contiguous balanced series

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1: 16.4 Argument Unity

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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

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§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

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§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

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§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

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

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

Referenced by:

§17.7(iii).

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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).

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ISBN 0-521-35049-2, MathReview (Shaun Cooper), ZentralBlatt

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(17.9.3), Erratum (V1.0.23) for Equation (17.9.3).

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Matrices, moments and quadrature with applications. Princeton Series in Applied Mathematics, Princeton University Press, Princeton, NJ.

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ISBN 978-0-691-14341-5, MathReview (G. A. Evans)

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§18.2(ix), §18.40(ii).

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

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Notes:

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

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MathReview (O. Widlund), ZentralBlatt

Referenced by:

§18.38(i).

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

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ISSN 0036-1410, Document, MathReview, ZentralBlatt

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§17.15.

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6: Bibliography S

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

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

Referenced by:

§5.11(ii).

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

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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.

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

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FullText, ZentralBlatt

Referenced by:

§4.13, §4.13.

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

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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.

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

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

Referenced by:

§17.3(i).

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7: 18.2 General Orthogonal Polynomials

… ►Between the systems $\{p_n(x)\}$ and $\{q_n(x)\}$ there are the contiguous relations … ►This says roughly that the series (18.2.25) has the same pointwise convergence behavior as the same series with $p_n(x)=T_n\left(x\right)$, a Chebyshev polynomial of the first kind, see Table 18.3.1. … ►for $x,y$ in the support of the orthogonality measure and $z$ such that the series in (18.2.41) converges absolutely for all these $x,y$. … ►where $f(t)$ and $u(t)$ are formal power series in $t$, with $f(0)=1$, $u(0)=0$ and $u^{\prime }(0)=1$. …If $v(s)$ is the formal power series such that $v(u(t))=t$ then a property equivalent to (18.2.45) with $c_n=1$ is that …

8: 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. …

9: 17.9 Further Transformations of ${}_{r+1}{}^{}\varphi _r^{}$ Functions

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

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

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

§17.9(iv) Bibasic Series

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10: Bibliography M

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

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External Links:

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

Referenced by:

§35.4(ii).

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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.

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

Referenced by:

§3.11(ii).

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

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

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§17.15.

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

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

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(25.6.5).

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

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

Referenced by:

§5.10.

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