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three-term 2ϕ1 transformation

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1: 1.14 Integral Transforms
§1.14 Integral Transforms
§1.14(i) Fourier Transform
§1.14(iii) Laplace Transform
Fourier Transform
Laplace Transform
2: 6.13 Zeros
Ci ( x ) and si ( x ) each have an infinite number of positive real zeros, which are denoted by c k , s k , respectively, arranged in ascending order of absolute value for k = 0 , 1 , 2 , . Values of c 1 and c 2 to 30D are given by MacLeod (1996b). … where α = k π for c k , and α = ( k + 1 2 ) π for s k . For these results, together with the next three terms in (6.13.2), see MacLeod (2002a). …
3: 20.14 Methods of Computation
The Fourier series of §20.2(i) usually converge rapidly because of the factors q ( n + 1 2 ) 2 or q n 2 , and provide a convenient way of calculating values of θ j ( z | τ ) . …For instance, the first three terms of (20.2.1) give the value of θ 1 ( 2 - i | i ) ( = θ 1 ( 2 - i , e - π ) ) to 12 decimal places. For values of | q | near 1 the transformations of §20.7(viii) can be used to replace τ with a value that has a larger imaginary part and hence a smaller value of | q | . …In theory, starting from any value of τ , a finite number of applications of the transformations τ τ + 1 and τ - 1 / τ will result in a value of τ with τ 3 / 2 ; see §23.18. In practice a value with, say, τ 1 / 2 , | q | 0.2 , is found quickly and is satisfactory for numerical evaluation.
4: 16.4 Argument Unity
There are two types of three-term identities for F 2 3 ’s. … The other three-term relations are extensions of Kummer’s relations for F 1 2 ’s given in §15.10(ii). … Balanced F 3 4 ( 1 ) series have transformation formulas and three-term relations. … One example of such a three-term relation is the recurrence relation (18.26.16) for Racah polynomials. … Relations between three solutions of three-term recurrence relations are given by Masson (1991). …
5: 2.9 Difference Equations
Many special functions that depend on parameters satisfy a three-term linear recurrence relation … s = 1 , 2 , 3 , . The construction fails if ρ 1 = ρ 2 , that is, when f 0 2 = 4 g 0 . … Then (2.9.1) has independent solutions w j ( n ) , j = 1 , 2 , such that … Alternatively, suppose that 2 g 1 = f 0 f 1 . …
6: 3.10 Continued Fractions
if the expansion of its n th convergent C n in ascending powers of z agrees with (3.10.7) up to and including the term in z n - 1 , n = 1 , 2 , 3 , . … We say that it is associated with the formal power series f ( z ) in (3.10.7) if the expansion of its n th convergent C n in ascending powers of z , agrees with (3.10.7) up to and including the term in z 2 n - 1 , n = 1 , 2 , 3 , . For the same function f ( z ) , the convergent C n of the Jacobi fraction (3.10.11) equals the convergent C 2 n of the Stieltjes fraction (3.10.6). … The A n and B n of (3.10.2) can be computed by means of three-term recurrence relations (1.12.5). … Then for n 2 , …
7: Bibliography D
  • B. Davies (1984) Integral Transforms and their Applications. 2nd edition, Applied Mathematical Sciences, Vol. 25, Springer-Verlag, New York.
  • A. Deaño, J. Segura, and N. M. Temme (2010) Computational properties of three-term recurrence relations for Kummer functions. J. Comput. Appl. Math. 233 (6), pp. 1505–1510.
  • S. R. Deans (1983) The Radon Transform and Some of Its Applications. A Wiley-Interscience Publication, John Wiley & Sons Inc., New York.
  • L. Debnath and D. Bhatta (2015) Integral transforms and their applications. Third edition, CRC Press, Boca Raton, FL.
  • G. Doetsch (1955) Handbuch der Laplace-Transformation. Bd. II. Anwendungen der Laplace-Transformation. 1. Abteilung. Birkhäuser Verlag, Basel und Stuttgart (German).
  • 8: Bibliography G
  • B. Gabutti and B. Minetti (1981) A new application of the discrete Laguerre polynomials in the numerical evaluation of the Hankel transform of a strongly decreasing even function. J. Comput. Phys. 42 (2), pp. 277–287.
  • 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.
  • W. Gautschi (1967) Computational aspects of three-term recurrence relations. SIAM Rev. 9 (1), pp. 24–82.
  • A. G. Gibbs (1973) Problem 72-21, Laplace transforms of Airy functions. SIAM Rev. 15 (4), pp. 796–798.
  • H. Gupta (1935) A table of partitions. Proc. London Math. Soc. (2) 39, pp. 142–149.
  • 9: 17.6 ϕ 1 2 Function
    §17.6(ii) ϕ 1 2 Transformations
    Heine’s Third Transformation
    Fine’s Second Transformation
    Fine’s Third Transformation
    Three-Term ϕ 1 2 Transformations
    10: 28.31 Equations of Whittaker–Hill and Ince
    Hill’s equation with three terms …It has been discussed in detail by Arscott (1967) for k 2 < 0 , and by Urwin and Arscott (1970) for k 2 > 0 . … Formal 2 π -periodic solutions can be constructed as Fourier series; compare §28.4: … For m 1 m 2 , …and also for all p 1 , p 2 , m 1 , m 2 , given by …