generalized hypergeometric differential equation

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31: Bibliography C

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Generalized hypergeometric functions and the evaluation of scalar one-loop integrals in Feynman diagrams. J. Comput. Appl. Math. 115 (1-2), pp. 93–99.

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External Links:

ISSN 0377-0427, Document, MathReview (Lorenzo L. Salcedo), ZentralBlatt

Referenced by:

§16.24(ii).

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Elementary Differential Equations. Clarendon Press, Oxford.

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External Links:

ISBN 0-19-853139-7, MathReview, ZentralBlatt

Referenced by:

§16.12.

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Algorithm 352: Characteristic values and associated solutions of Mathieu’s differential equation. Comm. ACM 12 (7), pp. 399–407.

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Notes:

Includes double-precision Fortran program, maximum accuracy 13D.

External Links:

ISSN 0001-0782, Document

Referenced by:

§28.36(ii), §28.36(iii).

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Theory of ordinary differential equations. McGraw-Hill Book Company, Inc., New York-Toronto-London.

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External Links:

MathReview (M. Zlámal)

Referenced by:

§1.18(ix), §1.18(x).

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Exact elliptic compactons in generalized Korteweg-de Vries equations. Complexity 11 (6), pp. 30–34.

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ISSN 1076-2787, Document, MathReview

Referenced by:

§19.6(i).

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32: Bibliography K

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On the zeros of some generalized hypergeometric functions. J. Math. Anal. Appl. 243 (2), pp. 249–260.

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External Links:

ISSN 0022-247X, Document, MathReview (M. K. Jain), ZentralBlatt

Referenced by:

§16.9.

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Approximation Formulae for Generalized Hypergeometric Functions for Large Values of the Parameters. J. B. Wolters, Groningen.

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External Links:

MathReview (L. J. Slater), ZentralBlatt

Referenced by:

§16.11(iii).

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On differential equations for Sobolev-type Laguerre polynomials. Trans. Amer. Math. Soc. 350 (1), pp. 347–393.

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ISSN 0002-9947, FullText, MathReview (Hans W. Volkmer), ZentralBlatt

Referenced by:

§18.36(ii).

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Fractional integral and generalized Stieltjes transforms for hypergeometric functions as transmutation operators. SIGMA Symmetry Integrability Geom. Methods Appl. 11, pp. Paper 074, 22.

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External Links:

ISSN 1815-0659, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

§15.14, §15.14.

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Some special cases of the generalized hypergeometric function ${}_{q+1}{}^{}F_q^{}$ . J. Comput. Appl. Math. 78 (1), pp. 79–95.

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External Links:

ISSN 0377-0427, Document, MathReview (Boro Döring), ZentralBlatt

Referenced by:

§16.7.

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33: Bibliography

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Asymptotics of solutions of the generalized sine-Gordon equation, the third Painlevé equation and the d’Alembert equation. Dokl. Akad. Nauk SSSR 280 (2), pp. 265–268 (Russian).

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Notes:

English translation: Soviet Math. Dokl. 31(1985), no. 1, pp. 45–47

External Links:

ISSN 0002-3264, MathReview (M. S. Agranovich), ZentralBlatt

Referenced by:

§32.11(iv).

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On the zeros of confluent hypergeometric functions. III. Characterization by means of nonlinear equations. Lett. Nuovo Cimento (2) 29 (11), pp. 353–358.

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ISSN 0024-1318, MathReview (James M. Horner)

Referenced by:

§13.9(i).

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Generalized elliptic integrals and modular equations. Pacific J. Math. 192 (1), pp. 1–37.

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ISSN 0030-8730, FullText, MathReview (Bruce C. Berndt), ZentralBlatt

Referenced by:

§19.35(i).

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Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.

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External Links:

ISBN 0-89871-412-5, MathReview (Michael Hanke), ZentralBlatt

Referenced by:

§3.7(i).

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Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials. Mem. Amer. Math. Soc. 54 (319), pp. iv+55.

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External Links:

ISSN 0065-9266, MathReview (Mourad E. H. Ismail), ZentralBlatt

Referenced by:

§18.21(ii), §18.22(ii), (18.28.24), §18.28(ii), Ch.18, Profile Richard A. Askey.

