basic solutions

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11: Bibliography

… ►

G. E. Andrews (1966a) On basic hypergeometric series, mock theta functions, and partitions. II. Quart. J. Math. Oxford Ser. (2) 17, pp. 132–143.

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External Links:: Document, MathReview (B. R. Bhonsle), ZentralBlatt
Referenced by:: §17.6(ii).
Encodings:: BibTeX

… ►

G. E. Andrews (1972) Summations and transformations for basic Appell series. J. London Math. Soc. (2) 4, pp. 618–622.

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External Links:: MathReview (R. N. Jain), ZentralBlatt
Referenced by:: §17.11.
Encodings:: BibTeX

►

G. E. Andrews (1974) Applications of basic hypergeometric functions. SIAM Rev. 16 (4), pp. 441–484.

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External Links:: Document, MathReview (L. J. Slater), ZentralBlatt
Referenced by:: §17.16, Ch.17.
Encodings:: BibTeX

… ►

R. Askey and J. Wilson (1985) 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.
Encodings:: BibTeX

… ►

R. Askey (1980) Some basic hypergeometric extensions of integrals of Selberg and Andrews. SIAM J. Math. Anal. 11 (6), pp. 938–951.

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External Links:: ISSN 0036-1410, Document, MathReview (W. A. Al-Salam), ZentralBlatt
Referenced by:: §17.13, §17.13.
Encodings:: BibTeX

…

12: Bibliography G

… ►

F. Gao and V. J. W. Guo (2013) Contiguous relations and summation and transformation formulae for basic hypergeometric series. J. Difference Equ. Appl. 19 (12), pp. 2029–2042.

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External Links:: ISSN 1023-6198, Document, FullText, MathReview (Thomas Britz), ZentralBlatt
Referenced by:: §17.7(iii).
Encodings:: BibTeX

… ►

L. Gårding (1947) 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.

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External Links:: FullText, MathReview (E. T. Copson), ZentralBlatt
Referenced by:: §35.2, §35.3(ii).
Encodings:: BibTeX

… ►

G. Gasper and M. Rahman (1990) Basic Hypergeometric Series. Encyclopedia of Mathematics and its Applications, Vol. 35, Cambridge University Press, Cambridge.

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Notes:: A revised and updated edition, with three new chapters, was published in 2004 by Cambridge University Press (Encyclopedia of Mathematics and its Applications, vol. 96).
External Links:: ISBN 0-521-35049-2, MathReview (Shaun Cooper), ZentralBlatt
Referenced by:: (17.9.3), Erratum (V1.0.23) for Equation (17.9.3).
Encodings:: BibTeX

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G. Gasper and M. Rahman (2004) Basic Hypergeometric Series. Second edition, Encyclopedia of Mathematics and its Applications, Vol. 96, Cambridge University Press, Cambridge.

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Notes:: With a foreword by Richard Askey
External Links:: ISBN 0-521-83357-4, MathReview (Shaun Cooper), ZentralBlatt
Referenced by:: §17.1, §17.1, §17.10, (17.11.1), (17.11.2), (17.11.3), §17.13, §17.16, (17.4.5), (17.4.6), (17.4.7), (17.4.8), §17.4(i), §17.5, §17.6(i), §17.6(ii), §17.6(iii), §17.6(iv), §17.6(v), §17.7(i), §17.7(ii), §17.7(iii), §17.8, (17.9.3), §17.9(i), §17.9(ii), §17.9(iii), Ch.17, §18.27(i), §18.27(i), §18.27(ii), §18.27(iii), §18.27(iv), (18.28.24), (18.28.25), §18.28(i), §18.28(ii), §18.28(v), §18.28(viii), §26.19, Erratum (V1.0.10) for Section 17.1, Erratum (V1.0.23) for Equation (17.9.3).
Encodings:: BibTeX

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R. A. Gustafson (1987) Multilateral summation theorems for ordinary and basic hypergeometric series in

{\rm U}(n)

. SIAM J. Math. Anal. 18 (6), pp. 1576–1596.

