three-term 2ϕ1 transformation

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11: 28.31 Equations of Whittaker–Hill and Ince

… ►Hill’s equation with three terms …It has been discussed in detail by Arscott (1967) for , and by Urwin and Arscott (1970) for $k^2>0$. … ►For $m_1\ne m_2$, …and also for all $p_1,p_2,m_1,m_2$, given by … ►All other periodic solutions behave as multiples of $\exp \left(\frac{1}{4}\xi \cos \left(2z\right)\right){(\cos z)}^{-p-2}$. …

12: 18.33 Polynomials Orthogonal on the Unit Circle

… ►After a quadratic transformation (18.2.23) this would express OP’s on $[-1,1]$ with an even orthogonality measure in terms of the ${\varphi }_n$. See Zhedanov (1998, §2). … ►For the hypergeometric function ${}_2{}^{}F_1^{}$ see §§15.1 and 15.2(i). … ►while combination of (18.33.27) and (18.33.23) gives the three-term recurrence relation … ►See Simon (2005a, p. 2, item (2)). …

13: 19.26 Addition Theorems

… ►where for $\gamma =\alpha ,\beta ,\sigma$, except that ${\sigma }^2-\theta$ can be 0, and … ►

$(p-x)(q-x)={(y-x)}^2,$

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$\xi =y^2/x,$

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$\eta -\xi =p+q-2y.$

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${\beta }^2-{\alpha }^2=(p-x)(p-y)(p-z),$

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14: 2.11 Remainder Terms; Stokes Phenomenon

… ►Taking $m=10$ in (2.11.2), the first three terms give us the approximation … ►

§2.11(vi) Direct Numerical Transformations

… ► … ►Similar improvements are achievable by Aitken’s ${\mathrm{\Delta }}^2$-process, Wynn’s $ϵ$-algorithm, and other acceleration transformations. … ►However, direct numerical transformations need to be used with care. …

15: 18.39 Applications in the Physical Sciences

… ►Here the term $-\frac{{\mathrm{\hslash }}^2}{2m}\frac{{\partial }^2}{{\partial x}^2}$ represents the quantum kinetic energy of a single particle of mass $m$, and $V(x)$ its potential energy. … ►The corresponding eigenfunction transform is a generalization of the Kontorovich–Lebedev transform §10.43(v), see Faraut (1982, §IV). … ►where ${\nabla }^2$ is the Laplacian (1.5.17). …where ${\mathrm{L}}^2$ is the (squared) angular momentum operator (14.30.12). … ►As this follows from the three term recursion of (18.39.46) it is referred to as the J-Matrix approach, see (3.5.31), to single and multi-channel scattering numerics. …

16: 14.15 Uniform Asymptotic Approximations

… ►For the interval the following asymptotic approximations hold when $\mu \to \mathrm{\infty }$, with $\nu$ ($\ge -\frac{1}{2}$) fixed, uniformly with respect to $x$: … ►The points $x={\left(1-{\alpha }^2\right)}^{1/2}$, $x=1$, and $x=\mathrm{\infty }$ are mapped to $y={\alpha }^2$, $y=0$, and $y=-\mathrm{\infty }$, respectively. … ►

§14.15(v) Large $\nu$, $(\nu +\frac{1}{2})\delta \le \mu \le (\nu +\frac{1}{2})/\delta$

… ►uniformly with respect to $x\in [0,1)$ and $\mu \in [\delta (\nu +\frac{1}{2}),\nu +\frac{1}{2}]$. … ►uniformly with respect to $x\in (-1,1)$ and $\mu \in [\nu +\frac{1}{2},(1/\delta )(\nu +\frac{1}{2})]$. …

17: 3.5 Quadrature

… ►All the monic orthogonal polynomials $\{p_n\}$ used with Gauss quadrature satisfy a three-term recurrence relation (§18.2(iv)): … ►

Example. Laplace Transform Inversion

… ►With the transformation $\zeta ={\lambda }^{2}t$, (3.5.42) becomes … ► ►Other contour integrals occur in standard integral transforms or their inverses, for example, Hankel transforms (§10.22(v)), Kontorovich–Lebedev transforms (§10.43(v)), and Mellin transforms (§1.14(iv)). …

18: 10.21 Zeros

… ►For the next three terms in (10.21.19) and the next two terms in (10.21.20) see Bickley et al. (1952, p. xxxvii) or Olver (1960, pp. xvii–xviii). … ►An error bound is included for the case $\nu \ge \frac{3}{2}$. … ►For numerical coefficients for $m=2,3,4,5$ see Olver (1951, Tables 3–6). … ►

Zeros of $H_n^{(1)}\left(nz\right)$, $H_n^{(2)}\left(nz\right)$, $H_n^{(1)}{}_{}{}^{\prime }\left(nz\right)$, $H_n^{(2)}{}_{}{}^{\prime }\left(nz\right)$

… ►The Riccati–Bessel functions are ${(\frac{1}{2}\pi x)}^{\frac{1}{2}}{J}_{\nu }\left(x\right)$ and ${(\frac{1}{2}\pi x)}^{\frac{1}{2}}{Y}_{\nu }\left(x\right)$. …

19: 15.14 Integrals

§15.14 Integrals

►The Mellin transform of the hypergeometric function of negative argument is given by … ►Fourier transforms of hypergeometric functions are given in Erdélyi et al. (1954a, §§1.14 and 2.14). Laplace transforms of hypergeometric functions are given in Erdélyi et al. (1954a, §4.21), Oberhettinger and Badii (1973, §1.19), and Prudnikov et al. (1992a, §3.37). …Hankel transforms of hypergeometric functions are given in Oberhettinger (1972, §1.17) and Erdélyi et al. (1954b, §8.17). …

20: 22.7 Landen Transformations

§22.7 Landen Transformations

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§22.7(i) Descending Landen Transformation

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§22.7(ii) Ascending Landen Transformation

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$k_2={\displaystyle \frac{2\sqrt{k}}{1+k}},$

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§22.7(iii) Generalized Landen Transformations

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