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1: 16.6 Transformations of Variable
§16.6 Transformations of Variable
Quadratic
Cubic
16.6.2 F 2 3 ( a , 2 b - a - 1 , 2 - 2 b + a b , a - b + 3 2 ; z 4 ) = ( 1 - z ) - a F 2 3 ( 1 3 a , 1 3 a + 1 3 , 1 3 a + 2 3 b , a - b + 3 2 ; - 27 z 4 ( 1 - z ) 3 ) .
2: Vadim B. Kuznetsov
Kuznetsov published papers on special functions and orthogonal polynomials, the quantum scattering method, integrable discrete many-body systems, separation of variables, Bäcklund transformation techniques, and integrability in classical and quantum mechanics. …
3: 16.16 Transformations of Variables
§16.16 Transformations of Variables
§16.16(i) Reduction Formulas
16.16.10 F 4 ( α , β ; γ , γ ; x , y ) = Γ ( γ ) Γ ( β - α ) Γ ( γ - α ) Γ ( β ) ( - y ) - α F 4 ( α , α - γ + 1 ; γ , α - β + 1 ; x y , 1 y ) + Γ ( γ ) Γ ( α - β ) Γ ( γ - β ) Γ ( α ) ( - y ) - β F 4 ( β , β - γ + 1 ; γ , β - α + 1 ; x y , 1 y ) .
4: 15.19 Methods of Computation
5: 15.8 Transformations of Variable
§15.8 Transformations of Variable
§15.8(i) Linear Transformations
§15.8(iii) Quadratic Transformations
§15.8(v) Cubic Transformations
6: 18.25 Wilson Class: Definitions
Table 18.25.1 lists the transformations of variable, orthogonality ranges, and parameter constraints that are needed in §18.2(i) for the Wilson polynomials W n ( x ; a , b , c , d ) , continuous dual Hahn polynomials S n ( x ; a , b , c ) , Racah polynomials R n ( x ; α , β , γ , δ ) , and dual Hahn polynomials R n ( x ; γ , δ , N ) .
Table 18.25.1: Wilson class OP’s: transformations of variable, orthogonality ranges, and parameter constraints.
p n ( x ) x = λ ( y ) Orthogonality range for y Constraints
7: 23.18 Modular Transformations
23.18.3 λ ( 𝒜 τ ) = λ ( τ ) ,
23.18.4 J ( 𝒜 τ ) = J ( τ ) .
23.18.5 η ( 𝒜 τ ) = ε ( 𝒜 ) ( - i ( c τ + d ) ) 1 / 2 η ( τ ) ,
8: 1.13 Differential Equations
Transformation of the Point at Infinity
Liouville Transformation
9: 31.7 Relations to Other Functions
Joyce (1994) gives a reduction in which the independent variable is transformed not polynomially or rationally, but algebraically. …
10: 23.15 Definitions
23.15.3 𝒜 τ = a τ + b c τ + d ,
23.15.5 f ( 𝒜 τ ) = c 𝒜 ( c τ + d ) f ( τ ) , τ > 0 ,