# contiguous balanced series

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##### 1: 16.4 Argument Unity
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###### Pfaff–Saalschütz Balanced Sum
โบBalanced ${{}_{4}F_{3}}\left(1\right)$ series have transformation formulas and three-term relations. … โบ โบContiguous 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
โบThe series (17.4.1) is said to be balanced or Saalschützian when it terminates, $r=s$, $z=q$, and … โบThe series (17.4.1) is said to be k-balanced when $r=s$ and … โบThe series (17.4.1) is said to be well-poised when $r=s$ and … โบThe 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
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###### §15.5(ii) Contiguous Functions
โบThe six functions $F\left(a\pm 1,b;c;z\right)$, $F\left(a,b\pm 1;c;z\right)$, $F\left(a,b;c\pm 1;z\right)$ are said to be contiguous to $F\left(a,b;c;z\right)$. … โบAn equivalent equation to the hypergeometric differential equation (15.10.1) is …Further contiguous relations include: …
##### 4: 16.3 Derivatives and Contiguous Functions
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###### §16.3(ii) Contiguous Functions
โบTwo generalized hypergeometric functions ${{}_{p}F_{q}}\left(\mathbf{a};\mathbf{b};z\right)$ are (generalized) contiguous if they have the same pair of values of $p$ and $q$, and corresponding parameters differ by integers. If $p\leq q+1$, then any $q+2$ distinct contiguous functions are linearly related. …
##### 5: Bibliography G
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• 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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• 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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• G. H. Golub and G. Meurant (2010) Matrices, moments and quadrature with applications. Princeton Series in Applied Mathematics, Princeton University Press, Princeton, NJ.
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• D. Gottlieb and S. A. Orszag (1977) Numerical Analysis of Spectral Methods: Theory and Applications. Society for Industrial and Applied Mathematics, Philadelphia, PA.
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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.
• ##### 6: 17.6 ${{}_{2}\phi_{1}}$ Function
โบNote that for several of the equations below, the constraints are included to guarantee that the infinite series representation (17.4.1) of the ${{}_{2}\phi_{1}}$ functions converges. … โบ
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##### 7: Bibliography S
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• F. W. Schäfke and A. Finsterer (1990) On Lindelöf’s error bound for Stirling’s series. J. Reine Angew. Math. 404, pp. 135–139.
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• I. J. Schwatt (1962) An Introduction to the Operations with Series. 2nd edition, Chelsea Publishing Co., New York.
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• T. C. Scott, G. Fee, and J. Grotendorst (2013) Asymptotic series of generalized Lambert $W$ function. ACM Commun. Comput. Algebra 47 (3), pp. 75–83.
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• J. Segura (2008) Interlacing of the zeros of contiguous hypergeometric functions. Numer. Algorithms 49 (1-4), pp. 387–407.
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• S. K. Suslov (2003) An Introduction to Basic Fourier Series. Developments in Mathematics, Vol. 9, Kluwer Academic Publishers, Dordrecht.
• ##### 8: 18.2 General Orthogonal Polynomials
โบBetween the systems $\{p_{n}(x)\}$ and $\{q_{n}(x)\}$ there are the contiguous relations … โบThis 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. … โบfor $x,y$ in the support of the orthogonality measure and $z$ such that the series in (18.2.41) converges absolutely for all these $x,y$. … โบwhere $f(t)$ 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 …
##### 9: 17.16 Mathematical Applications
###### §17.16 Mathematical Applications
โบMany 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. …
##### 10: 17.9 Further Transformations of ${{}_{r+1}\phi_{r}}$ Functions
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###### Sears’ Balanced${{}_{4}\phi_{3}}$ Transformations
โบprovided that the series expansions of both $\phi$’s terminate. โบ