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1: 16.6 Transformations of Variable
§16.6 Transformations of Variable
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Quadratic
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Cubic
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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
β–ΊKuznetsov 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
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§16.16(i) Reduction Formulas
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16.16.10 F 4 ⁑ ( Ξ± , Ξ² ; Ξ³ , Ξ³ ; x , y ) = Ξ“ ⁑ ( Ξ³ ) ⁒ Ξ“ ⁑ ( Ξ² Ξ± ) Ξ“ ⁑ ( Ξ³ Ξ± ) ⁒ Ξ“ ⁑ ( Ξ² ) ⁒ ( y ) Ξ± ⁒ F 4 ⁑ ( Ξ± , Ξ± Ξ³ + 1 ; Ξ³ , Ξ± Ξ² + 1 ; x y , 1 y ) + Ξ“ ⁑ ( Ξ³ ) ⁒ Ξ“ ⁑ ( Ξ± Ξ² ) Ξ“ ⁑ ( Ξ³ Ξ² ) ⁒ Ξ“ ⁑ ( Ξ± ) ⁒ ( y ) Ξ² ⁒ F 4 ⁑ ( Ξ² , Ξ² Ξ³ + 1 ; Ξ³ , Ξ² Ξ± + 1 ; x y , 1 y ) .
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4: 15.19 Methods of Computation
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5: 15.8 Transformations of Variable
§15.8 Transformations of Variable
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§15.8(i) Linear Transformations
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§15.8(iii) Quadratic Transformations
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§15.8(v) Cubic Transformations
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6: 18.25 Wilson Class: Definitions
β–ΊTable 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 ; Ξ± , Ξ² , Ξ³ , Ξ΄ ) , and dual Hahn polynomials R n ⁑ ( x ; Ξ³ , Ξ΄ , N ) . β–Ί
Table 18.25.1: Wilson class OP’s: transformations of variable, orthogonality ranges, and parameter constraints.
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p n ⁑ ( x ) x = λ ⁒ ( y ) Orthogonality range for y Constraints
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7: 23.18 Modular Transformations
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23.18.3 Ξ» ⁑ ( π’œ Ο„ ) = Ξ» ⁑ ( Ο„ ) ,
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23.18.4 J ⁑ ( π’œ Ο„ ) = J ⁑ ( Ο„ ) .
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23.18.5 Ξ· ⁑ ( π’œ Ο„ ) = Ξ΅ ⁑ ( π’œ ) ⁒ ( i ⁒ ( c ⁒ Ο„ + d ) ) 1 / 2 ⁒ Ξ· ⁑ ( Ο„ ) ,
8: 1.13 Differential Equations
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Transformation of the Point at Infinity
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Liouville Transformation
9: 31.7 Relations to Other Functions
β–ΊJoyce (1994) gives a reduction in which the independent variable is transformed not polynomially or rationally, but algebraically. …
10: 23.15 Definitions
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23.15.3 π’œ Ο„ = a ⁒ Ο„ + b c ⁒ Ο„ + d ,
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23.15.5 f ⁑ ( π’œ Ο„ ) = c π’œ ⁒ ( c ⁒ Ο„ + d ) β„“ ⁒ f ⁑ ( Ο„ ) , ⁑ Ο„ > 0 ,