generalized hypergeometric differential equation

SearchAdvancedHelp

(0.011 seconds)

21—30 of 55 matching pages

21: 19.16 Definitions

… ►

§19.16(ii) $R_{-a}(𝐛;𝐳)$

►All elliptic integrals of the form (19.2.3) and many multiple integrals, including (19.23.6) and (19.23.6_5), are special cases of a multivariate hypergeometric function … ►For generalizations and further information, especially representation of the $R$-function as a Dirichlet average, see Carlson (1977b). ►

§19.16(iii) Various Cases of $R_{-a}(𝐛;𝐳)$

…

22: 2.6 Distributional Methods

… ►Although divergent, these integrals may be interpreted in a generalized sense. … ►To each function in this equation, we shall assign a tempered distribution (i. … ►Corresponding results for the generalized Stieltjes transform … ►These equations again hold only in the sense of distributions. … ►For rigorous derivations of these results and also order estimates for ${\delta }_n(x)$, see Wong (1979) and Wong (1989, Chapter 6).

23: Errata

… ►

In Equation (1.13.4), the determinant form of the two-argument Wronskian

1.13.4

was added as an equality. In ¶Wronskian (in §1.13(i)), immediately below Equation (1.13.4), a sentence was added indicating that in general the $n$-argument Wronskian is given by $𝒲\left\{w_1(z),\mathrm{\dots },w_n(z)\right\}=det\left[w_k^{(j-1)}(z)\right]$, where $1\le j,k\le n$. Immediately below Equation (1.13.4), a sentence was added giving the definition of the $n$-argument Wronskian. It is explained just above (1.13.5) that this equation is often referred to as Abel’s identity. Immediately below Equation (1.13.5), a sentence was added explaining how it generalizes for $n$th-order differential equations. A reference to Ince (1926, §5.2) was added.

… ►

These equations have been generalized to include the additional cases of $\partial J_{-\nu }\left(z\right)/\partial \nu$, $\partial I_{-\nu }\left(z\right)/\partial \nu$, respectively.

… ►

There were clarifications made in the conditions on the parameter $a$ in $U(a,b,z)$ of those equations.

… ►

The Wronskian was generalized to include both associated Legendre and Ferrers functions.

… ►

has been generalized to cover an additional case.

…

24: Bibliography W

… ►

Caustics: Mathematical curves generated by light shined through rippled plastic. Scientific American 249, pp. 146–153.

ⓘ

Referenced by:

§36.14(ii).

Encodings:

BibTeX

… ►

Asymptotic Expansions for Ordinary Differential Equations. Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney.

ⓘ

Notes:

Reprinted by Krieger, New York, 1976 and Dover, New York, 1987.

External Links:

MathReview (N. D. Kazarinoff), ZentralBlatt

Referenced by:

Ch.2, Ch.9.

Encodings:

BibTeX

… ►

Some transformations of generalized hypergeometric series. Proc. London Math. Soc. (2) 26 (2), pp. 257–272.

ⓘ

External Links:

Document, ZentralBlatt

Referenced by:

§16.6.

Encodings:

BibTeX

… ►

Asymptotic solutions of a fourth order differential equation. Stud. Appl. Math. 118 (2), pp. 133–152.

ⓘ

External Links:

ISSN 0022-2526, Document, MathReview Entry

Referenced by:

§2.8(vi).

Encodings:

BibTeX

… ►

The asymptotic expansion of the generalized hypergeometric function. Proc. London Math. Soc. (2) 46, pp. 389–408.

ⓘ

External Links:

ISSN 0024-6115, Document, MathReview (M. C. Gray), ZentralBlatt

Referenced by:

§16.11(ii).

Encodings:

BibTeX

…

25: 18.17 Integrals

… ►Formulas (18.17.12) and (18.17.13) are fractional generalizations of the differentiation formulas given in (Erdélyi et al., 1953b, §10.9(15)). … ►For the confluent hypergeometric function ${}_1{}^{}F_1^{}$ see (16.2.1) and Chapter 13. … ►This generalizes (18.17.34). For the hypergeometric function ${}_2{}^{}F_1^{}$ see §§15.1 and 15.2(i). … ►For the generalized hypergeometric function ${}_2{}^{}F_2^{}$ see (16.2.1). …

26: 33.23 Methods of Computation

… ►

§33.23(i) Methods for the Confluent Hypergeometric Functions

… ►

§33.23(iii) Integration of Defining Differential Equations

►When numerical values of the Coulomb functions are available for some radii, their values for other radii may be obtained by direct numerical integration of equations (33.2.1) or (33.14.1), provided that the integration is carried out in a stable direction (§3.7). … ►Curtis (1964a, §10) describes the use of series, radial integration, and other methods to generate the tables listed in §33.24. … ►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). …

