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1: 16.4 Argument Unity
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Pfaff–Saalschütz Balanced Sum
โ–บBalanced F 3 4 โก ( 1 ) 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
§15.5 Derivatives and Contiguous Functions
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§15.5(ii) Contiguous Functions
โ–บThe six functions F โก ( a ± 1 , b ; c ; z ) , F โก ( a , b ± 1 ; c ; z ) , F โก ( a , b ; c ± 1 ; z ) are said to be contiguous to F โก ( a , b ; c ; z ) . … โ–บAn 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
โ–บTwo generalized hypergeometric functions F q p โก ( ๐š ; ๐› ; z ) are (generalized) contiguous if they have the same pair of values of p and q , and corresponding parameters differ by integers. If p 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 U โข ( n ) . SIAM J. Math. Anal. 18 (6), pp. 1576–1596.
  • 6: 17.6 ฯ• 1 2 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 ฯ• 1 2 functions converges. … โ–บ
    §17.6(iii) Contiguous Relations
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    Heine’s Contiguous Relations
    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 โก ( x ) } and { q n โข ( 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 โก ( x ) = T n โก ( x ) , 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 โข ( t ) and u โข ( t ) are formal power series in t , with f โข ( 0 ) = 1 , u โข ( 0 ) = 0 and u โข ( 0 ) = 1 . …If v โข ( s ) is the formal power series such that v โข ( u โข ( 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 ฯ• r r + 1 Functions
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    Transformations of ฯ• 2 3 -Series
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    Sears’ Balanced ฯ• 3 4 Transformations
    โ–บprovided that the series expansions of both ฯ• ’s terminate. โ–บ
    §17.9(iv) Bibasic Series