# three-term 2ϕ1 transformation

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##### 2: 6.13 Zeros
$\operatorname{Ci}\left(x\right)$ and $\operatorname{si}\left(x\right)$ 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 $k=0,1,2,\dots$. Values of $c_{1}$ and $c_{2}$ to 30D are given by MacLeod (1996b). … where $\alpha=k\pi$ for $c_{k}$, and $\alpha=(k+\frac{1}{2})\pi$ 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 $q^{(n+\frac{1}{2})^{2}}$ or $q^{n^{2}}$, and provide a convenient way of calculating values of $\theta_{j}\left(z\middle|\tau\right)$. …For instance, the first three terms of (20.2.1) give the value of $\theta_{1}\left(2-i\middle|i\right)$ ($=\theta_{1}\left(2-i,e^{-\pi}\right)$) to 12 decimal places. For values of $\left|q\right|$ near $1$ the transformations of §20.7(viii) can be used to replace $\tau$ with a value that has a larger imaginary part and hence a smaller value of $\left|q\right|$. …In theory, starting from any value of $\tau$, a finite number of applications of the transformations $\tau\to\tau+1$ and $\tau\to-1/\tau$ will result in a value of $\tau$ with $\Im\tau\geq\sqrt{3}/2$; see §23.18. In practice a value with, say, $\Im\tau\geq 1/2$, $\left|q\right|\leq 0.2$, is found quickly and is satisfactory for numerical evaluation.
##### 4: 16.4 Argument Unity
There are two types of three-term identities for ${{}_{3}F_{2}}$’s. … The other three-term relations are extensions of Kummer’s relations for ${{}_{2}F_{1}}$’s given in §15.10(ii). … Balanced ${{}_{4}F_{3}}\left(1\right)$ 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: 4.13 Lambert $W$-Function
where ${\rm ln}_{k}(z)=\ln\left(z\right)+2\pi\mathrm{i}k$. …
$d_{0}=-1,\quad d_{1}=\sqrt{2},\quad d_{2}=-\tfrac{2}{3},\quad d_{3}=\tfrac{11}% {36}\sqrt{2},\quad d_{4}=-\tfrac{43}{135},$
where $\xi_{k}=\ln\left(z\right)+2\pi\mathrm{i}k$. …
##### 6: 2.9 Difference Equations
Many special functions that depend on parameters satisfy a three-term linear recurrence relation … $s=1,2,3,\dots$. The construction fails if $\rho_{1}=\rho_{2}$, that is, when $f_{0}^{2}=4g_{0}$. … Then (2.9.1) has independent solutions $w_{j}(n)$, $j=1,2$, such that … If $\alpha_{2}-\alpha_{1}=0,1,2,\dots$, then (2.9.12) applies only in the case $j=1$. …
##### 7: 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 $z^{n-1}$, $n=1,2,3,\dots$. … 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 $z^{2n-1}$, $n=1,2,3,\dots$. For the same function $f(z)$, the convergent $C_{n}$ of the Jacobi fraction (3.10.11) equals the convergent $C_{2n}$ 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 $n\geq 2$, …
##### 8: 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).
• ##### 9: 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.