Jacobi theta functions

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31: 31.16 Mathematical Applications

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31.16.1 $${\mathrm{𝐻𝑝}}_{n,m}\left(x\right){\mathrm{𝐻𝑝}}_{n,m}\left(y\right)=\sum _{j=0}^n{A}_j{\sin }^{2j}\theta P_{n-j}^{(\gamma +\delta +2j-1,ϵ-1)}\left(\cos \left(2\theta \right)\right)P_j^{(\delta -1,\gamma -1)}\left(\cos \left(2\varphi \right)\right),$$

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Symbols:

${\mathrm{𝐻𝑝}}_{n,m}(a,q_{n,m};-n,\beta ,\gamma ,\delta ;z)$: Heun polynomials, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $\cos z$: cosine function, $\sin z$: sine function, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $j$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer, $a$: complex parameter, $q$: real or complex parameter, $\beta$: real or complex parameter, $\theta$: angle, $\varphi$: angle, $x$: variable, $y$: variable and $A_j$: coefficients

Referenced by:

§31.16(ii), §31.16(ii)

Permalink:

http://dlmf.nist.gov/31.16.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §31.16(ii), §31.16 and Ch.31

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32: Errata

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22.19.2 $$\sin \left(\frac{1}{2}\theta (t)\right)=\sin \left(\frac{1}{2}\alpha \right)\mathrm{sn}(t+K,\sin \left(\frac{1}{2}\alpha \right))$$

Originally the first argument to the function $\mathrm{sn}$ was given incorrectly as $t$. The correct argument is $t+K$.

Reported 2014-03-05 by Svante Janson.

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22.19.3 $$\theta (t)=2\mathrm{am}(t\sqrt{E/2},\sqrt{2/E})$$

Originally the first argument to the function $\mathrm{am}$ was given incorrectly as $t$. The correct argument is $t\sqrt{E/2}$.

Reported 2014-03-05 by Svante Janson.

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33: Bibliography M

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Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions, and Schur functions. Ramanujan J. 6 (1), pp. 7–149.

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External Links:

Document, ZentralBlatt

Referenced by:

§27.13(iv).

Encodings:

BibTeX

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New infinite families of exact sums of squares formulas, Jacobi elliptic functions, and Ramanujan’s tau function. Proc. Nat. Acad. Sci. U.S.A. 93 (26), pp. 15004–15008.

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External Links:

ISSN 0027-8424, MathReview (Ken Ono), ZentralBlatt

Referenced by:

§27.13(iv).

Encodings:

BibTeX

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Balanced ${}_3{}^{}\Theta _2^{}$ summation theorems for $U(n)$ basic hypergeometric series. Adv. Math. 131 (1), pp. 93–187.

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External Links:

ISSN 0001-8708, Document, MathReview (Jiang Zeng), ZentralBlatt

Referenced by:

§17.15.

Encodings:

BibTeX

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Tata Lectures on Theta. I. Birkhäuser Boston Inc., Boston, MA.

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External Links:

ISBN 3-7643-3109-7, Document, MathReview (M. Kh. Gizatullin), ZentralBlatt

Referenced by:

§21.3(i), §21.3(ii), §21.5(i), §21.6(i), Ch.21.

Encodings:

BibTeX

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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

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34: 19.25 Relations to Other Functions

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§19.25(iv) Theta Functions

►For relations of symmetric integrals to theta functions, see §20.9(i). … ►If ${\mathrm{cs}}^2(u,k)\ge 0$, then …where we assume $0\le x^2\le 1$ if $x=\mathrm{sn}$, $\mathrm{cn}$, or $\mathrm{cd}$; $x^2\ge 1$ if $x=\mathrm{ns}$, $\mathrm{nc}$, or $\mathrm{dc}$; $x$ real if $x=\mathrm{cs}$ or $\mathrm{sc}$; $k^{\prime }\le x\le 1$ if $x=\mathrm{dn}$; $1\le x\le 1/k^{\prime }$ if $x=\mathrm{nd}$; $x^2\ge k_{}^{\prime }{}_{}{}^2$ if $x=\mathrm{ds}$; $0\le x^2\le 1/k_{}^{\prime }{}_{}{}^2$ if $x=\mathrm{sd}$. … ► …

35: 18.18 Sums

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18.18.9 $$P_n\left(\cos {\theta }_1\cos {\theta }_2+\sin {\theta }_1\sin {\theta }_2\cos \varphi \right)=P_n\left(\cos {\theta }_1\right)P_n\left(\cos {\theta }_2\right)+2\sum _{\mathrm{\ell }=1}^n\frac{(n-\mathrm{\ell })!(n+\mathrm{\ell })!}{2^{2\mathrm{\ell }}{(n!)}^2}{(\sin {\theta }_1)}^{\mathrm{\ell }}{P}_{n-\mathrm{\ell }}^{(\mathrm{\ell },\mathrm{\ell })}\left(\cos {\theta }_1\right){(\sin {\theta }_2)}^{\mathrm{\ell }}{P}_{n-\mathrm{\ell }}^{(\mathrm{\ell },\mathrm{\ell })}\left(\cos {\theta }_2\right)\cos \left(\mathrm{\ell }\varphi \right).$$

