generalized hypergeometric differential equation

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11: Bibliography B

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Products of generalized hypergeometric series. Proc. London Math. Soc. (2) 28 (2), pp. 242–254.

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External Links:

Document, ZentralBlatt

Referenced by:

§16.12.

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Transformations of generalized hypergeometric series. Proc. London Math. Soc. (2) 29 (2), pp. 495–502.

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External Links:

Document, ZentralBlatt

Referenced by:

§16.6.

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Generalized Hypergeometric Series. Stechert-Hafner, Inc., New York.

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External Links:

MathReview

Referenced by:

§16.4(iii), §16.4(iii), §16.4(iii), §16.4(iii), §16.4(iii), §17.1, §17.4(i), Ch.17.

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Generalized hypergeometric functions at unit argument. Proc. Amer. Math. Soc. 114 (1), pp. 145–153.

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ISSN 0002-9939, FullText, MathReview (R. K. Raina), ZentralBlatt

Referenced by:

§16.4(ii), §16.8(ii).

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The Numerical Analysis of Ordinary Differential Equations. Runge-Kutta and General Linear Methods. John Wiley & Sons Ltd., Chichester.

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ISBN 0-471-91046-5, MathReview (Peter Alfeld), ZentralBlatt

Referenced by:

§3.7(v).

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12: Bibliography O

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Hyperasymptotic solutions of second-order linear differential equations. I. Methods Appl. Anal. 2 (2), pp. 173–197.

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ISSN 1073-2772, FullText, MathReview (A. D. Wood), ZentralBlatt

Referenced by:

§10.17(v), §10.40(iv), §13.29(i), §13.7(iii), §2.11(v), §9.7(v).

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On the calculation of Stokes multipliers for linear differential equations of the second order. Methods Appl. Anal. 2 (3), pp. 348–367.

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ISSN 1073-2772, Pdf, MathReview (Archibald D. MacGillivray), ZentralBlatt

Referenced by:

§2.7(ii).

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Second-order differential equations with fractional transition points. Trans. Amer. Math. Soc. 226, pp. 227–241.

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External Links:

FullText, MathReview (Anthony Leung), ZentralBlatt

Referenced by:

§2.8(v).

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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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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).

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Applications of Lie Groups to Differential Equations. 2nd edition, Graduate Texts in Mathematics, Vol. 107, Springer-Verlag, New York.

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ISBN 0-387-94007-3; 0-387-95000-1, MathReview, ZentralBlatt

Referenced by:

§32.13(i).

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13: 15.17 Mathematical Applications

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§15.17(i) Differential Equations

… ►The logarithmic derivatives of some hypergeometric functions for which quadratic transformations exist (§15.8(iii)) are solutions of Painlevé equations. … … ►Harmonic analysis can be developed for the Jacobi transform either as a generalization of the Fourier-cosine transform (§1.14(ii)) or as a specialization of a group Fourier transform. … ►These monodromy groups are finite iff the solutions of Riemann’s differential equation are all algebraic. …

14: Bibliography N

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Confluent hypergeometric equations and related solvable potentials in quantum mechanics. J. Math. Phys. 41 (12), pp. 7964–7996.

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ISSN 0022-2488, Document, MathReview, ZentralBlatt

Referenced by:

§13.28(i).

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Uniform Asymptotic Approximations of Solutions of Second-order Linear Differential Equations, with a Coalescing Simple Turning Point and Simple Pole. Ph.D. Thesis, University of Maryland, College Park, MD.

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Referenced by:

§2.8(vi).

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A separating surface for the Painlevé differential equation $x^{\prime \prime }=x^2-t$ . J. Math. Anal. Appl. 193 (3), pp. 817–831.

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ISSN 0022-247X, Document, MathReview (W. S. Loud), ZentralBlatt

Referenced by:

§32.17.

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Tables of Generalized Airy Functions for the Asymptotic Solution of the Differential Equations $ϵ{(p{y}^{\prime })}^{\prime }+(q+ϵr)y=f$ . Pergamon Press, Oxford.

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Notes:

D. E. Brown, translator.

External Links:

MathReview, ZentralBlatt

Referenced by:

4th item.

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The asymptotic behavior of the general real solution of the third Painlevé equation. Dokl. Akad. Nauk SSSR 283 (5), pp. 1161–1165 (Russian).

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Notes:

In Russian. English translation: Sov. Math., Dokl. 30(1988), no. 8, pp. 666–668.

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ISSN 0002-3264, MathReview (Vojislav Marić), ZentralBlatt

Referenced by:

§32.11(iv).

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15: 35.7 Gaussian Hypergeometric Function of Matrix Argument

§35.7 Gaussian Hypergeometric Function of Matrix Argument

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§35.7(iii) Partial Differential Equations

… ►Subject to the conditions (a)–(c), the function $f(\mathbf{T})={}_{2}F_1(a,b;c;\mathbf{T})$ is the unique solution of each partial differential equation … ►Systems of partial differential equations for the ${}_{0}F_1$ (defined in §35.8) and ${}_{1}F_1$ functions of matrix argument can be obtained by applying (35.8.9) and (35.8.10) to (35.7.9). …

16: 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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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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A class of generalized hypergeometric summations. J. Comput. Appl. Math. 87 (1), pp. 79–85.

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External Links:

ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

§16.10.

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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.

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On the choice of standard solutions for a homogeneous linear differential equation of the second order. Quart. J. Mech. Appl. Math. 3 (2), pp. 225–235.

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External Links:

Document, MathReview (W. R. Wasow), ZentralBlatt

Referenced by:

§12.2(vi), §2.7(iv), §2.7(iv).

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17: 32.10 Special Function Solutions

… ►For certain combinations of the parameters, ${\text{P}}_{\text{II}}$–${\text{P}}_{\text{VI}}$ have particular solutions expressible in terms of the solution of a Riccati differential equation, which can be solved in terms of special functions defined in other chapters. … ►

§32.10(iv) Fourth Painlevé Equation

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§32.10(vi) Sixth Painlevé Equation

… ►where the fundamental periods $2{\varphi }_1$ and $2{\varphi }_2$ are linearly independent functions satisfying the hypergeometric equation … ►

18: 19.16 Definitions

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§19.16(ii) $R_{-a}(\mathbf{b};\mathbf{z})$

►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 $R_{-a}(\mathbf{b};\mathbf{z})$

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19: 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).

20: Errata

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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.

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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.

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

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

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has been generalized to cover an additional case.

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