contiguous relations (Heine)

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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. Examples are provided by the following recurrence relations: …

3: 17.6 ${}_2{}^{}\varphi _1^{}$ Function

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Heine’s First Transformation

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Heine’s Second Tranformation

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Heine’s Third Transformation

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§17.6(iii) Contiguous Relations

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Heine’s Contiguous Relations

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4: 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. …

5: 18.9 Recurrence Relations and Derivatives

§18.9 Recurrence Relations and Derivatives

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§18.9(i) Recurrence Relations

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

… ►Further $n$-th derivative formulas relating two different Jacobi polynomials can be obtained from §15.5(i) by substitution of (18.5.7). … ►and the structure relation …

6: 18.11 Relations to Other Functions

§18.11 Relations to Other Functions

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Ultraspherical

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Laguerre

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Hermite

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§18.11(ii) Formulas of Mehler–Heine Type

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7: 18.2 General Orthogonal Polynomials

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§18.2(iii) Standardization and Related Constants

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§18.2(iv) Recurrence Relations

… ►the monic recurrence relations (18.2.8) and (18.2.10) take the form … ►are OP’s with orthogonality relation …Between the systems $\{p_n(x)\}$ and $\{q_n(x)\}$ there are the contiguous relations …

8: 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).

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Some elementary inequalities relating to the gamma and incomplete gamma function. J. Math. Phys. 38 (1), pp. 77–81.

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External Links:

MathReview (H. O. Pollak), ZentralBlatt

Referenced by:

§5.6(i), §6.8, §7.8, §8.10.

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Computational aspects of three-term recurrence relations. SIAM Rev. 9 (1), pp. 24–82.

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External Links:

Document, MathReview (E. Frank), ZentralBlatt

Referenced by:

§10.74(iv), §14.32, §3.10(iii), §3.6(iii).

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Questions of Numerical Condition Related to Polynomials. In Studies in Numerical Analysis, G. H. Golub (Ed.), pp. 140–177.

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External Links:

MathReview Entry, ZentralBlatt

Referenced by:

§3.8(vi).

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Computation of Bessel and Airy functions and of related Gaussian quadrature formulae. BIT 42 (1), pp. 110–118.

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External Links:

ISSN 0006-3835, Document, MathReview (Khélifa Trimèche), ZentralBlatt

Referenced by:

§10.74(iii), §9.17(iii).

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9: 14.28 Sums

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§14.28(ii) Heine’s Formula

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10: 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).

Encodings:

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

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External Links:

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

Referenced by:

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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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.

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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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