contiguous relations

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1: 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. …

2: 15.5 Derivatives and Contiguous Functions

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§15.5(ii) Contiguous Functions

… ►Further contiguous relations include: …

3: 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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4: 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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5: 18.2 General Orthogonal Polynomials

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

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

Referenced by:

§17.7(iii).

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7: 16.3 Derivatives and Contiguous Functions

… ►If $p\le q+1$, then any $q+2$ distinct contiguous functions are linearly related. …

8: Bibliography S

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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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ISSN 0036-1410, Document, MathReview (Ernest G. Kalnins), ZentralBlatt

Referenced by:

§28.32(i).

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Chebyshev expansions for the error and related functions. Math. Comp. 32 (144), pp. 1232–1240.

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FullText, Document, MathReview (L. Fox), ZentralBlatt

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3rd item.

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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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ISSN 0025-5718, Document, MathReview (Boro Döring), ZentralBlatt

Referenced by:

§10.74(vi).

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

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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.

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Structure of avoided crossings for eigenvalues related to equations of Heun’s class. J. Phys. A 30 (2), pp. 673–687.

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ISSN 0305-4470, Document, MathReview (Éric Delabaere), ZentralBlatt

Referenced by:

§31.13.

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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 ${𝒞}_{\nu +1}\left(z\right)$. … … ►The functions ${\rho }_{\nu }(t)$ and ${\sigma }_{\nu }(t)$ are related to the inverses of the phase functions ${\theta }_{\nu }\left(x\right)$ and ${\varphi }_{\nu }\left(x\right)$ defined in §10.18(i): if $\nu \ge 0$, then … ►

${\varphi }_{\nu }\left(y_{\nu ,m}^{\prime }\right)=m\pi ,$ $m=1,2,\mathrm{\dots }$.

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