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34: 14.19 Toroidal (or Ring) Functions

… ►This form of the differential equation arises when Laplace’s equation is transformed into toroidal coordinates $(\eta ,\theta ,\varphi )$, which are related to Cartesian coordinates $(x,y,z)$ by … ►

§14.19(ii) Hypergeometric Representations

►With $𝐅$ as in §14.3 and $\xi >0$, ►

14.19.2 $$P_{\nu -\frac{1}{2}}^{\mu }\left(\cosh \xi \right)=\frac{\mathrm{\Gamma }\left(\frac{1}{2}-\mu \right)}{{\pi }^{1/2}{\left(1-{\mathrm{e}}^{-2\xi }\right)}^{\mu }{\mathrm{e}}^{(\nu +(1/2))\xi }}𝐅(\frac{1}{2}-\mu ,\frac{1}{2}+\nu -\mu ;1-2\mu ;1-{\mathrm{e}}^{-2\xi }),$$ $\mu \ne \frac{1}{2},\frac{3}{2},\frac{5}{2},\mathrm{\dots }$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $\mu$: general order, $\nu$: general degree and $\xi >0$: variable

Referenced by:

§14.19(ii), Erratum (V1.0.1) for Equation (14.19.2)

Permalink:

http://dlmf.nist.gov/14.19.E2

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Errata (effective with 1.0.1):

Originally the argument to $𝐅$ was incorrect and the condition on $\mu$ too weak:

$$P_{\nu -\frac{1}{2}}^{\mu }\left(\cosh \xi \right)=\frac{\mathrm{\Gamma }\left(1-2\mu \right)2^{2\mu }}{\mathrm{\Gamma }\left(1-\mu \right){\left(1-{\mathrm{e}}^{-2\xi }\right)}^{\mu }{\mathrm{e}}^{(\nu +(1/2))\xi }}𝐅(\frac{1}{2}-\mu ,\frac{1}{2}+\nu -\mu ;1-2\mu ;{\mathrm{e}}^{-2\xi }),$$ $\mu \ne \frac{1}{2}$.

Also, the factor multiplying $𝐅$ was rewritten to clarify the poles, which determine the correct condition on $\mu$.

Reported 2010-11-02 by Alvaro Valenzuela

See also:

Annotations for §14.19(ii), §14.19 and Ch.14

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14.19.3 $${𝑸}_{\nu -\frac{1}{2}}^{\mu }\left(\cosh \xi \right)=\frac{{\pi }^{1/2}{\left(1-{\mathrm{e}}^{-2\xi }\right)}^{\mu }}{{\mathrm{e}}^{(\nu +(1/2))\xi }}𝐅(\mu +\frac{1}{2},\nu +\mu +\frac{1}{2};\nu +1;{\mathrm{e}}^{-2\xi }).$$

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Symbols:

${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $\mu$: general order, $\nu$: general degree and $\xi >0$: variable

Permalink:

http://dlmf.nist.gov/14.19.E3

Encodings:

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See also:

Annotations for §14.19(ii), §14.19 and Ch.14

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35: 18.27 $q$-Hahn Class

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§18.27(i) Introduction

… ►For the notation of $q$-hypergeometric functions see §§17.2 and 17.4(i). … ►They are defined by their $q$-hypergeometric representations, followed by their orthogonality properties. … ►

§18.27(ii) $q$-Hahn Polynomials

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Discrete $q$-Hermite II

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36: 18.28 Askey–Wilson Class

… ►For the notation of $q$-hypergeometric functions see §§17.2 and 17.4(i). ►

§18.28(ii) Askey–Wilson Polynomials

… ►More generally, … ►

$q$-Difference Equation

… ►Genest et al. (2016) showed that these polynomials coincide with the nonsymmetric Wilson polynomials in Groenevelt (2007).

