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21: Bibliography

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Multiple series Rogers-Ramanujan type identities. Pacific J. Math. 114 (2), pp. 267–283.

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External Links:

ISSN 0030-8730, MathReview (David M. Bressoud), ZentralBlatt

Referenced by:

§17.12.

Encodings:

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22: 14.7 Integer Degree and Order

… ►For $n=0,1,2,\mathrm{\dots }$, … ►

§14.7(ii) Rodrigues-Type Formulas

►For $m=0,1,2,\mathrm{\dots }$, and $n=0,1,2,\mathrm{\dots }$, … ►When $x>1$, (14.7.19) applies with . … ►Lastly, when $x>1$, (14.7.21) applies with $|h|>x+{\left(x^2-1\right)}^{1/2}$. …

23: 16.4 Argument Unity

… ►There are two types of three-term identities for ${}_3{}^{}F_2^{}$’s. …

24: 7.24 Approximations

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Luke (1969b, pp. 323–324) covers $\frac{1}{2}\sqrt{\pi }\mathrm{erf}x$ and ${\mathrm{e}}^{x^2}F\left(x\right)$ for $-3\le x\le 3$ (the Chebyshev coefficients are given to 20D); $\sqrt{\pi }x{\mathrm{e}}^{x^2}\mathrm{erfc}x$ and $2xF\left(x\right)$ for $x\ge 3$ (the Chebyshev coefficients are given to 20D and 15D, respectively). Coefficients for the Fresnel integrals are given on pp. 328–330 (20D).

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Schonfelder (1978) gives coefficients of Chebyshev expansions for $x^{-1}\mathrm{erf}x$ on $0\le x\le 2$, for $x{\mathrm{e}}^{x^2}\mathrm{erfc}x$ on $[2,\mathrm{\infty })$, and for ${\mathrm{e}}^{x^2}\mathrm{erfc}x$ on $[0,\mathrm{\infty })$ (30D).

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Shepherd and Laframboise (1981) gives coefficients of Chebyshev series for $(1+2x){\mathrm{e}}^{x^2}\mathrm{erfc}x$ on $(0,\mathrm{\infty })$ (22D).

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§7.24(iii) Padé-Type Expansions

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Luke (1969b, vol. 2, pp. 422–435) gives main diagonal Padé approximations for $F\left(z\right)$, $\mathrm{erf}z$, $\mathrm{erfc}z$, $C\left(z\right)$, and $S\left(z\right)$; approximate errors are given for a selection of $z$-values.

25: Bibliography H

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An Euler-Maclaurin-type formula involving conjugate Bernoulli polynomials and an application to $\zeta (2m+1)$ . Commun. Appl. Anal. 1 (1), pp. 15–32.

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External Links:

ISSN 1083-2564, MathReview (Pavel Guerzhoy), ZentralBlatt

Referenced by:

§24.16(iii).

Encodings:

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A Boole-type Formula involving Conjugate Euler Polynomials. In Charlemagne and his Heritage. 1200 Years of Civilization and Science in Europe, Vol. 2 (Aachen, 1995), P.L. Butzer, H. Th. Jongen, and W. Oberschelp (Eds.), pp. 361–375.

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External Links:

ISBN 2-503-50674-7, MathReview (Bruce C. Berndt), ZentralBlatt

Referenced by:

§24.16(iii).

Encodings:

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26: 29.12 Definitions

… ►In the fourth column the variable $z$ and modulus $k$ of the Jacobian elliptic functions have been suppressed, and $P({\mathrm{sn}}^2)$ denotes a polynomial of degree $n$ in ${\mathrm{sn}}^2(z,k)$ (different for each type). …

27: 18.2 General Orthogonal Polynomials

… ►If these $x_k$ satisfy then Szegő type asymptotics outside $[-1,1]$ can be given for the corresponding OP’s, see Simon (2011, Corollary 3.7.2 and following). …

28: 19.7 Connection Formulas

… ►For two further transformations of this type see Erdélyi et al. (1953b, p. 316). … ►There are three relations connecting $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$ and $\mathrm{\Pi }(\varphi ,{\omega }^2,k)$, where ${\omega }^2$ is a rational function of ${\alpha }^2$. If $k^2$ and ${\alpha }^2$ are real, then both integrals are circular cases or both are hyperbolic cases (see §19.2(ii)). … ►Let $c={\csc }^2\varphi \ne {\alpha }^2$. …Since $k^2\le c$ we have ${\alpha }^2{\omega }^2\le c$; hence ${\alpha }^2>c$ implies . …

29: Bibliography O

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Uniform Airy-type expansions of integrals. SIAM J. Math. Anal. 25 (2), pp. 304–321.

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External Links:

ISSN 0036-1410, Document, MathReview, ZentralBlatt

Referenced by:

§2.4(v).

Encodings:

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30: Bibliography M

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Euler-type transformations for the generalized hypergeometric function ${}_{r+2}{}^{}F_{r+1}^{}(x)$ . Z. Angew. Math. Phys. 62 (1), pp. 31–45.

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External Links:

ISSN 0044-2275, Document, FullText, MathReview (Mridula Garg), ZentralBlatt

Referenced by:

§16.6, §16.6.

Encodings:

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On a Kummer-type transformation for the generalized hypergeometric function ${}_2{}^{}F_2^{}$ . J. Comput. Appl. Math. 157 (2), pp. 507–509.

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External Links:

ISSN 0377-0427, Document, MathReview Entry, ZentralBlatt

Referenced by:

§16.6.

Encodings:

BibTeX

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