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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

►The six functions $F(a\pm 1,b;c;z)$, $F(a,b\pm 1;c;z)$, $F(a,b;c\pm 1;z)$ are said to be contiguous to $F(a,b;c;z)$. … ►An 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

►Two generalized hypergeometric functions ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ are (generalized) contiguous if they have the same pair of values of $p$ and $q$, and corresponding parameters differ by integers. If $p\le q+1$, then any $q+2$ distinct contiguous functions are linearly related. …

3: 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. ►Contiguous balanced series have parameters shifted by an integer but still balanced. … … ►See Bailey (1964, §§4.3(7) and 7.6(1)) for the transformation formulas and Wilson (1978) for contiguous relations. …

4: 18.9 Recurrence Relations and Derivatives

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§18.9(ii) Contiguous Relations in the Parameters and the Degree

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5: 17.6 ${}_2{}^{}\varphi _1^{}$ Function

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§17.6(iii) Contiguous Relations

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Heine’s Contiguous Relations

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6: Bibliography G

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Contiguous relations and summation and transformation formulae for basic hypergeometric series. J. Difference Equ. Appl. 19 (12), pp. 2029–2042.

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External Links:

ISSN 1023-6198, Document, FullText, MathReview (Thomas Britz), ZentralBlatt

Referenced by:

§17.7(iii).

Encodings:

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7: 18.2 General Orthogonal Polynomials

… ►Between the systems $\{p_n(x)\}$ and $\{q_n(x)\}$ there are the contiguous relations …

8: Bibliography S

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Interlacing of the zeros of contiguous hypergeometric functions. Numer. Algorithms 49 (1-4), pp. 387–407.

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External Links:

ISSN 1017-1398, Document, FullText, MathReview, ZentralBlatt

Referenced by:

§13.9(i), §13.9(i), §13.9(ii), §13.9(ii), §15.13, §15.13.

Encodings:

BibTeX

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9: 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 ${𝒞}_{\nu }\left(z\right)$ and the contiguous function ${𝒞}_{\nu +1}\left(z\right)$. … …