37: 13.8 Asymptotic Approximations for Large Parameters

… ►To obtain approximations for $M(a,b,z)$ and $U(a,b,z)$ that hold as $b\to \mathrm{\infty }$, with $a>\frac{1}{2}-b$ and $z>0$ combine (13.14.4), (13.14.5) with §13.20(i). … ►For asymptotic approximations to $M(a,b,x)$ and $U(a,b,x)$ as $a\to -\mathrm{\infty }$ that hold uniformly with respect to $x\in (0,\mathrm{\infty })$ and bounded positive values of $(b-1)/\left|a\right|$, combine (13.14.4), (13.14.5) with §§13.21(ii), 13.21(iii). … ►

13.8.11 $$U(a,b,z)\sim 2{\left(z/a\right)}^{(1-b)/2}\frac{{\mathrm{e}}^{z/2}}{\mathrm{\Gamma }\left(a\right)}\left(K_{b-1}\left(2\sqrt{az}\right)\sum _{s=0}^{\mathrm{\infty }}\frac{p_s(z)}{a^s}+\sqrt{z/a}{K}_b\left(2\sqrt{az}\right)\sum _{s=0}^{\mathrm{\infty }}\frac{q_s(z)}{a^s}\right),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $U(a,b,z)$: Kummer confluent hypergeometric function, $\sim$: Poincaré asymptotic expansion, $\mathrm{e}$: base of natural logarithm, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $s$: nonnegative integer and $z$: complex variable

Referenced by:

§13.8(iii)

Permalink:

http://dlmf.nist.gov/13.8.E11

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Addition (effective with 1.0.11):

The whole paragraph containing this equation has been added.

See also:

Annotations for §13.8(iii), §13.8 and Ch.13

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13.8.16 $$(k+1)c_{k+1}(z)+\sum _{s=0}^k\left(\frac{b{B}_{s+1}}{(s+1)!}+\frac{z(s+1)B_{s+2}}{(s+2)!}\right)c_{k-s}(z)=0,$$ $k=0,1,2,\mathrm{\dots }$.

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Symbols:

$B_n$: Bernoulli numbers, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $(a,b)$: open interval, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $s$: nonnegative integer and $z$: complex variable

Referenced by:

(13.8.16), §13.8(iii)

Permalink:

http://dlmf.nist.gov/13.8.E16

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Addition (effective with 1.0.11):

The whole paragraph containing this equation has been added. In Temme (2015, Equation 10.3.28) the generating function $f(z,t)$ for $c_k(z)$ is given. It satisfies,

$$\frac{\partial f}{\partial t}=\left(b\left(\frac{1}{t}-\frac{1}{{\mathrm{e}}^t-1}\right)-z\left(\frac{1}{t^2}-\frac{{\mathrm{e}}^t}{{\left({\mathrm{e}}^t-1\right)}^2}\right)\right)f.$$

For (13.8.16) combine this differentialequation with $f(z,t)={\sum }_{k=0}^{\mathrm{\infty }}{c}_k(z)t^k$ and (24.4.1).

See also:

Annotations for §13.8(iii), §13.8 and Ch.13

… ►For generalizations in which $z$ is also allowed to be large see Temme and Veling (2022).

38: 8.19 Generalized Exponential Integral

§8.19 Generalized Exponential Integral

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§8.19(ii) Graphics

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§8.19(vi) Relation to Confluent Hypergeometric Function

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§8.19(ix) Inequalities

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§8.19(xi) Further Generalizations

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39: 19.23 Integral Representations

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19.23.6_5 $$R_G(x,y,z)=\frac{1}{4\pi }{\int }_0^{2\pi }{\int }_0^{\pi }{\left(x{\sin }^2\theta {\cos }^2\varphi +y{\sin }^2\theta {\sin }^2\varphi +z{\cos }^2\theta \right)}^{1/2}\sin \theta d\theta d\varphi ,$$

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Symbols:

$R_G(x,y,z)$: symmetric elliptic integral of second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function and $\varphi$: real or complex argument

Referenced by:

(19.16.3), (19.16.3), §19.16(i), §19.16(ii), §19.16(ii), §19.20(ii), §19.23, §19.23, §19.25(vi), Erratum (V1.1.0) for Rearrangement

Permalink:

http://dlmf.nist.gov/19.23.E6_5

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Rearrangement (effective with 1.1.0):

This equation, which was originally (19.16.3), was moved here.

See also:

Annotations for §19.23 and Ch.19

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19.23.8 $$R_{-a}(𝐛;𝐳)=\frac{2}{\mathrm{B}(b_1,b_2)}{\int }_0^{\pi /2}{(z_1{\cos }^2\theta +z_2{\sin }^2\theta )}^{-a}{(\cos \theta )}^{2b_1-1}{(\sin \theta )}^{2b_2-1}d\theta ,$$ $b_1,b_2>0$; $\mathrm{\Re }{z}_1,\mathrm{\Re }{z}_2>0$.

