contiguous functions

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1: 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. …

2: 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 $F(a,b;c;z)$. … ►By repeated applications of (15.5.11)–(15.5.18) any function $F(a+k,b+\mathrm{\ell };c+m;z)$, in which $k,\mathrm{\ell },m$ are integers, can be expressed as a linear combination of $F(a,b;c;z)$ and any one of its contiguous functions, with coefficients that are rational functions of $a,b,c$, and $z$. ►An equivalent equation to the hypergeometric differential equation (15.10.1) is …

3: 16.4 Argument Unity

… ► … ► … ►See Bailey (1964, §§4.3(7) and 7.6(1)) for the transformation formulas and Wilson (1978) for contiguous relations. …

4: 17.6 ${}_2{}^{}\varphi _1^{}$ Function

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Heine’s Contiguous Relations

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5: 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 ${𝒞}_{\nu +1}\left(z\right)$. … …

6: 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.

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7: 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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Integrals of three Bessel functions and Legendre functions. I. J. Math. Phys. 26 (4), pp. 633–644.

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ISSN 0022-2488, Document, MathReview (B. D. Agrawal), ZentralBlatt

Referenced by:

§10.22(iv), §10.43(iv).

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Integrals of three Bessel functions and Legendre functions. II. J. Math. Phys. 26 (4), pp. 645–655.

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External Links:

ISSN 0022-2488, Document, MathReview (B. D. Agrawal), ZentralBlatt

Referenced by:

§10.22(iv), §10.43(iv).

Encodings:

BibTeX

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Mathieu functions. Trans. Camb. Philos. Soc. 23, pp. 303–336.

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External Links:

ZentralBlatt

Referenced by:

§28.26(i), §28.26(i), §28.8(iii).

Encodings:

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Staircase polygons, elliptic integrals, Heun functions, and lattice Green functions. Phys. Rev. E 47 (4), pp. R2233–R2236.

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External Links:

Document

Referenced by:

§19.35(ii).

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8: 18.2 General Orthogonal Polynomials

… ►Between the systems $\{p_n(x)\}$ and $\{q_n(x)\}$ there are the contiguous relations … ►For OP’s $\{p_n(x)\}$ on $ℝ$ with respect to an even weight function $w(x)$ we have … ►Under further conditions on the weight function there is an equiconvergence theorem, see Szegő (1975, Theorem 13.1.2). … ►

Monotonic Weight Functions

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