Jacobi?Abel addition theorem

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1: 28.27 Addition Theorems

§28.27 Addition Theorems

►Addition theorems provide important connections between Mathieu functions with different parameters and in different coordinate systems. They are analogous to the addition theorems for Bessel functions (§10.23(ii)) and modified Bessel functions (§10.44(ii)). …

2: 30.10 Series and Integrals

… ►For an addition theorem, see Meixner and Schäfke (1954, p. 300) and King and Van Buren (1973). …

3: 10.44 Sums

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§10.44(i) Multiplication Theorem

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§10.44(ii) Addition Theorems

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Neumann’s Addition Theorem

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Graf’s and Gegenbauer’s Addition Theorems

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4: 22.18 Mathematical Applications

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§22.18(iv) Elliptic Curves and the Jacobi–Abel Addition Theorem

… ►For any two points $(x_1,y_1)$ and $(x_2,y_2)$ on this curve, their sum $(x_3,y_3)$, always a third point on the curve, is defined by the Jacobi–Abel addition law …With the identification $x=\mathrm{sn}(z,k)$, $y=d(\mathrm{sn}(z,k))/dz$, the addition law (22.18.8) is transformed into the addition theorem (22.8.1); see Akhiezer (1990, pp. 42, 45, 73–74) and McKean and Moll (1999, §§2.14, 2.16). …

5: 13.13 Addition and Multiplication Theorems

§13.13 Addition and Multiplication Theorems

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§13.13(i) Addition Theorems for $M(a,b,z)$

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§13.13(ii) Addition Theorems for $U(a,b,z)$

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13.13.12 $${\mathrm{e}}^y{\left(\frac{x+y}{x}\right)}^{1-b}\sum _{n=0}^{\mathrm{\infty }}\frac{{(-y)}^n}{n!x^n}U(a-n,b-n,x),$$ .

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Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $n$: nonnegative integer, $x$: real variable and $y$: real variable

Permalink:

http://dlmf.nist.gov/13.13.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §13.13(ii), §13.13 and Ch.13

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§13.13(iii) Multiplication Theorems for $M(a,b,z)$ and $U(a,b,z)$

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6: 10.23 Sums

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§10.23(i) Multiplication Theorem

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§10.23(ii) Addition Theorems

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Neumann’s Addition Theorem

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Graf’s and Gegenbauer’s Addition Theorems

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7: 14.28 Sums

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§14.28(i) Addition Theorem

… ►For generalizations in terms of Gegenbauer and Jacobi polynomials, see Theorem 2. 1 in Cohl (2013b) and Theorem 1 in Cohl (2013a) respectively. …

8: 13.26 Addition and Multiplication Theorems

§13.26 Addition and Multiplication Theorems

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§13.26(i) Addition Theorems for $M_{\kappa ,\mu }\left(z\right)$

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§13.26(ii) Addition Theorems for $W_{\kappa ,\mu }\left(z\right)$

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§13.26(iii) Multiplication Theorems for $M_{\kappa ,\mu }\left(z\right)$ and $W_{\kappa ,\mu }\left(z\right)$

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9: 12.13 Sums

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§12.13(i) Addition Theorems

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10: 23.23 Tables

… ►05, and in the case of $\mathrm{\wp }\left(z\right)$ the user may deduce values for complex $z$ by application of the addition theorem (23.10.1). …