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Symbols:

$R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(𝐛;𝐳)$: multivariate hypergeometric function, $\mathrm{B}(a,b)$: beta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $\mathrm{\Re }$: real part and $\sin z$: sine function

Referenced by:

§19.2(iv), §19.23, §19.23, §19.23

Permalink:

http://dlmf.nist.gov/19.23.E8

Encodings:

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See also:

Annotations for §19.23 and Ch.19

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19.23.9 $$R_{-a}(𝐛;𝐳)=\frac{4\mathrm{\Gamma }\left(b_1+b_2+b_3\right)}{\mathrm{\Gamma }\left(b_1\right)\mathrm{\Gamma }\left(b_2\right)\mathrm{\Gamma }\left(b_3\right)}{\int }_0^{\pi /2}{\int }_0^{\pi /2}{\left(\sum _{j=1}^3{z}_j{l}_j^2\right)}^{-a}\prod _{j=1}^3{l}_j^{2b_j-1}\sin \theta d\theta d\varphi ,$$ $b_j>0$, $\mathrm{\Re }{z}_j>0$.

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Symbols:

$R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(𝐛;𝐳)$: multivariate hypergeometric function, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $\mathrm{\Re }$: real part, $\sin z$: sine function, $l$: nonnegative integer and $\varphi$: real or complex argument

Referenced by:

§19.23, §19.23

Permalink:

http://dlmf.nist.gov/19.23.E9

Encodings:

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See also:

Annotations for §19.23 and Ch.19

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19.23.10 $$R_{-a}(𝐛;𝐳)=\frac{1}{\mathrm{B}(a,a^{\prime })}{\int }_0^1{u}^{a-1}{(1-u)}^{a^{\prime }-1}\prod _{j=1}^n{(1-u+u{z}_j)}^{-b_j}du,$$ $a,a^{\prime }>0$; $a+a^{\prime }={\sum }_{j=1}^n{b}_j$; $z_j\in ℂ\setminus (-\mathrm{\infty },0]$.

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Symbols:

$R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(𝐛;𝐳)$: multivariate hypergeometric function, $\mathrm{B}(a,b)$: beta function, $ℂ$: complex plane, $dx$: differential of $x$ , $\in$: element of, $\int$: integral, $(a,b]$: half-closed interval, $\setminus$: set subtraction and $n$: nonnegative integer

Referenced by:

§19.19, §19.23, §19.23

Permalink:

http://dlmf.nist.gov/19.23.E10

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See also:

Annotations for §19.23 and Ch.19

►For generalizations of (19.23.6_5) and (19.23.8) see Carlson (1964, (6.2), (6.12), and (6.1)).

40: Bibliography D

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Unification of one-dimensional Fokker-Planck equations beyond hypergeometrics: Factorizer solution method and eigenvalue schemes. Phys. Rev. E (3) 57 (1), pp. 252–275.

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External Links:

ISSN 1063-651X, Document, MathReview (Victor I. Stepanov), ZentralBlatt

Referenced by:

§31.17(ii).

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Stability of Runge-Kutta Methods for Stiff Nonlinear Differential Equations. CWI Monographs, Vol. 2, North-Holland Publishing Co., Amsterdam.

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External Links:

ISBN 0-444-87634-0, MathReview (A. de Castro Brzezicki), ZentralBlatt

Referenced by:

§3.7(v).

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Error bounds for exponentially improved asymptotic solutions of ordinary differential equations having irregular singularities of rank one. Methods Appl. Anal. 3 (1), pp. 109–134.

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External Links:

ISSN 1073-2772, Pdf, MathReview (Roderick Wong), ZentralBlatt

Referenced by:

§2.11(v).

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Convergent expansions for solutions of linear ordinary differential equations having a simple turning point, with an application to Bessel functions. Stud. Appl. Math. 107 (3), pp. 293–323.

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External Links:

ISSN 0022-2526, Document, MathReview, ZentralBlatt

Referenced by:

§10.20(ii), §2.8(i).

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Olver’s error bound methods applied to linear ordinary differential equations having a simple turning point. Anal. Appl. (Singap.) 12 (4), pp. 385–402.

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External Links:

ISSN 0219-5305, Document, Link, MathReview (Thomas Mbah Acho), ZentralBlatt

Referenced by:

5th item, §14.32.

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