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1: 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 (1994) Algorithm 726: ORTHPOL — a package of routines for generating orthogonal polynomials and Gauss-type quadrature rules. ACM Trans. Math. Software 20 (1), pp. 21–62.
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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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  • 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.
  • 2: 8 Incomplete Gamma and Related
    Functions
    Chapter 8 Incomplete Gamma and Related Functions
    3: 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: …
    4: 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: …
    5: 20 Theta Functions
    Chapter 20 Theta Functions
    6: 28 Mathieu Functions and Hill’s Equation
    Chapter 28 Mathieu Functions and Hill’s Equation
    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: 26.4 Lattice Paths: Multinomial Coefficients and Set Partitions
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    §26.4(i) Definitions
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    Table 26.4.1: Multinomials and partitions.
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    n m Ξ» M 1 M 2 M 3
    5 2 2 1 , 3 1 10 20 10
    5 3 1 2 , 3 1 20 20 10
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    §26.4(iii) Recurrence Relation