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External Links:: ISSN 0036-1410, Document, MathReview, ZentralBlatt
Referenced by:: §17.15.
Encodings:: BibTeX

…

13: 29.3 Definitions and Basic Properties

§29.3 Definitions and Basic Properties

… ►For each pair of values of

\nu

and

k

there are four infinite unbounded sets of real eigenvalues

h

for which equation (29.2.1) has even or odd solutions with periods

2K

4K

. … ►If

\nu

is a nonnegative integer and

2p\leq\nu

, then the continued fraction on the right-hand side of (29.3.10) terminates, and (29.3.10) has only the solutions (29.3.9) with

2m\leq\nu

. If

\nu

is a nonnegative integer and

2p>\nu

, then (29.3.10) has only the solutions (29.3.9) with

2m>\nu

. …

14: 28.20 Definitions and Basic Properties

§28.20 Definitions and Basic Properties

… ►

§28.20(ii) Solutions $\operatorname{Ce}_{\nu}$ , $\operatorname{Se}_{\nu}$ , $\operatorname{Me}_{\nu}$ , $\operatorname{Fe}_{n}$ , $\operatorname{Ge}_{n}$

… ►

§28.20(iii) Solutions ${\operatorname{M}^{(j)}_{\nu}}$

… ►It follows that (28.20.1) has independent and unique solutions

{\operatorname{M}^{(3)}_{\nu}}\left(z,h\right)

{\operatorname{M}^{(4)}_{\nu}}\left(z,h\right)

such that … ►

§28.20(v) Solutions $\operatorname{Ie}_{n}$ , $\operatorname{Io}_{n}$ , $\operatorname{Ke}_{n}$ , $\operatorname{Ko}_{n}$

…

15: 27.13 Functions

… ►The basic problem is that of expressing a given positive integer

n

as a sum of integers from some prescribed set

S

whose members are primes, squares, cubes, or other special integers. … ►Every even integer $n>4$ is the sum of two odd primes. In this case,

S(n)

is the number of solutions of the equation

n=p+q

, where

p

and

q

are odd primes. … ►has nonnegative integer solutions for all

n\geq 1

. …Similarly,

G\left(k\right)

denotes the smallest

m

for which (27.13.1) has nonnegative integer solutions for all sufficiently large

n

. … ►For a given integer

k\geq 2

the function

r_{k}\left(n\right)

is defined as the number of solutions of the equation …

16: Bibliography K

… ►

K. Kajiwara and Y. Ohta (1996) Determinant structure of the rational solutions for the Painlevé II equation. J. Math. Phys. 37 (9), pp. 4693–4704.

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External Links:: ISSN 0022-2488, Document, MathReview (Andrei A. Kapaev), ZentralBlatt
Referenced by:: §32.8(ii).
Encodings:: BibTeX

►

K. Kajiwara and Y. Ohta (1998) Determinant structure of the rational solutions for the Painlevé IV equation. J. Phys. A 31 (10), pp. 2431–2446.

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External Links:: ISSN 0305-4470, Document, MathReview (Andrei A. Kapaev), ZentralBlatt
Referenced by:: §32.8(iv).
Encodings:: BibTeX

… ►

V. Kourganoff (1952) Basic Methods in Transfer Problems. Radiative Equilibrium and Neutron Diffusion. Oxford University Press, Oxford.

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Notes:: With the collaboration of I. W. Busbridge
External Links:: MathReview (S. Chandrasekhar), ZentralBlatt
Referenced by:: §8.19(x), §8.24(iii).
Encodings:: BibTeX

… ►

C. Krattenthaler (1993) HYP and HYPQ. Mathematica packages for the manipulation of binomial sums and hypergeometric series respectively

q

-binomial sums and basic hypergeometric series. Séminaire Lotharingien de Combinatoire 30, pp. 61–76.

