# contiguous relations (Heine)

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##### 1: 15.5 Derivatives and Contiguous Functions
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###### §15.5(ii) Contiguous Functions
โบThe six functions $F\left(a\pm 1,b;c;z\right)$, $F\left(a,b\pm 1;c;z\right)$, $F\left(a,b;c\pm 1;z\right)$ are said to be contiguous to $F\left(a,b;c;z\right)$. … โบAn equivalent equation to the hypergeometric differential equation (15.10.1) is …Further contiguous relations include: …
##### 2: 16.3 Derivatives and Contiguous Functions
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###### §16.3(ii) Contiguous Functions
โบTwo generalized hypergeometric functions ${{}_{p}F_{q}}\left(\mathbf{a};\mathbf{b};z\right)$ are (generalized) contiguous if they have the same pair of values of $p$ and $q$, and corresponding parameters differ by integers. If $p\leq q+1$, then any $q+2$ distinct contiguous functions are linearly related. Examples are provided by the following recurrence relations: …
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##### 4: 16.4 Argument Unity
โบThe 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. โบContiguous balanced series have parameters shifted by an integer but still balanced. … … โบSee Bailey (1964, §§4.3(7) and 7.6(1)) for the transformation formulas and Wilson (1978) for contiguous relations. …
##### 5: 18.9 Recurrence Relations and Derivatives
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###### §18.9(i) Recurrence Relations
โบwith initial values $p_{0}(x)=1$ and $p_{1}(x)=A_{0}x+B_{0}$. … โบ โบ
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##### 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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• W. Gautschi (1959b) Some elementary inequalities relating to the gamma and incomplete gamma function. J. Math. Phys. 38 (1), pp. 77–81.
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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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• W. Gautschi (2002a) Computation of Bessel and Airy functions and of related Gaussian quadrature formulae. BIT 42 (1), pp. 110–118.
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##### 9: Bibliography S
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• 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.
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• J. L. Schonfelder (1978) Chebyshev expansions for the error and related functions. Math. Comp. 32 (144), pp. 1232–1240.
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• 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.
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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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• 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.
• ##### 10: Bibliography V
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• G. Valent (1986) An integral transform involving Heun functions and a related eigenvalue problem. SIAM J. Math. Anal. 17 (3), pp. 688–703.
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• H. Volkmer (1999) Expansions in products of Heine-Stieltjes polynomials. Constr. Approx. 15 (4), pp. 467–480.
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• H. Volkmer (1982) Integral relations for Lamé functions. SIAM J. Math. Anal. 13 (6), pp. 978–987.