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Graf addition theorem

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1: 10.44 Sums
§10.44(i) Multiplication Theorem
§10.44(ii) Addition Theorems
Neumann’s Addition Theorem
Graf’s and Gegenbauer’s Addition Theorems
2: 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)). …
3: 10.23 Sums
§10.23(ii) Addition Theorems
Neumann’s Addition Theorem
Graf’s and Gegenbauer’s Addition Theorems
See accompanying text
Figure 10.23.1: Graf’s and Gegenbauer’s addition theorems. Magnify
4: 30.10 Series and Integrals
For an addition theorem, see Meixner and Schäfke (1954, p. 300) and King and Van Buren (1973). …
5: 22.18 Mathematical Applications
§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 = sn ( z , k ) , y = d ( sn ( z , k ) ) / d z , 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). …
6: 13.13 Addition and Multiplication Theorems
§13.13 Addition and Multiplication Theorems
§13.13(i) Addition Theorems for M ( a , b , z )
§13.13(ii) Addition Theorems for U ( a , b , z )
13.13.12 e y ( x + y x ) 1 b n = 0 ( y ) n n ! x n U ( a n , b n , x ) , | y | < | x | .
§13.13(iii) Multiplication Theorems for M ( a , b , z ) and U ( a , b , z )
7: 14.28 Sums
§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
§13.26(i) Addition Theorems for M κ , μ ( z )
§13.26(ii) Addition Theorems for W κ , μ ( z )
§13.26(iii) Multiplication Theorems for M κ , μ ( z ) and W κ , μ ( z )
9: 12.13 Sums
§12.13(i) Addition Theorems
10: 23.23 Tables
05, and in the case of ( z ) the user may deduce values for complex z by application of the addition theorem (23.10.1). …