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1: 16.3 Derivatives and Contiguous Functions
§16.3 Derivatives and Contiguous Functions
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§16.3(ii) Contiguous Functions
โ–บTwo generalized hypergeometric functions F q p โก ( ๐š ; ๐› ; 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. …
2: 15.5 Derivatives and Contiguous Functions
§15.5 Derivatives and Contiguous Functions
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§15.5(ii) Contiguous Functions
โ–บThe six functions F โก ( a ± 1 , b ; c ; z ) , F โก ( a , b ± 1 ; c ; z ) , F โก ( a , b ; c ± 1 ; z ) are said to be contiguous to F โก ( a , b ; c ; z ) . … โ–บBy repeated applications of (15.5.11)–(15.5.18) any function F โก ( a + k , b + โ„“ ; c + m ; z ) , in which k , โ„“ , m are integers, can be expressed as a linear combination of F โก ( a , b ; c ; z ) and any one of its contiguous functions, with coefficients that are rational functions of a , b , c , and z . โ–บAn equivalent equation to the hypergeometric differential equation (15.10.1) is …
3: 16.4 Argument Unity
โ–บโ–บโ–บSee Bailey (1964, §§4.3(7) and 7.6(1)) for the transformation formulas and Wilson (1978) for contiguous relations. …
4: 17.6 ฯ• 1 2 Function
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Heine’s Contiguous Relations
5: 10.21 Zeros
โ–บThe positive zeros of any two real distinct cylinder functions of the same order are interlaced, as are the positive zeros of any real cylinder function ๐’ž ฮฝ โก ( z ) and the contiguous function ๐’ž ฮฝ + 1 โก ( z ) . … …
6: Bibliography S
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  • J. Segura (2008) Interlacing of the zeros of contiguous hypergeometric functions. Numer. Algorithms 49 (1-4), pp. 387–407.
  • 7: 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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  • A. Gervois and H. Navelet (1985a) Integrals of three Bessel functions and Legendre functions. I. J. Math. Phys. 26 (4), pp. 633–644.
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  • A. Gervois and H. Navelet (1985b) Integrals of three Bessel functions and Legendre functions. II. J. Math. Phys. 26 (4), pp. 645–655.
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  • S. Goldstein (1927) Mathieu functions. Trans. Camb. Philos. Soc. 23, pp. 303–336.
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  • A. J. Guttmann and T. Prellberg (1993) Staircase polygons, elliptic integrals, Heun functions, and lattice Green functions. Phys. Rev. E 47 (4), pp. R2233–R2236.
  • 8: 18.2 General Orthogonal Polynomials
    โ–บBetween the systems { p n โก ( x ) } and { q n โข ( x ) } there are the contiguous relations … โ–บFor OP’s { p n โก ( x ) } on โ„ with respect to an even weight function w โก ( x ) we have … โ–บUnder further conditions on the weight function there is an equiconvergence theorem, see Szegล‘ (1975, Theorem 13.1.2). … โ–บ
    Monotonic Weight Functions
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