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31: Bibliography L

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Some Differential Equations Satisfied by Hypergeometric Functions. In Approximation and Computation (West Lafayette, IN, 1993), Internat. Ser. Numer. Math., Vol. 119, pp. 371–381.

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External Links:

MathReview (M. E. Muldoon), ZentralBlatt

Referenced by:

§16.8(ii).

Encodings:

BibTeX

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32: Bibliography O

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Exponentially-improved asymptotic solutions of ordinary differential equations I: The confluent hypergeometric function. SIAM J. Math. Anal. 24 (3), pp. 756–767.

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External Links:

ISSN 0036-1410, Document, MathReview (Hussain S. Nur), ZentralBlatt

Referenced by:

§10.17(v), §13.7(iii), (9.7.18), (9.7.19), (9.7.20), (9.7.21), (9.7.22), (9.7.23), §9.7(v).

Encodings:

BibTeX

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33: Errata

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17.5.1 $${}_0{}^{}\varphi _0^{}(-;-;q,z)=\sum _{n=0}^{\mathrm{\infty }}\frac{{(-1)}^n{q}^{\left(\genfrac{}{}{0.0pt}{}{n}{2}\right)}{z}^n}{{(q;q)}_n}={(z;q)}_{\mathrm{\infty }}$$

The constraint originally given by is not necessary and has been removed.

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17.5.5 $${}_1{}^{}\varphi _1^{}(\genfrac{}{}{0pt}{}{a}{c};q,c/a)=\frac{{(c/a;q)}_{\mathrm{\infty }}}{{(c;q)}_{\mathrm{\infty }}}$$

The constraint originally given by is not necessary and has been removed.

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17.7.1 $${}_2{}^{}\varphi _2^{}(\genfrac{}{}{0pt}{}{a,q/a}{-q,b};q,-b)=\frac{{(ab,bq/a;q^2)}_{\mathrm{\infty }}}{{(b;q)}_{\mathrm{\infty }}}$$

The constraint originally given by is not necessary and has been removed.

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13.2.7 $$U(-m,b,z)={(-1)}^m{\left(b\right)}_{m}M(-m,b,z)={(-1)}^m\sum _{s=0}^m\left(\genfrac{}{}{0.0pt}{}{m}{s}\right){\left(b+s\right)}_{m-s}{(-z)}^s$$

The equality $U(-m,b,z)={(-1)}^m{\left(b\right)}_{m}M(-m,b,z)$ has been added to the original equation to express an explicit connection between the two standard solutions of Kummer’s equation. Note also that the notation $a=-n$ has been changed to $a=-m$.

Reported 2015-02-10 by Adri Olde Daalhuis.

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13.18.7 $$W_{-\frac{1}{4},\pm \frac{1}{4}}\left(z^2\right)={\mathrm{e}}^{\frac{1}{2}{z}^2}\sqrt{\pi z}\mathrm{erfc}\left(z\right)$$

Originally the left-hand side was given correctly as $W_{-\frac{1}{4},-\frac{1}{4}}\left(z^2\right)$; the equation is true also for $W_{-\frac{1}{4},+\frac{1}{4}}\left(z^2\right)$.

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34: 16.25 Methods of Computation

§16.25 Methods of Computation

►Methods for computing the functions of the present chapter include power series, asymptotic expansions, integral representations, differential equations, and recurrence relations. They are similar to those described for confluent hypergeometric functions, and hypergeometric functions in §§13.29 and 15.19. There is, however, an added feature in the numerical solution of differential equations and difference equations (recurrence relations). …

35: Howard S. Cohl

… ►Cohl has published papers in orthogonal polynomials and special functions, and is particularly interested in fundamental solutions of linear partial differential equations on Riemannian manifolds, associated Legendre functions, generalized and basic hypergeometric functions, eigenfunction expansions of fundamental solutions in separable coordinate systems for linear partial differential equations, orthogonal polynomial generating function and generalized expansions, and $q$-series. …

36: 13.18 Relations to Other Functions

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13.18.7 $$W_{-\frac{1}{4},\pm \frac{1}{4}}\left(z^2\right)={\mathrm{e}}^{\frac{1}{2}{z}^2}\sqrt{\pi z}\mathrm{erfc}\left(z\right).$$

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Symbols:

$W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $\mathrm{e}$: base of natural logarithm and $z$: complex variable

Referenced by:

Erratum (V1.0.3) for Equation (13.18.7)

Permalink:

http://dlmf.nist.gov/13.18.E7

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.3):

Originally the left-hand side was given correctly as $W_{-\frac{1}{4},-\frac{1}{4}}\left(z^2\right)$; the equation is true also for $W_{-\frac{1}{4},+\frac{1}{4}}\left(z^2\right)$.

