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1: 18.9 Recurrence Relations and Derivatives
§18.9(ii) Contiguous Relations in the Parameters and the Degree
2: 16.4 Argument Unity
The 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. … See Bailey (1964, §§4.3(7) and 7.6(1)) for the transformation formulas and Wilson (1978) for contiguous relations. …
3: 17.6 ϕ 1 2 Function
§17.6(iii) Contiguous Relations
Heine’s Contiguous Relations
4: 15.5 Derivatives and Contiguous Functions
§15.5(ii) Contiguous Functions
Further contiguous relations include: …
5: Bibliography G
  • 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.
  • 6: 16.3 Derivatives and Contiguous Functions
    If p q + 1 , then any q + 2 distinct contiguous functions are linearly related. …
    7: Bibliography S
  • D. Schmidt and G. Wolf (1979) A method of generating integral relations by the simultaneous separability of generalized Schrödinger equations. SIAM J. Math. Anal. 10 (4), pp. 823–838.
  • J. L. Schonfelder (1978) Chebyshev expansions for the error and related functions. Math. Comp. 32 (144), pp. 1232–1240.
  • J. Segura (2001) Bounds on differences of adjacent zeros of Bessel functions and iterative relations between consecutive zeros. Math. Comp. 70 (235), pp. 1205–1220.
  • J. Segura (2008) Interlacing of the zeros of contiguous hypergeometric functions. Numer. Algorithms 49 (1-4), pp. 387–407.
  • S. Yu. Slavyanov and N. A. Veshev (1997) Structure of avoided crossings for eigenvalues related to equations of Heun’s class. J. Phys. A 30 (2), pp. 673–687.
  • 8: 10.21 Zeros
    The 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 𝒞 ν ( z ) and the contiguous function 𝒞 ν + 1 ( z ) . … … The functions ρ ν ( t ) and σ ν ( t ) are related to the inverses of the phase functions θ ν ( x ) and ϕ ν ( x ) defined in §10.18(i): if ν 0 , then …
    ϕ ν ( y ν , m ) = m π , m = 1 , 2 , .