relation to hypergeometric differential equation

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11: Bibliography M

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Painlevé-type differential equations for the recurrence coefficients of semi-classical orthogonal polynomials. J. Comput. Appl. Math. 57 (1-2), pp. 215–237.

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External Links:

ISSN 0377-0427, Document, MathReview (Leonid B. Golinskiĭ), ZentralBlatt

Referenced by:

§32.15.

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On reducing the Heun equation to the hypergeometric equation. J. Differential Equations 213 (1), pp. 171–203.

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ISSN 0022-0396, Document, MathReview (Wolfgang Bühring), ZentralBlatt

Referenced by:

§31.7(i).

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Quadratic relations for confluent hypergeometric functions. Tohoku Math. J. (2) 52 (4), pp. 489–513.

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ISSN 0040-8735, Document, MathReview (Takeshi Sasaki), ZentralBlatt

Referenced by:

§13.12.

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An Introduction to the Fractional Calculus and Fractional Differential Equations. A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York.

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External Links:

ISBN 0-471-58884-9, MathReview (K. C. Gupta), ZentralBlatt

Referenced by:

§1.15(vi), Erratum (V1.0.14) for Subsections 1.15(vi), 1.15(vii), 2.6(iii).

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A new symmetry related to $\mathrm{𝑆𝑈}(n)$ for classical basic hypergeometric series. Adv. in Math. 57 (1), pp. 71–90.

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ISSN 0001-8708, Document, MathReview (George E. Andrews), ZentralBlatt

Referenced by:

§17.15.

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12: Errata

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

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

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Section: 15.9(v) Complete Elliptic Integrals. Equations: (11.11.9_5), (11.11.13_5), Intermediate equality in (15.4.27) which relates to $F(a,a;a+1;\frac{1}{2})$, (15.4.34), (19.5.4_1), (19.5.4_2) and (19.5.4_3).

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The Kronecker delta symbols have been moved furthest to the right, as is common convention for orthogonality relations.

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Following a suggestion from James McTavish on 2017-04-06, the recurrence relation $u_k=\frac{(6k-5)(6k-3)(6k-1)}{(2k-1)216k}{u}_{k-1}$ was added to Equation (9.7.2).

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A number of additions and changes have been made to the metadata to reflect new and changed references as well as to how some equations have been derived.

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13: 18.34 Bessel Polynomials

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§18.34(i) Definitions and Recurrence Relation

►For the confluent hypergeometric function ${}_1{}^{}F_1^{}$ and the generalized hypergeometric function ${}_2{}^{}F_0^{}$, the Laguerre polynomial $L_n^{(\alpha )}$ and the Whittaker function $W_{\kappa ,\mu }$ see §16.2(ii), §16.2(iv), (18.5.12), and (13.14.3), respectively. … ►where ${𝗄}_n$ is a modified spherical Bessel function (10.49.9), and … … ►

§18.34(iii) Other Properties

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14: 13.29 Methods of Computation

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§13.29(ii) Differential Equations

►A comprehensive and powerful approach is to integrate the differential equations (13.2.1) and (13.14.1) by direct numerical methods. … ►The recurrence relations in §§13.3(i) and 13.15(i) can be used to compute the confluent hypergeometric functions in an efficient way. … ►normalizing relation … ►normalizing relation …

15: Bibliography B

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Ordinary differential equations. Fourth edition, John Wiley & Sons, Inc., New York.

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External Links:

ISBN 0-471-86003-4, MathReview (Michal Čverčko)

Referenced by:

§1.13(viii), §1.13(viii), §1.18(iv).

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Tables Relating to the Radial Mathieu Functions. Vol. 1: Functions of the First Kind. U.S. Government Printing Office, Washington, D.C..

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External Links:

MathReview (C. J. Bouwkamp)

Referenced by:

1st item.

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Bessel functions and modular relations of higher type and hyperbolic differential equations. Comm. Sém. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.] 1952 (Tome Supplementaire), pp. 12–20.

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External Links:

MathReview (S. C. van Veen), ZentralBlatt

Referenced by:

§35.5(i).

Encodings:

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Uniform asymptotic solutions of a class of second-order linear differential equations having a turning point and a regular singularity, with an application to Legendre functions. SIAM J. Math. Anal. 17 (2), pp. 422–450.

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External Links:

ISSN 0036-1410, Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§14.15(iv), §14.15(iv), §14.26, §2.8(vi).

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Numerical Methods for Ordinary Differential Equations. John Wiley & Sons Ltd., Chichester.

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External Links:

ISBN 0-471-96758-0, MathReview (Ian Gladwell), ZentralBlatt

Referenced by:

§32.17.

