numerically satisfactory solutions

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21: 15.10 Hypergeometric Differential Equation

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§15.10(i) Fundamental Solutions

… ►They are also numerically satisfactory (§2.7(iv)) in the neighborhood of the corresponding singularity. … ► ►

§15.10(ii) Kummer’s 24 Solutions and Connection Formulas

►The three pairs of fundamental solutions given by (15.10.2), (15.10.4), and (15.10.6) can be transformed into 18 other solutions by means of (15.8.1), leading to a total of 24 solutions known as Kummer’s solutions. …

22: 2.9 Difference Equations

… ►As in the case of differential equations (§§2.7(iii), 2.7(iv)) recessive solutions are unique and dominant solutions are not; furthermore, one member of a numerically satisfactory pair has to be recessive. …

23: 14.20 Conical (or Mehler) Functions

… ►Solutions are known as conical or Mehler functions. For and $\tau >0$, a numerically satisfactory pair of real conical functions is ${𝖯}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x\right)$ and ${𝖯}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(-x\right)$. … ►Another real-valued solution ${\widehat{𝖰}}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x\right)$ of (14.20.1) was introduced in Dunster (1991). …It is an important companion solution to ${𝖯}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x\right)$ when $\tau$ is large; compare §§14.20(vii), 14.20(viii), and 10.25(iii). … ►Lastly, for the range , $P_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x\right)$ is a real-valued solution of (14.20.1); in terms of $Q_{-\frac{1}{2}\pm \mathrm{i}\tau }^{\mu }\left(x\right)$ (which are complex-valued in general): …