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1: 30.2 Differential Equations

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§30.2(ii) Other Forms

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2: 29.2 Differential Equations

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§29.2(ii) Other Forms

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3: Guide to Searching the DLMF

… ►Note that the first form may match other functions $K$ than the Bessel $K$ function, so if you are sure you want Bessel $K$, you might as well enter one of the other 3 forms. …

4: Mathematical Introduction

… ►Other examples are: (a) the notation for the Ferrers functions—also known as associated Legendre functions on the cut—for which existing notations can easily be confused with those for other associated Legendre functions (§14.1); (b) the spherical Bessel functions for which existing notations are unsymmetric and inelegant (§§10.47(i) and 10.47(ii)); and (c) elliptic integrals for which both Legendre’s forms and the more recent symmetric forms are treated fully (Chapter 19). … ►For equations or other technical information that appeared previously in AMS 55, the DLMF usually includes the corresponding AMS 55 equation number, or other form of reference, together with corrections, if needed. …

5: 18.40 Methods of Computation

… ►Orthogonal polynomials can be computed from their explicit polynomial form by Horner’s scheme (§1.11(i)). Usually, however, other methods are more efficient, especially the numerical solution of difference equations (§3.6) and the application of uniform asymptotic expansions (when available) for OP’s of large degree. ►However, for applications in which the OP’s appear only as terms in series expansions (compare §18.18(i)) the need to compute them can be avoided altogether by use instead of Clenshaw’s algorithm (§3.11(ii)) and its straightforward generalization to OP’s other than Chebyshev. …

6: 2.11 Remainder Terms; Stokes Phenomenon

… ►In the transition through $\theta =\pi$, $\mathrm{erfc}\left(\sqrt{\frac{1}{2}\rho }c(\theta )\right)$ changes very rapidly, but smoothly, from one form to the other; compare the graph of its modulus in Figure 2.11.1 in the case $\rho =100$. …

7: 3.6 Linear Difference Equations

… ►For further information, including a more general form of normalizing condition, other examples, convergence proofs, and error analyses, see Olver (1967a), Olver and Sookne (1972), and Wimp (1984, Chapter 6). …

8: 22.15 Inverse Functions

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§22.15(ii) Representations as Elliptic Integrals

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9: 18.34 Bessel Polynomials

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18.34.8 $$\underset{\alpha \to \mathrm{\infty }}{lim}\frac{P_n^{(\alpha ,a-\alpha -2)}\left(1+\alpha x\right)}{P_n^{(\alpha ,a-\alpha -2)}\left(1\right)}=y_n(x;a).$$

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Symbols:

$y_n(x;a)$: Bessel polynomial, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.34.E8

Encodings:

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See also:

Annotations for §18.34(iii), §18.34 and Ch.18

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10: Notices

… ►Bulk copying, reproduction, or redistribution in any form is not permitted. Excessive downloading which results in bandwidth degradation for other users is not permitted and may result in having offending subnets blocked. … ►

Within each of the DLMF chapters themselves we will provide a list of research software for the functions discussed in that chapter. The purpose of these listings is to provide references to the research literature on the engineering of software for special functions. To qualify for listing, the development of the software must have been the subject of a research paper published in the peer-reviewed literature. If such software is available online for free download we will provide a link to the software.

In general, we will not index other software within DLMF chapters unless the software is unique in some way, such as being the only known software for computing a particular function.

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