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1: 15.5 Derivatives and Contiguous Functions
§15.5 Derivatives and Contiguous Functions
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§15.5(ii) Contiguous Functions
β–ΊThe six functions F ⁑ ( a ± 1 , b ; c ; z ) , F ⁑ ( a , b ± 1 ; c ; z ) , F ⁑ ( a , b ; c ± 1 ; z ) are said to be contiguous to F ⁑ ( a , b ; c ; z ) . … β–ΊAn equivalent equation to the hypergeometric differential equation (15.10.1) is …Further contiguous relations include: …
2: 16.3 Derivatives and Contiguous Functions
§16.3 Derivatives and Contiguous Functions
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§16.3(ii) Contiguous Functions
β–ΊTwo generalized hypergeometric functions F q p ⁑ ( 𝐚 ; 𝐛 ; z ) are (generalized) contiguous if they have the same pair of values of p and q , and corresponding parameters differ by integers. If p q + 1 , then any q + 2 distinct contiguous functions are linearly related. Examples are provided by the following recurrence relations: …
3: 8 Incomplete Gamma and Related
Functions
Chapter 8 Incomplete Gamma and Related Functions
4: Bibliography G
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  • F. Gao and V. J. W. Guo (2013) Contiguous relations and summation and transformation formulae for basic hypergeometric series. J. Difference Equ. Appl. 19 (12), pp. 2029–2042.
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  • W. Gautschi (1967) Computational aspects of three-term recurrence relations. SIAM Rev. 9 (1), pp. 24–82.
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  • W. Gautschi (1984) Questions of Numerical Condition Related to Polynomials. In Studies in Numerical Analysis, G. H. Golub (Ed.), pp. 140–177.
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  • A. Gil, J. Segura, and N. M. Temme (2014) Algorithm 939: computation of the Marcum Q-function. ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.
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  • Ya. I. GranovskiΔ­, I. M. Lutzenko, and A. S. Zhedanov (1992) Mutual integrability, quadratic algebras, and dynamical symmetry. Ann. Phys. 217 (1), pp. 1–20.
  • 5: 16.4 Argument Unity
    β–ΊThe characterizing properties (18.22.2), (18.22.10), (18.22.19), (18.22.20), and (18.26.14) of the Hahn and Wilson class polynomials are examples of the contiguous relations mentioned in the previous three paragraphs. β–ΊContiguous balanced series have parameters shifted by an integer but still balanced. … … β–ΊSee Bailey (1964, §§4.3(7) and 7.6(1)) for the transformation formulas and Wilson (1978) for contiguous relations. …
    6: 18.9 Recurrence Relations and Derivatives
    §18.9 Recurrence Relations and Derivatives
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    §18.9(i) Recurrence Relations
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    §18.9(ii) Contiguous Relations in the Parameters and the Degree
    β–ΊFurther n -th derivative formulas relating two different Jacobi polynomials can be obtained from §15.5(i) by substitution of (18.5.7). … β–Ίand the structure relation
    7: Bibliography S
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  • K. L. Sala (1989) Transformations of the Jacobian amplitude function and its calculation via the arithmetic-geometric mean. SIAM J. Math. Anal. 20 (6), pp. 1514–1528.
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  • J. Segura (2008) Interlacing of the zeros of contiguous hypergeometric functions. Numer. Algorithms 49 (1-4), pp. 387–407.
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  • A. Sharples (1967) Uniform asymptotic forms of modified Mathieu functions. Quart. J. Mech. Appl. Math. 20 (3), pp. 365–380.
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  • J. R. Stembridge (1995) A Maple package for symmetric functions. J. Symbolic Comput. 20 (5-6), pp. 755–768.
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  • F. Stenger (1993) Numerical Methods Based on Sinc and Analytic Functions. Springer Series in Computational Mathematics, Vol. 20, Springer-Verlag, New York.
  • 8: GergΕ‘ Nemes
    β–ΊAs of September 20, 2021, Nemes performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 25 Zeta and Related Functions. …
    9: Wolter Groenevelt
    β–ΊGroenevelt’s research interests is in special functions and orthogonal polynomials and their relations with representation theory and interacting particle systems. β–ΊAs of September 20, 2022, Groenevelt performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 18 Orthogonal Polynomials. …
    10: 33.24 Tables
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  • Abramowitz and Stegun (1964, Chapter 14) tabulates F 0 ⁑ ( Ξ· , ρ ) , G 0 ⁑ ( Ξ· , ρ ) , F 0 ⁑ ( Ξ· , ρ ) , and G 0 ⁑ ( Ξ· , ρ ) for Ξ· = 0.5 ⁒ ( .5 ) ⁒ 20 and ρ = 1 ⁒ ( 1 ) ⁒ 20 , 5S; C 0 ⁑ ( Ξ· ) for Ξ· = 0 ⁒ ( .05 ) ⁒ 3 , 6S.

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  • Curtis (1964a) tabulates P β„“ ⁑ ( Ο΅ , r ) , Q β„“ ⁑ ( Ο΅ , r ) 33.1), and related functions for β„“ = 0 , 1 , 2 and Ο΅ = 2 ⁒ ( .2 ) ⁒ 2 , with x = 0 ⁒ ( .1 ) ⁒ 4 for Ο΅ < 0 and x = 0 ⁒ ( .1 ) ⁒ 10 for Ο΅ 0 ; 6D.