recessive%20solutions

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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: 2.9 Difference Equations

§2.9 Difference Equations

… ►If $|{\rho }_2|>|{\rho }_1|$, or if $|{\rho }_2|=|{\rho }_1|$ and $\mathrm{\Re }{\alpha }_2>\mathrm{\Re }{\alpha }_1$, then $w_1(n)$ is recessive and $w_2(n)$ is dominant as $n\to \mathrm{\infty }$. 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. … ►But there is an independent solution … ►

3: 13.29 Methods of Computation

… ►As described in §3.7(ii), to insure stability the integration path must be chosen in such a way that as we proceed along it the wanted solution grows in magnitude at least as fast as all other solutions of the differential equation. … ►with recessive solution ►

13.29.2 $$y(n)=z^{-n-\mu -\frac{1}{2}}{M}_{\kappa ,n+\mu }\left(z\right),$$

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Defines:

$y(n)$: solution (locally)

Symbols:

$M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $n$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.29.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §13.29(iv), §13.29(iv), §13.29 and Ch.13

… ►

13.29.4 $$y(n)=1+O\left(n^{-1}\right),$$ $n\to \mathrm{\infty }$.

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Symbols:

$O\left(x\right)$: order not exceeding, $n$: nonnegative integer and $y(n)$: solution

Permalink:

http://dlmf.nist.gov/13.29.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §13.29(iv), §13.29(iv), §13.29 and Ch.13

… ►with recessive solution …

4: 2.7 Differential Equations

… ►has twice-continuously differentiable solutions … ► $w_1(x)$ is a recessive (or subdominant) solution as $x\to a_1+$, and $w_4(x)$ is a dominant solution as $x\to a_1+$. … ►as $x\to +\mathrm{\infty }$, $w_2(x)$ being recessive and $w_3(x)$ dominant. … ►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 $\mathrm{\Re }z\to +\mathrm{\infty }$. …In a neighborhood, or sectorial neighborhood of a singularity, one member has to be recessive. …

5: 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. …

6: 27.15 Chinese Remainder Theorem

… ►The Chinese remainder theorem states that a system of congruences $x\equiv a_1(mod{m}_1),\mathrm{\dots },x\equiv a_k(mod{m}_k)$, always has a solution if the moduli are relatively prime in pairs; the solution is unique (mod $m$), where $m$ is the product of the moduli. … ►Their product $m$ has 20 digits, twice the number of digits in the data. …These numbers, in turn, are combined by the Chinese remainder theorem to obtain the final result $(modm)$, which is correct to 20 digits. …

7: 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). …

8: 33.14 Definitions and Basic Properties

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§33.14(i) Coulomb Wave Equation

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§33.14(ii) Regular Solution $f(ϵ,\mathrm{\ell };r)$

►The function $f(ϵ,\mathrm{\ell };r)$ is recessive (§2.7(iii)) at $r=0$, and is defined by … ►

§33.14(iii) Irregular Solution $h(ϵ,\mathrm{\ell };r)$

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§33.14(iv) Solutions $s(ϵ,\mathrm{\ell };r)$ and $c(ϵ,\mathrm{\ell };r)$

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9: 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 . …

10: 30.8 Expansions in Series of Ferrers Functions

… ►Then the set of coefficients $a_{n,k}^m({\gamma }^2)$, $k=-R,-R+1,-R+2,\mathrm{\dots }$ is the solution of the difference equation … ►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 …