contiguous balanced series

SearchAdvancedHelp

(0.003 seconds)

1—10 of 277 matching pages

1: 16.4 Argument Unity

… ►

Pfaff–Saalschütz Balanced Sum

… ►Balanced ${}_4{}^{}F_3^{}\left(1\right)$ series have transformation formulas and three-term relations. … ► ►Contiguous balanced series have parameters shifted by an integer but still balanced. … …

2: 17.4 Basic Hypergeometric Functions

… ►

§17.4(iv) Classification

►The series (17.4.1) is said to be balanced or Saalschützian when it terminates, $r=s$, $z=q$, and … ►The series (17.4.1) is said to be k-balanced when $r=s$ and … ►The series (17.4.1) is said to be well-poised when $r=s$ and … ►The series (17.4.1) is said to be very-well-poised when $r=s$, (17.4.11) is satisfied, and …

3: 15.5 Derivatives and Contiguous Functions

§15.5 Derivatives and Contiguous Functions

… ►

§15.5(ii) Contiguous Functions

►The six functions $F(a\pm 1,b;c;z)$, $F(a,b\pm 1;c;z)$, $F(a,b;c\pm 1;z)$ are said to be contiguous to $F(a,b;c;z)$. … ►An equivalent equation to the hypergeometric differential equation (15.10.1) is …Further contiguous relations include: …

4: 16.3 Derivatives and Contiguous Functions

§16.3 Derivatives and Contiguous Functions

… ►

§16.3(ii) Contiguous Functions

►Two generalized hypergeometric functions ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ are (generalized) contiguous if they have the same pair of values of $p$ and $q$, and corresponding parameters differ by integers. If $p\le q+1$, then any $q+2$ distinct contiguous functions are linearly related. …

5: Bibliography G

… ►

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

ⓘ

External Links:

ISSN 1023-6198, Document, FullText, MathReview (Thomas Britz), ZentralBlatt

Referenced by:

§17.7(iii).

Encodings:

BibTeX

… ►

Basic Hypergeometric Series. Encyclopedia of Mathematics and its Applications, Vol. 35, Cambridge University Press, Cambridge.

ⓘ

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

►

Basic Hypergeometric Series. Second edition, Encyclopedia of Mathematics and its Applications, Vol. 96, Cambridge University Press, Cambridge.

ⓘ

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(ii), §18.27(iii), §18.27(iv), §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

… ►

Numerical Analysis of Spectral Methods: Theory and Applications. Society for Industrial and Applied Mathematics, Philadelphia, PA.

ⓘ

Notes:

CBMS-NSF Regional Conference Series in Applied Mathematics, No. 26

External Links:

MathReview (O. Widlund), ZentralBlatt

Referenced by:

§18.38(i).

Encodings:

BibTeX

… ►

Multilateral summation theorems for ordinary and basic hypergeometric series in $\mathrm{U}(n)$ . SIAM J. Math. Anal. 18 (6), pp. 1576–1596.

ⓘ

External Links:

ISSN 0036-1410, Document, MathReview, ZentralBlatt

Referenced by:

§17.15.

Encodings:

BibTeX

…

6: Bibliography S

… ►

On Lindelöf’s error bound for Stirling’s series. J. Reine Angew. Math. 404, pp. 135–139.

ⓘ

External Links:

ISSN 0075-4102, MathReview (Bruce C. Berndt), ZentralBlatt

Referenced by:

§5.11(ii).

Encodings:

BibTeX

… ►

An Introduction to the Operations with Series. 2nd edition, Chelsea Publishing Co., New York.

ⓘ

Notes:

This is a reprint, with corrections, of the first edition [University of Pennsylvania Press, Philadelphia, Pa., 1924].

External Links:

MathReview, ZentralBlatt

Referenced by:

§24.1, §24.6.

Encodings:

BibTeX

… ►

Asymptotic series of generalized Lambert $W$ function. ACM Commun. Comput. Algebra 47 (3), pp. 75–83.

ⓘ

External Links:

FullText

Referenced by:

§4.13, §4.13.

Encodings:

BibTeX

… ►

Interlacing of the zeros of contiguous hypergeometric functions. Numer. Algorithms 49 (1-4), pp. 387–407.

ⓘ

External Links:

ISSN 1017-1398, Document, FullText, MathReview, ZentralBlatt

Referenced by:

§13.9(i), §13.9(i), §13.9(ii), §13.9(ii), §15.13, §15.13.

Encodings:

BibTeX

… ►

An Introduction to Basic Fourier Series. Developments in Mathematics, Vol. 9, Kluwer Academic Publishers, Dordrecht.

ⓘ

External Links:

ISBN 1-4020-1221-7, MathReview (P. D. F. Ion), ZentralBlatt

Referenced by:

§17.3(i).

Encodings:

BibTeX

…

7: 17.16 Mathematical Applications

§17.16 Mathematical Applications

►Many special cases of $q$-series arise in the theory of partitions, a topic treated in §§27.14(i) and 26.9. In Lie algebras Lepowsky and Milne (1978) and Lepowsky and Wilson (1982) laid foundations for extensive interaction with $q$-series. …

8: 17.9 Further Transformations of ${}_{r+1}{}^{}\varphi _r^{}$ Functions

… ►

Transformations of ${}_3{}^{}\varphi _2^{}$-Series

… ►

Sears’ Balanced ${}_4{}^{}\varphi _3^{}$ Transformations

… ►provided that the series expansions of both $\varphi$’s terminate. ►

§17.9(iv) Bibasic Series

…

9: Bibliography M

… ►

Siegel’s modular forms and Dirichlet series. Lecture Notes in Mathematics, Vol. 216, Springer-Verlag, Berlin.

ⓘ

External Links:

Document, MathReview (K.-B. Gundlach), ZentralBlatt

Referenced by:

§35.4(ii).

Encodings:

BibTeX

… ►

On the convergence of the Chebyshev series for functions possessing a singularity in the range of representation. SIAM J. Numer. Anal. 3 (3), pp. 390–409.

ⓘ

External Links:

ISSN 1095-7170, Document, MathReview (Y. L. Luke), ZentralBlatt

Referenced by:

§3.11(ii).

Encodings:

BibTeX

… ►

Balanced ${}_3{}^{}\Theta _2^{}$ summation theorems for $U(n)$ basic hypergeometric series. Adv. Math. 131 (1), pp. 93–187.

ⓘ

External Links:

ISSN 0001-8708, Document, MathReview (Jiang Zeng), ZentralBlatt

Referenced by:

§17.15.

Encodings:

BibTeX

… ►

On the evaluation of some multiple series. J. London Math. Soc. (2) 33, pp. 368–371.

ⓘ

External Links:

Document, MathReview (V. F. Cowling), ZentralBlatt

Referenced by:

(25.6.5).

Encodings:

BibTeX

… ►

A new Stirling series as continued fraction. Numer. Algorithms 56 (1), pp. 17–26.

ⓘ

External Links:

ISSN 1017-1398, Document, FullText, MathReview, ZentralBlatt

Referenced by:

§5.10.

Encodings:

BibTeX

…

10: 18.9 Recurrence Relations and Derivatives

… ►

§18.9(ii) Contiguous Relations in the Parameters and the Degree

…