Encodings:

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16: 33.2 Definitions and Basic Properties

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§33.2(i) Coulomb Wave Equation

… ►This differential equation has a regular singularity at $\rho =0$ with indices $\mathrm{\ell }+1$ and $-\mathrm{\ell }$, and an irregular singularity of rank 1 at $\rho =\mathrm{\infty }$ (§§2.7(i), 2.7(ii)). … ►The function $F_{\mathrm{\ell }}(\eta ,\rho )$ is recessive (§2.7(iii)) at $\rho =0$, and is defined by …where $M_{\kappa ,\mu }\left(z\right)$ and $M(a,b,z)$ are defined in §§13.14(i) and 13.2(i), and … ►The functions $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$ are defined by …

17: 33.23 Methods of Computation

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§33.23(i) Methods for the Confluent Hypergeometric Functions

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§33.23(iii) Integration of Defining Differential Equations

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§33.23(iv) Recurrence Relations

►In a similar manner to §33.23(iii) the recurrence relations of §§33.4 or 33.17 can be used for a range of values of the integer $\mathrm{\ell }$, provided that the recurrence is carried out in a stable direction (§3.6). … ►Noble (2004) obtains double-precision accuracy for $W_{-\eta ,\mu }\left(2\rho \right)$ for a wide range of parameters using a combination of recurrence techniques, power-series expansions, and numerical quadrature; compare (33.2.7). …

18: 17.13 Integrals

§17.13 Integrals

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17.13.1 $${\int }_{-c}^d\frac{{(-qx/c;q)}_{\mathrm{\infty }}{(qx/d;q)}_{\mathrm{\infty }}}{{(-ax/c;q)}_{\mathrm{\infty }}{(bx/d;q)}_{\mathrm{\infty }}}{d}_{q}x=\frac{(1-q){(q;q)}_{\mathrm{\infty }}{(ab;q)}_{\mathrm{\infty }}cd{(-c/d;q)}_{\mathrm{\infty }}{(-d/c;q)}_{\mathrm{\infty }}}{{(a;q)}_{\mathrm{\infty }}{(b;q)}_{\mathrm{\infty }}(c+d){(-bc/d;q)}_{\mathrm{\infty }}{(-ad/c;q)}_{\mathrm{\infty }}},$$

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Symbols:

$\int$: integral, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $d_{q}x$: $q$-differential, $q$: complex base and $x$: real variable

Permalink:

http://dlmf.nist.gov/17.13.E1

Encodings:

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See also:

Annotations for §17.13 and Ch.17

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Ramanujan’s Integrals

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17.13.3 $${\int }_0^{\mathrm{\infty }}{t}^{\alpha -1}\frac{{(-t{q}^{\alpha +\beta };q)}_{\mathrm{\infty }}}{{(-t;q)}_{\mathrm{\infty }}}dt=\frac{\mathrm{\Gamma }\left(\alpha \right)\mathrm{\Gamma }\left(1-\alpha \right){\mathrm{\Gamma }}_q\left(\beta \right)}{{\mathrm{\Gamma }}_q\left(1-\alpha \right){\mathrm{\Gamma }}_q\left(\alpha +\beta \right)},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\int$: integral, ${\mathrm{\Gamma }}_q\left(z\right)$: $q$-gamma function, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial) and $q$: complex base

Referenced by:

Erratum (V1.0.1) for Equation (17.13.3)

Permalink:

http://dlmf.nist.gov/17.13.E3

Encodings:

pMML, png, TeX

Errata (effective with 1.0.1):

Originally the differential was identified incorrectly as a $q$-differential; the correct differential is $dt$.

See also:

Annotations for §17.13, §17.13 and Ch.17

… ►Askey (1980) conjectured extensions of the foregoing integrals that are closely related to Macdonald (1982). …

19: 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

Encodings:

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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:

pMML, png, TeX

See also:

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

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20: 28.8 Asymptotic Expansions for Large $q$

… ►For recurrence relations for the coefficients in these expansions see Frenkel and Portugal (2001, §3). … ►For recurrence relations for the coefficients in these expansions see Frenkel and Portugal (2001, §4 and §5). … ►Barrett (1981) supplies asymptotic approximations for numerically satisfactory pairs of solutions of both Mathieu’s equation (28.2.1) and the modified Mathieu equation (28.20.1). …It is stated that corresponding uniform approximations can be obtained for other solutions, including the eigensolutions, of the differential equations by application of the results, but these approximations are not included. … ►For related results see Langer (1934) and Sharples (1967, 1971). …