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Notes:: Also published in J. Symbolic Comput. (1995), v. 20, no. 5–6, pp. 737–744.
External Links:: Code, Document, MathReview (G. Gasper), ZentralBlatt
Referenced by:: 1st item.
Encodings:: BibTeX

… ►

Y. A. Kravtsov (1964) Asymptotic solution of Maxwell’s equations near caustics. Izv. Vuz. Radiofiz. 7, pp. 1049–1056.

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Referenced by:: §36.14(i).
Encodings:: BibTeX

…

17: 28.12 Definitions and Basic Properties

§28.12 Definitions and Basic Properties

… ►In consequence, for the Floquet solutions

w(z)

the factor

e^{\pi\mathrm{i}\nu}

in (28.2.14) is no longer

\pm 1

. … ►The Floquet solution with respect to

\nu

is denoted by

\operatorname{me}_{\nu}\left(z,q\right)

. …The other eigenfunction is

\operatorname{me}_{\nu}\left(-z,q\right)

, a Floquet solution with respect to

-\nu

with

a=\lambda_{\nu}\left(q\right)

. …They have the following pseudoperiodic and orthogonality properties: …

18: Bibliography M

… ►

R. S. Maier (2007) The 192 solutions of the Heun equation. Math. Comp. 76 (258), pp. 811–843.

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External Links:: ISSN 0025-5718, Document, MathReview (Wadim Zudilin), ZentralBlatt (Masaaki Yoshida)
Referenced by:: §31.3(iii).
Encodings:: BibTeX

… ►

M. Mazzocco (2001a) Rational solutions of the Painlevé VI equation. J. Phys. A 34 (11), pp. 2281–2294.

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External Links:: ISSN 0305-4470, Document, MathReview (Shun Shimomura), ZentralBlatt
Referenced by:: §32.8(vi).
Encodings:: BibTeX

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M. Mazzocco (2001b) Picard and Chazy solutions to the Painlevé VI equation. Math. Ann. 321 (1), pp. 157–195.

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External Links:: ISSN 0025-5831, Document, MathReview, ZentralBlatt
Referenced by:: §32.9(iii).
Encodings:: BibTeX

… ►

S. C. Milne (1985c) A new symmetry related to

\mathit{SU}(n)

for classical basic hypergeometric series. Adv. in Math. 57 (1), pp. 71–90.

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External Links:: ISSN 0001-8708, Document, MathReview (George E. Andrews), ZentralBlatt
Referenced by:: §17.15.
Encodings:: BibTeX

… ►

S. C. Milne (1997) Balanced

\sideset{{}_{3}}{{}_{2}}{\mathop{\Theta}}

summation theorems for

U(n)

basic hypergeometric series. Adv. Math. 131 (1), pp. 93–187.

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External Links:: ISSN 0001-8708, Document, MathReview (Jiang Zeng), ZentralBlatt
Referenced by:: §17.15.
Encodings:: BibTeX

…

19: 10.47 Definitions and Basic Properties

§10.47 Definitions and Basic Properties

… ►

§10.47(ii) Standard Solutions

… ►

§10.47(iii) Numerically Satisfactory Pairs of Solutions

►For (10.47.1) numerically satisfactory pairs of solutions are given by Table 10.2.1 with the symbols

J

Y

H

, and

\nu

replaced by

\mathsf{j}

\mathsf{y}

\mathsf{h}

, and

n

, respectively. ►For (10.47.2) numerically satisfactory pairs of solutions are

{\mathsf{i}^{(1)}_{n}}\left(z\right)

and

\mathsf{k}_{n}\left(z\right)

in the right half of the

z

-plane, and

{\mathsf{i}^{(1)}_{n}}\left(z\right)

and

\mathsf{k}_{n}\left(-z\right)

in the left half of the

z

-plane. …

20: 13.2 Definitions and Basic Properties

§13.2 Definitions and Basic Properties

… ►

Standard Solutions

►The first two standard solutions are: … ►

§13.2(v) Numerically Satisfactory Solutions

… ► …