# recessive solutions

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## 1—10 of 11 matching pages

##### 1: 3.6 Linear Difference Equations

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►Then ${w}_{n}$ is said to be a

*recessive*(equivalently,*minimal*or*distinguished*)*solution*as $n\to \mathrm{\infty}$, and it is unique except for a constant factor. … … ►Because the recessive solution of a homogeneous equation is the fastest growing solution in the backward direction, it occurred to J. … ►See also Gautschi (1967) and Gil et al. (2007a, Chapter 4) for the computation of recessive solutions via continued fractions. … ►It is applicable equally to the computation of the recessive solution of the homogeneous equation (3.6.3) or the computation of any solution ${w}_{n}$ of the inhomogeneous equation (3.6.1) for which the conditions of §3.6(iv) are satisfied. …##### 2: 13.29 Methods of Computation

##### 3: 30.16 Methods of Computation

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►The coefficients ${a}_{n,r}^{m}({\gamma}^{2})$ are computed as the recessive solution of (30.8.4) (§3.6), and normalized via (30.8.5).
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##### 4: 2.9 Difference Equations

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►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.
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##### 5: 30.8 Expansions in Series of Ferrers Functions

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►The set of coefficients $a^{\prime}{}_{n,k}{}^{m}({\gamma}^{2})$, $k=-N-1,-N-2,\mathrm{\dots}$, is the recessive solution of (30.8.4) as $k\to -\mathrm{\infty}$ that is normalized by
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##### 6: 2.7 Differential Equations

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2.7.30
$${w}_{1}(x)/{w}_{4}(x)\to 0,$$
$x\to {a}_{1}+$,

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${w}_{1}(x)$ is a *recessive*(or*subdominant*) solution as $x\to {a}_{1}+$, and ${w}_{4}(x)$ is a*dominant*solution as $x\to {a}_{1}+$. … ►The solutions ${w}_{1}(z)$ and ${w}_{2}(z)$ are respectively recessive and dominant as $\mathrm{\Re}z\to -\mathrm{\infty}$, and*vice versa*as $\mathrm{\Re}z\to +\mathrm{\infty}$. …##### 7: 29.6 Fourier Series

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►When $\nu \ne 2n$, where $n$ is a nonnegative integer, it follows from §2.9(i) that for any value of $H$ the system (29.6.4)–(29.6.6) has a unique recessive solution
${A}_{0},{A}_{2},{A}_{4},\mathrm{\dots}$; furthermore
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##### 8: 14.2 Differential Equations

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###### §14.2(i) Legendre’s Equation

… ►###### §14.2(ii) Associated Legendre Equation

… ►###### §14.2(iii) Numerically Satisfactory Solutions

… ►When $\mu -\nu \ne 0,-1,-2,\mathrm{\dots}$, and $\mu +\nu \ne -1,-2,-3,\mathrm{\dots}$, ${\mathsf{P}}_{\nu}^{-\mu}\left(x\right)$ and ${\mathsf{P}}_{\nu}^{-\mu}\left(-x\right)$ are linearly independent, and when $\mathrm{\Re}\mu \ge 0$ they are recessive at $x=1$ and $x=-1$, respectively. … ►When $\mathrm{\Re}\mu \ge 0$ and $\mathrm{\Re}\nu \ge -\frac{1}{2}$, ${P}_{\nu}^{-\mu}\left(x\right)$ and ${\mathit{Q}}_{\nu}^{\mu}\left(x\right)$ are linearly independent, and recessive at $x=1$ and $x=\mathrm{\infty}$, respectively. …##### 9: 9.13 Generalized Airy Functions

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►The function on the right-hand side is recessive in the sector $-(2j-1)\pi /m\le \mathrm{ph}z\le (2j+1)\pi /m$, and is therefore an essential member of any numerically satisfactory pair of solutions in this region.
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##### 10: 33.2 Definitions and Basic Properties

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