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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 , 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 n , r m ( γ 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 n , k m ( γ 2 ) , k = - N - 1 , - N - 2 , , is the recessive solution of (30.8.4) as k - that is normalized by …
6: 2.7 Differential Equations
2.7.30 w 1 ( x ) / w 4 ( x ) 0 , x a 1 + ,
w 1 ( x ) is a recessive (or subdominant) solution as x a 1 + , and w 4 ( x ) is a dominant solution as x a 1 + . … The solutions w 1 ( z ) and w 2 ( z ) are respectively recessive and dominant as z - , and vice versa as z + . …
7: 29.6 Fourier Series
When ν 2 n , 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 , ; furthermore …
8: 14.2 Differential Equations
§14.2(i) Legendre’s Equation
§14.2(ii) Associated Legendre Equation
§14.2(iii) Numerically Satisfactory Solutions
When μ - ν 0 , - 1 , - 2 , , and μ + ν - 1 , - 2 , - 3 , , P ν - μ ( x ) and P ν - μ ( - x ) are linearly independent, and when μ 0 they are recessive at x = 1 and x = - 1 , respectively. … When μ 0 and ν - 1 2 , P ν - μ ( x ) and Q ν μ ( x ) are linearly independent, and recessive at x = 1 and x = , respectively. …
9: 9.13 Generalized Airy Functions
The function on the right-hand side is recessive in the sector - ( 2 j - 1 ) π / m ph z ( 2 j + 1 ) π / 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(i) Coulomb Wave Equation
§33.2(ii) Regular Solution F ( η , ρ )
The function F ( η , ρ ) is recessive2.7(iii)) at ρ = 0 , and is defined by …
§33.2(iii) Irregular Solutions G ( η , ρ ) , H ± ( η , ρ )
As in the case of F ( η , ρ ) , the solutions H ± ( η , ρ ) and G ( η , ρ ) are analytic functions of ρ when 0 < ρ < . …