Legendre relation

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11: 19.7 Connection Formulas

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Legendre’s Relation

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§19.7(iii) Change of Parameter of $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$

… ►The third relation (missing from the literature of Legendre’s integrals) maps each circular region onto the other and each hyperbolic region onto the other: …

12: 14.7 Integer Degree and Order

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§14.7(i) $\mu =0$

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13: 18.9 Recurrence Relations and Derivatives

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Table 18.9.1: Classical OP’s: recurrence relations (18.9.1).

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$p_n(x)$	$A_n$	$B_n$	$C_n$
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Table 18.9.2: Classical OP’s: recurrence relations (18.9.2_1).

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$p_n(x)$	$a_n$	$b_n$	$c_n$
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14: 14.3 Definitions and Hypergeometric Representations

§14.3 Definitions and Hypergeometric Representations

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§14.3(ii) Interval

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§14.3(iii) Alternative Hypergeometric Representations

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14.3.14 $$w_2(\nu ,\mu ,x)=\frac{2^{\mu }\mathrm{\Gamma }\left(\frac{1}{2}\nu +\frac{1}{2}\mu +1\right)}{\mathrm{\Gamma }\left(\frac{1}{2}\nu -\frac{1}{2}\mu +\frac{1}{2}\right)}x{\left(1-x^2\right)}^{-\mu /2}𝐅(\frac{1}{2}-\frac{1}{2}\nu -\frac{1}{2}\mu ,\frac{1}{2}\nu -\frac{1}{2}\mu +1;\frac{3}{2};x^2).$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $x$: real variable, $\mu$: general order, $\nu$: general degree and $w_2$: function $w_1$,

Referenced by:

§10.59, §14.3(i), §14.5(i)

Permalink:

http://dlmf.nist.gov/14.3.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §14.3(iii), §14.3 and Ch.14

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§14.3(iv) Relations to Other Functions

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15: 34.3 Basic Properties: $\mathit{3}j$ Symbol

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§34.3(vii) Relations to Legendre Polynomials and Spherical Harmonics

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16: 14.10 Recurrence Relations and Derivatives

§14.10 Recurrence Relations and Derivatives

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17: Bibliography D

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The Legendre Relation for Elliptic Integrals. In Paul Halmos: Celebrating 50 Years of Mathematics, J. H. Ewing and F. W. Gehring (Eds.), pp. 305–315.

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External Links:

ISBN 0-387-97509-8, MathReview (J. M. H. Peters), ZentralBlatt

Referenced by:

§19.7(i).

Encodings:

BibTeX

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18: 19.5 Maclaurin and Related Expansions

§19.5 Maclaurin and Related Expansions

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19: 15.9 Relations to Other Functions

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Legendre

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§15.9(iv) Associated Legendre Functions; Ferrers Functions

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20: 14.21 Definitions and Basic Properties

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§14.21(iii) Properties

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