# recessive solutions

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##### 1: 3.6 Linear Difference Equations
Then $w_{n}$ is said to be a recessive (equivalently, minimal or distinguished) solution as $n\to\infty$, 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
with recessive solutionwith recessive solution
##### 3: 30.16 Methods of Computation
The coefficients $a^{m}_{n,r}(\gamma^{2})$ are computed as the recessive solution of (30.8.4) (§3.6), and normalized via (30.8.5). …
##### 4: 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. …
##### 5: 30.8 Expansions in Series of Ferrers Functions
The set of coefficients ${a^{\prime}}^{m}_{n,k}(\gamma^{2})$, $k=-N-1,-N-2,\dots$, is the recessive solution of (30.8.4) as $k\to-\infty$ that is normalized by …
##### 6: 2.7 Differential Equations
2.7.30 $w_{1}(x)/w_{4}(x)\to 0,$ $x\to a_{1}+$,
$w_{1}(x)$ 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 $\Re z\to-\infty$, and vice versa as $\Re z\to+\infty$. …
##### 7: 29.6 Fourier Series
When $\nu\neq 2n$, 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 $A_{0},A_{2},A_{4},\dots$; furthermore …
##### 8: 14.2 Differential Equations
###### §14.2(iii) Numerically Satisfactory Solutions
When $\mu-\nu\neq 0,-1,-2,\dots$, and $\mu+\nu\neq-1,-2,-3,\dots$, $\mathsf{P}^{-\mu}_{\nu}\left(x\right)$ and $\mathsf{P}^{-\mu}_{\nu}\left(-x\right)$ are linearly independent, and when $\Re\mu\geq 0$ they are recessive at $x=1$ and $x=-1$, respectively. … When $\Re\mu\geq 0$ and $\Re\nu\geq-\frac{1}{2}$, $P^{-\mu}_{\nu}\left(x\right)$ and $\boldsymbol{Q}^{\mu}_{\nu}\left(x\right)$ are linearly independent, and recessive at $x=1$ and $x=\infty$, respectively. …
##### 9: 9.13 Generalized Airy Functions
The function on the right-hand side is recessive in the sector $-(2j-1)\pi/m\leq\operatorname{ph}z\leq(2j+1)\pi/m$, and is therefore an essential member of any numerically satisfactory pair of solutions in this region. …
##### 10: 33.2 Definitions and Basic Properties
###### §33.2(ii) Regular Solution$F_{\ell}\left(\eta,\rho\right)$
The function $F_{\ell}\left(\eta,\rho\right)$ is recessive2.7(iii)) at $\rho=0$, and is defined by …
###### §33.2(iii) Irregular Solutions$G_{\ell}\left(\eta,\rho\right),{H^{\pm}_{\ell}}\left(\eta,\rho\right)$
As in the case of $F_{\ell}\left(\eta,\rho\right)$, the solutions ${H^{\pm}_{\ell}}\left(\eta,\rho\right)$ and $G_{\ell}\left(\eta,\rho\right)$ are analytic functions of $\rho$ when $0<\rho<\infty$. …