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21: 12.10 Uniform Asymptotic Expansions for Large Parameter

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12.10.11 $$(t^2-1)u_s^{\prime }(t)-3st{u}_s(t)=r_{s-1}(t),$$

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Symbols:

$s$: nonnegative integer, $u_s(t)$: solution and $r_s(t)$: polynomial

Permalink:

http://dlmf.nist.gov/12.10.E11

Encodings:

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See also:

Annotations for §12.10(ii), §12.10 and Ch.12

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12.10.12 $$8r_s(t)=(3t^2+2)u_s(t)-12(s+1)t{r}_{s-1}(t)+4(t^2-1)r_{s-1}^{\prime }(t),$$

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Defines:

$r_s(t)$: polynomial (locally)

Symbols:

$s$: nonnegative integer and $u_s(t)$: solution

Permalink:

http://dlmf.nist.gov/12.10.E12

Encodings:

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See also:

Annotations for §12.10(ii), §12.10 and Ch.12

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12.10.13 $$v_s(t)=u_s(t)+\frac{1}{2}t{u}_{s-1}(t)-r_{s-2}(t).$$

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Symbols:

$s$: nonnegative integer, $u_s(t)$: solution, $v_s(t)$: solution and $r_s(t)$: polynomial

Permalink:

http://dlmf.nist.gov/12.10.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §12.10(ii), §12.10 and Ch.12

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22: 2.8 Differential Equations with a Parameter

§2.8 Differential Equations with a Parameter

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§2.8(i) Classification of Cases

►Many special functions satisfy an equation of the form … ►

§2.8(vi) Coalescing Transition Points

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23: 10.20 Uniform Asymptotic Expansions for Large Order

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10.20.10 $$A_k(\zeta )=\sum _{j=0}^{2k}{(\frac{3}{2})}^j{v}_j{\zeta }^{-3j/2}{U}_{2k-j}\left({(1-z^2)}^{-\frac{1}{2}}\right),$$

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Defines:

$A_k(\zeta )$: coefficients (locally)

Symbols:

$k$: nonnegative integer, $z$: complex variable, $\zeta (z)$: solution, $v_k$: constants and $U_k(p)$: polynomial coefficient

A&S Ref:

9.3.40

Referenced by:

§10.20(i), §10.41(v), §10.74(i)

Permalink:

http://dlmf.nist.gov/10.20.E10

Encodings:

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See also:

Annotations for §10.20(i), §10.20(i), §10.20 and Ch.10

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10.20.11 $$B_k(\zeta )=-{\zeta }^{-\frac{1}{2}}\sum _{j=0}^{2k+1}{(\frac{3}{2})}^j{u}_j{\zeta }^{-3j/2}{U}_{2k-j+1}\left({(1-z^2)}^{-\frac{1}{2}}\right),$$

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Defines:

$B_k(\zeta )$: coefficients (locally)

Symbols:

$k$: nonnegative integer, $z$: complex variable, $\zeta (z)$: solution, $u_k$: constants and $U_k(p)$: polynomial coefficient

A&S Ref:

9.3.40

Referenced by:

§10.21(viii), §10.41(v)

Permalink:

http://dlmf.nist.gov/10.20.E11

Encodings:

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See also:

Annotations for §10.20(i), §10.20(i), §10.20 and Ch.10

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10.20.12 $$C_k(\zeta )=-{\zeta }^{\frac{1}{2}}\sum _{j=0}^{2k+1}{(\frac{3}{2})}^j{v}_j{\zeta }^{-3j/2}{V}_{2k-j+1}\left({(1-z^2)}^{-\frac{1}{2}}\right),$$

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Defines:

$C_k(\zeta )$: coefficients (locally)

Symbols:

$k$: nonnegative integer, $z$: complex variable, $\zeta (z)$: solution, $v_k$: constants and $V_k(p)$: polynomial coefficient

A&S Ref:

9.3.46

Referenced by:

§10.21(viii)

Permalink:

http://dlmf.nist.gov/10.20.E12

Encodings:

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See also:

Annotations for §10.20(i), §10.20(i), §10.20 and Ch.10

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10.20.13 $$D_k(\zeta )=\sum _{j=0}^{2k}{(\frac{3}{2})}^j{u}_j{\zeta }^{-3j/2}{V}_{2k-j}\left({(1-z^2)}^{-\frac{1}{2}}\right).$$

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Defines:

$D_k(\zeta )$: coefficients (locally)

Symbols:

$k$: nonnegative integer, $z$: complex variable, $\zeta (z)$: solution, $u_k$: constants and $V_k(p)$: polynomial coefficient

A&S Ref:

9.3.46

Referenced by:

§10.20(i), §10.74(i)

Permalink:

http://dlmf.nist.gov/10.20.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §10.20(i), §10.20(i), §10.20 and Ch.10

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24: 32.8 Rational Solutions

§32.8 Rational Solutions

… ►Special rational solutions of ${\text{P}}_{\text{III}}$ are … ►These solutions have the form … ►These rational solutions have the form … ►

25: 18.30 Associated OP’s

… ►The corecursive orthogonal polynomials, $p_n^{(0)}(x)$, these being linearly independent solutions of the recurrence for the $p_n(x)$, are defined as follows: …

26: 2.9 Difference Equations

§2.9 Difference Equations

… ►Formal solutions are … ►

§2.9(ii) Coincident Characteristic Values

… ►But there is an independent solution … ►

27: Bibliography Z

… ►

Doron Zeilberger’s Maple Packages and Programs Department of Mathematics, Rutgers University, New Jersey.

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Notes:

Includes hypergeometric and $q$-hypergeometric summation, solution of difference equations (for example, by Petkovšek’s algorithm), combinatorial algorithms, and (package LUC) evaluation of zonal polynomials.

External Links:

Home Page

Referenced by:

§ ‣ Software Cross Index, § ‣ Software Cross Index, L. Lapointe and L. Vinet (1996).

Encodings:

BibTeX

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28: 29.7 Asymptotic Expansions

… ►Müller (1966a, b) found three formal asymptotic expansions for a fundamental system of solutions of (29.2.1) (and (29.11.1)) as $\nu \to \mathrm{\infty }$, one in terms of Jacobian elliptic functions and two in terms of Hermite polynomials. …

29: Peter A. Clarkson

… ►He is a member of the editorial boards of nine international journals and has served as Chair, Vice-Chair, and Secretary of the SIAM Activity Group on Orthogonal Polynomials and Special Functions. … ► Kruskal, he developed the “direct method” for determining symmetry solutions of partial differential equations in New similarity reductions of the Boussinesq equation (with M. …

30: 18.39 Applications in the Physical Sciences

… ►The same solutions as in paragraph c), above, appear frequently in the literature in terms of associated Laguerre polynomials, which are referred to here as associated Coulomb–Laguerre polynomials to avoid confusion with the more recent meaning of ‘associated’ of §18.30. … ►Bound state solutions to the relativistic Dirac Equation, for this same problem of a single electron attracted by a nucleus with $Z$ protons, involve Laguerre polynomials of fractional index. …