See also:

Annotations for §13.18(ii), §13.18 and Ch.13

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37: Bibliography Z

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Doron Zeilberger’s Maple Packages and Programs Department of Mathematics, Rutgers University, New Jersey.

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Notes:

Includes hypergeometric and $q$-hypergeometric summation, solution of difference equations (for example, by Petkovšek’s algorithm), combinatorial algorithms, and (package LUC) evaluation of zonal polynomials.

External Links:

Home Page

Referenced by:

§ ‣ Software Cross Index, § ‣ Software Cross Index, L. Lapointe and L. Vinet (1996).

Encodings:

BibTeX

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38: Frank W. J. Olver

… ►He is particularly known for his extensive work in the study of the asymptotic solution of differential equations, i. …, the behavior of solutions as the independent variable, or some parameter, tends to infinity, and in the study of the particular solutions of differential equations known as special functions (e. …, Bessel functions, hypergeometric functions, Legendre functions). …

39: 18.38 Mathematical Applications

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18.38.3 $$\sum _{m=0}^n{P}_m^{(\alpha ,0)}\left(x\right)=\frac{{\left(\alpha +2\right)}_n}{n!}{}_3{}^{}F_2^{}(\genfrac{}{}{0pt}{}{-n,n+\alpha +2,\frac{1}{2}(\alpha +1)}{\alpha +1,\frac{1}{2}(\alpha +3)};\frac{1}{2}(1-x))\ge 0,$$ $x\ge -1$, $\alpha \ge -2$, $n=0,1,\mathrm{\dots }$,

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Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Proved:

Askey and Gasper (1976, Theorem 3, p.717)(proved)

Referenced by:

§18.14(iv), §18.18(viii), Erratum (V1.2.0) for Equation (18.38.3)

Permalink:

http://dlmf.nist.gov/18.38.E3

Encodings:

pMML, png, TeX

Modification (effective with 1.2.0):

This equation was updated to include the value of the sum in terms of the ${}_3{}^{}F_2^{}$ function. Also the constraint was previously $\alpha >-1$.

See also:

Annotations for §18.38(ii), §18.38(ii), §18.38 and Ch.18

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40: 17.4 Basic Hypergeometric Functions

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17.4.2 $$\underset{q\to 1-}{lim}{}_{r+1}{}^{}\varphi _s^{}(\genfrac{}{}{0pt}{}{q^{a_0},q^{a_1},\mathrm{\dots },q^{a_r}}{q^{b_1},\mathrm{\dots },q^{b_s}};q,{(q-1)}^{s-r}z)={}_{r+1}{}^{}F_s^{}(\genfrac{}{}{0pt}{}{a_0,a_1,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};z).$$

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Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, ${}_{r+1}{}^{}\varphi _s^{}(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_{r+1}{}^{}\varphi _s^{}(\genfrac{}{}{0pt}{}{a_0,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: basic hypergeometric (or $q$-hypergeometric) function, $q$: complex base, $r$: nonnegative integer, $s$: nonnegative integer and $z$: complex variable

Source:

Koekoek et al. (2010, p. 15)

Referenced by:

Erratum (V1.1.11) for Equation (17.4.2)

Permalink:

http://dlmf.nist.gov/17.4.E2

Encodings:

pMML, png, TeX

Generalization (effective with 1.1.11):

This limit relation which was previously accurate for ${}_{r+1}{}^{}\varphi _r^{}$ was updated to be accurate for the more general ${}_{r+1}{}^{}\varphi _s^{}$.

See also:

Annotations for §17.4(i), §17.4 and Ch.17

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