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 $n\to \mathrm{\infty }$, 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

… ►with recessive solution … ►with recessive solution …

3: 30.16 Methods of Computation

… ►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). …

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 $a_{}^{\prime }{}_{n,k}{}^m({\gamma }^2)$, $k=-N-1,-N-2,\mathrm{\dots }$, is the recessive solution of (30.8.4) as $k\to -\mathrm{\infty }$ that is normalized by …

6: 2.7 Differential Equations

… ►

2.7.30 $$w_1(x)/w_4(x)\to 0,$$ $x\to a_1+$,

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Symbols:

$(a_1,a_2)$: interval and $w_j(z)$: solutions

Permalink:

http://dlmf.nist.gov/2.7.E30

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Annotations for §2.7(iii), §2.7(iii), §2.7 and Ch.2

► $w_1(x)$ 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

… ►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 …

8: 14.2 Differential Equations

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§14.2(i) Legendre’s Equation

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§14.2(ii) Associated Legendre Equation

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§14.2(iii) Numerically Satisfactory Solutions

… ►When $\mu -\nu \ne 0,-1,-2,\mathrm{\dots }$, and $\mu +\nu \ne -1,-2,-3,\mathrm{\dots }$, ${𝖯}_{\nu }^{-\mu }\left(x\right)$ and ${𝖯}_{\nu }^{-\mu }\left(-x\right)$ 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 ${𝑸}_{\nu }^{\mu }\left(x\right)$ are linearly independent, and recessive at $x=1$ and $x=\mathrm{\infty }$, respectively. …

9: 9.13 Generalized Airy Functions

… ►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. …

10: 33.2 Definitions and Basic Properties

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§33.2(i) Coulomb Wave Equation

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§33.2(ii) Regular Solution $F_{\mathrm{\ell }}(\eta ,\rho )$

►The function $F_{\mathrm{\ell }}(\eta ,\rho )$ is recessive (§2.7(iii)) at $\rho =0$, and is defined by … ►

§33.2(iii) Irregular Solutions $G_{\mathrm{\ell }}(\eta ,\rho ),H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$

… ►As in the case of $F_{\mathrm{\ell }}(\eta ,\rho )$, the solutions $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$ and $G_{\mathrm{\ell }}(\eta ,\rho )$ are analytic functions of $\rho$ when . …