) is to determine the least number

\theta_{0}

such that the error term in (27.11.2) is

O\left(x^{\theta}\right)

for all

\theta>\theta_{0}

. Huxley (2003) proves that

\theta_{0}\leq\frac{131}{416}

. ►Equations (27.11.3)–(27.11.11) list further asymptotic formulas related to some of the functions listed in §27.2. …

18: 18.34 Bessel Polynomials

… ►

§18.34(i) Definitions and Recurrence Relation

… ►where

\mathsf{k}_{n}

is a modified spherical Bessel function (10.49.9), and … Sometimes the polynomials

\theta_{n}(x;a,b)

are called reverse Bessel polynomials. … … ►The full system satisfies orthogonality with respect to a (not positive definite) moment functional; see Evans et al. (1993, (2.7)) for the simple expression of the moments

\mu_{n}

. …

19: 9.9 Zeros

… ►

§9.9(ii) Relation to Modulus and Phase

… ►

§9.9(iii) Derivatives With Respect to $k$

… ►

§9.9(iv) Asymptotic Expansions

… ►

§9.9(v) Tables

►Tables 9.9.1 and 9.9.2 give 10D values of the first ten real zeros of

\operatorname{Ai}

\operatorname{Ai}'

\operatorname{Bi}

\operatorname{Bi}'

, together with the associated values of the derivative or the function. …

20: 9.8 Modulus and Phase

… ►(These definitions of

\theta\left(x\right)

and

\phi\left(x\right)

differ from Abramowitz and Stegun (1964, Chapter 10), and agree more closely with those used in Miller (1946) and Olver (1997b, Chapter 11).) …