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1: 16.22 Asymptotic Expansions

… ►Asymptotic expansions of $G_{p,q}^{m,n}(z;𝐚;𝐛)$ for large $z$ are given in Luke (1969a, §§5.7 and 5.10) and Luke (1975, §5.9). …

2: 13.8 Asymptotic Approximations for Large Parameters

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§13.8(i) Large $|b|$, Fixed $a$ and $z$

… ►When the foregoing results are combined with Kummer’s transformation (13.2.39), an approximation is obtained for the case when $|b|$ is large, and $|b-a|$ and $|z|$ are bounded. ►

§13.8(ii) Large $b$ and $z$, Fixed $a$ and $b/z$

… ►For other asymptotic expansions for large $b$ and $z$ see López and Pagola (2010). … ►

§13.8(iv) Large $a$ and $b$

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3: 15.12 Asymptotic Approximations

… ►For large $b$ and $c$ with $c>b+1$ see López and Pagola (2011). … ►If , then as $\lambda \to \mathrm{\infty }$ with $|\mathrm{ph}\lambda |\le \pi -\delta$, … ►If , then as $\lambda \to \mathrm{\infty }$ with $|\mathrm{ph}\lambda |\le \pi -\delta$, … ►If , then as $\lambda \to \mathrm{\infty }$ with $|\mathrm{ph}\lambda |\le \frac{1}{2}\pi -\delta$, … ►For other extensions, see Wagner (1986), Temme (2003) and Temme (2015, Chapters 12 and 28).

4: 28.16 Asymptotic Expansions for Large $q$

§28.16 Asymptotic Expansions for Large $q$

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5: Bibliography L

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The confluent hypergeometric functions $M(a,b;z)$ and $U(a,b;z)$ for large $b$ and $z$ . J. Comput. Appl. Math. 233 (6), pp. 1570–1576.

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External Links:

ISSN 0377-0427, Document, FullText, MathReview, ZentralBlatt

Referenced by:

§13.8(ii), §13.8(ii).

Encodings:

BibTeX

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Large degree asymptotics of generalized Bernoulli and Euler polynomials. J. Math. Anal. Appl. 363 (1), pp. 197–208.

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External Links:

ISSN 0022-247X, Document, Link, MathReview (Richard B. Paris), ZentralBlatt

Referenced by:

§24.16(i), §24.16(i).

Encodings:

BibTeX

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6: Bibliography Y

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Computation of Kummer functions $U(a,b,x)$ for large argument $x$ by using the $\tau$-method. Trans. Inform. Process. Soc. Japan 36 (10), pp. 2335–2342 (Japanese).

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Notes:

Japanese with English summary. Double-precision Fortran.

External Links:

ISSN 0387-5806, FullText, MathReview

Referenced by:

§13.32(ii).

Encodings:

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7: Bibliography H

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Lamé polynomials of large order. SIAM J. Math. Anal. 8 (5), pp. 800–842.

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External Links:

Document, MathReview (F. M. Arscott), ZentralBlatt

Referenced by:

§29.16.

Encodings:

BibTeX

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On the resurgence properties of the uniform asymptotic expansion of Bessel functions of large order. Proc. Roy. Soc. London Ser. A 455, pp. 3917–3930.

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External Links:

ISSN 1364-5021, Document, MathReview (A. D. Wood), ZentralBlatt

Referenced by:

§10.20(ii).

Encodings:

BibTeX

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8: Bibliography O

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Uniform asymptotic expansions for hypergeometric functions with large parameters. I. Analysis and Applications (Singapore) 1 (1), pp. 111–120.

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External Links:

ISSN 0219-5305, Document, MathReview, ZentralBlatt

Referenced by:

§15.12(iii).

Encodings:

BibTeX

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Uniform asymptotic expansions for hypergeometric functions with large parameters. II. Analysis and Applications (Singapore) 1 (1), pp. 121–128.

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External Links:

ISSN 0219-5305, Document, MathReview, ZentralBlatt

Referenced by:

§15.12(iii).

Encodings:

BibTeX

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Uniform asymptotic expansions for hypergeometric functions with large parameters. III. Analysis and Applications (Singapore) 8 (2), pp. 199–210.

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External Links:

ISSN 0219-5305, Document, MathReview (Javier Segura), ZentralBlatt

Referenced by:

§15.12(iii).

Encodings:

BibTeX

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Uniform asymptotic expansions for Weber parabolic cylinder functions of large orders. J. Res. Nat. Bur. Standards Sect. B 63B, pp. 131–169.

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External Links:

MathReview (N. D. Kazarinoff), ZentralBlatt

Referenced by:

§12.10(i), §12.10(ii), §12.10(iii), §12.10(iv), §12.10(v), §12.10(vii), §12.10(vii), §12.11(iii), §12.11(iii), §12.14(ix), §12.14(ix), §12.14(ix), §12.14(xi), Ch.12, §18.16(v).

Encodings:

BibTeX

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9: 8.18 Asymptotic Expansions of $I_x(a,b)$

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Large $a$, Fixed $b$

… ►For asymptotic expansions for large values of $a$ and/or $b$ of the $x$-solution of the equation …

10: 13.9 Zeros

… ►For fixed $a,b\in ℂ$ the large $z$-zeros of $M(a,b,z)$ satisfy … ►For fixed $b$ and $z$ in $ℂ$ the large $a$-zeros of $M(a,b,z)$ are given by … ►For fixed $b$ and $z$ in $ℂ$ the large $a$-zeros of $U(a,b,z)$ are given by …