27: 18.35 Pollaczek Polynomials

… ►

§18.35(i) Definition and Hypergeometric Representation

… ►we have the explicit representations …For type 1 take $\lambda =\frac{1}{2}$ and for Gauss’ hypergeometric function $F$ see (15.2.1). … ►For type 3 orthogonality (18.35.5) generalizes to …For Gauss’ hypergeometric function $F$ see (15.2.1). …

28: Bibliography S

… ►

Generalized Hypergeometric Functions. Cambridge University Press, Cambridge.

ⓘ

External Links:

ISBN 9780521090612, MathReview (S. C. van Veen), ZentralBlatt

Referenced by:

Ch.15, §16.2(iii), §16.4(ii), §16.4(iii), §16.4(v), Ch.16, §17.1, §17.4(i), Ch.17.

Encodings:

BibTeX

… ►

On the logarithmic solutions of the generalized hypergeometric equation when $p=q+1$ . Bull. Amer. Math. Soc. 45 (8), pp. 629–636.

ⓘ

External Links:

Document, MathReview (H. Bateman), ZentralBlatt

Referenced by:

§16.8(ii).

Encodings:

BibTeX

►

Relations among the fundamental solutions of the generalized hypergeometric equation when $p=q+1$. II. Logarithmic cases. Bull. Amer. Math. Soc. 45 (12), pp. 927–935.

ⓘ

External Links:

Document, MathReview (C. S. Meijer), ZentralBlatt

Referenced by:

§16.8(ii).

Encodings:

BibTeX

… ►

Computation of angular momentum coefficients using sets of generalized hypergeometric functions. Comput. Phys. Comm. 22 (2-3), pp. 297–302.

ⓘ

External Links:

Document

Referenced by:

§34.13.

Encodings:

BibTeX

… ►

Error bounds for asymptotic solutions of differential equations. II. The general case. J. Res. Nat. Bur. Standards Sect. B 70B, pp. 187–210.

ⓘ

External Links:

MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§2.7(ii).

Encodings:

BibTeX

…

29: 18.38 Mathematical Applications

… ►

Differential Equations: Spectral Methods

… ►This process has been generalized to spectral methods for solving partial differential equations. … ►

Quadrature “Extended” to Pseudo-Spectral (DVR) Representations of Operators in One and Many Dimensions

… ►The Askey–Gasper inequality …For the generalized hypergeometric function ${}_3{}^{}F_2^{}$ see (16.2.1). …

30: Bibliography G

… ►

The solution of Cauchy’s problem for two totally hyperbolic linear differential equations by means of Riesz integrals. Ann. of Math. (2) 48 (4), pp. 785–826.

ⓘ

External Links:

FullText, MathReview (E. T. Copson), ZentralBlatt

Referenced by:

§35.2, §35.3(ii).

Encodings:

BibTeX

… ►

Computing solutions of the modified Bessel differential equation for imaginary orders and positive arguments. ACM Trans. Math. Software 30 (2), pp. 145–158.

ⓘ

External Links:

ISSN 0098-3500, Document, MathReview, ZentralBlatt

Referenced by:

§10.74(viii).

Encodings:

BibTeX

… ►

Linear Differential Equations and Group Theory from Riemann to Poincaré. 2nd edition, Birkhäuser Boston Inc., Boston, MA.

ⓘ

External Links:

ISBN 0-8176-3837-7, MathReview, ZentralBlatt

Referenced by:

§15.17(v).

Encodings:

BibTeX

… ►

Special classes of solutions of Painlevé equations. Differ. Uravn. 18 (3), pp. 419–429 (Russian).

ⓘ

Notes:

In Russian. English translation: Differential Equations 18(3), pp. 317–326.

External Links:

ISSN 0374-0641, MathReview, ZentralBlatt

Referenced by:

§32.10(iv), §32.10(v), §32.8(iii), §32.8(v), §32.9(ii).

Encodings:

BibTeX

►

Theory of Painlevé’s equations. Differ. Uravn. 11 (11), pp. 373–376 (Russian).

ⓘ

Notes:

In Russian. English translation: Differential Equations 11(11), pp. 285–287

External Links:

MathReview (Z. Nehari), ZentralBlatt

Referenced by:

§32.7(iii), §32.7(vi).

Encodings:

BibTeX

…