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Symbols:

$P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $P_n\left(x\right)$: Legendre polynomial, $\cos z$: cosine function, $!$: factorial (as in $n!$), $\sin z$: sine function, $\mathrm{\ell }$: nonnegative integer and $n$: nonnegative integer

Referenced by:

§14.30(iii), §18.17(ii), §18.18(ii), §18.18(ii), §18.18(ii)

Permalink:

http://dlmf.nist.gov/18.18.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §18.18(ii), §18.18(ii), §18.18 and Ch.18

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36: 31.2 Differential Equations

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31.2.5 $$z={\sin }^2\theta ,$$

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Symbols:

$\sin z$: sine function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/31.2.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §31.2(iii), §31.2 and Ch.31

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31.2.6 $$\frac{d^{2}w}{{d\theta }^2}+\left((2\gamma -1)\cot \theta -(2\delta -1)\tan \theta -\frac{ϵ\sin \left(2\theta \right)}{a-{\sin }^2\theta }\right)\frac{dw}{d\theta }+4\frac{\alpha \beta {\sin }^2\theta -q}{a-{\sin }^2\theta }w=0.$$

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Symbols:

$\cot z$: cotangent function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\sin z$: sine function, $\tan z$: tangent function, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $a$: complex parameter, $q$: real or complex parameter, $\alpha$: real or complex parameter and $\beta$: real or complex parameter

Permalink:

http://dlmf.nist.gov/31.2.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §31.2(iii), §31.2 and Ch.31

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Jacobi’s Elliptic Form

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$z={\mathrm{sn}}^2(\zeta ,k).$

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31.2.8 $$\frac{d^{2}w}{{d\zeta }^2}+\left((2\gamma -1)\frac{\mathrm{cn}\zeta \mathrm{dn}\zeta }{\mathrm{sn}\zeta }-(2\delta -1)\frac{\mathrm{sn}\zeta \mathrm{dn}\zeta }{\mathrm{cn}\zeta }-(2ϵ-1)k^2\frac{\mathrm{sn}\zeta \mathrm{cn}\zeta }{\mathrm{dn}\zeta }\right)\frac{dw}{d\zeta }+4k^2(\alpha \beta {\mathrm{sn}}^2\zeta -q)w=0.$$

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Symbols:

$\mathrm{cn}(z,k)$: Jacobian elliptic function, $\mathrm{dn}(z,k)$: Jacobian elliptic function, $\mathrm{sn}(z,k)$: Jacobian elliptic function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $q$: real or complex parameter, $\alpha$: real or complex parameter, $\beta$: real or complex parameter, $\zeta$: change of variable and $k$: modulus

Referenced by:

§31.7(ii)

Permalink:

http://dlmf.nist.gov/31.2.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §31.2(iv), §31.2(iv), §31.2 and Ch.31

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37: 18.17 Integrals

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Jacobi

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Jacobi

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Jacobi

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Jacobi

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Jacobi

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38: Bibliography B

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The generating function of Jacobi polynomials. J. London Math. Soc. 13, pp. 8–12.

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External Links:

ISSN 0024-6107; 1469-7750/e, Document, ZentralBlatt

Referenced by:

§18.18(vii).

Encodings:

BibTeX

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Abelian Functions: Abel’s Theorem and the Allied Theory of Theta Functions. Cambridge University Press, Cambridge.

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Notes:

Reprint of the 1897 original, with a foreword by Igor Krichever.

External Links:

ISBN 0-521-49877-5, MathReview (John B. Little), ZentralBlatt

Referenced by:

§20.9(ii).

Encodings:

BibTeX

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The Bounds for the Error Term of an Asymptotic Approximation of Jacobi Polynomials. In Orthogonal Polynomials and Their Applications (Segovia, 1986), Lecture Notes in Math., Vol. 1329, pp. 203–221.

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External Links:

MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§18.15(i), §18.15(i).

Encodings:

BibTeX

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A Brief Introduction to Theta Functions. Athena Series: Selected Topics in Mathematics, Holt, Rinehart and Winston, New York.

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External Links:

MathReview (W. J. LeVeque), ZentralBlatt

Referenced by:

§20.10(i), §20.10(ii), §20.11(i), §20.13, §20.5(ii), §20.7(viii), §20.8.

Encodings:

BibTeX

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A cubic counterpart of Jacobi’s identity and the AGM. Trans. Amer. Math. Soc. 323 (2), pp. 691–701.

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External Links:

ISSN 0002-9947, FullText, MathReview (Bruce C. Berndt), ZentralBlatt

Referenced by:

§15.8(v).

Encodings:

BibTeX

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39: 18.12 Generating Functions

§18.12 Generating Functions

►The $z$-radii of convergence will depend on $x$, and in first instance we will assume $x\in [-1,1]$ for Jacobi, ultraspherical, Chebyshev and Legendre, $x\in [0,\mathrm{\infty })$ for Laguerre, and $x\in ℝ$ for Hermite. … ►

Jacobi

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Ultraspherical

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Legendre

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40: 15.9 Relations to Other Functions

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Jacobi

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§15.9(ii) Jacobi Function

… ►The Jacobi transform is defined as …with inverse … …