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31: 18.15 Asymptotic Approximations

… ►The case

M=1

of (18.15.1) goes back to Darboux. … ►where

J_{\nu}\left(z\right)

is the Bessel function (§10.2(ii)), and …For higher coefficients see Baratella and Gatteschi (1988), and for another estimate of the error term in a related expansion see Wong and Zhao (2003). … ►Here

J_{\nu}\left(z\right)

denotes the Bessel function (§10.2(ii)),

\operatorname{env}\mskip-2.0muJ_{\nu}\left(z\right)

denotes its envelope (§2.8(iv)), and

\delta

is again an arbitrary small positive constant. … ►With

\mu=\sqrt{2n+1}

the expansions in Chapter 12 are for the parabolic cylinder function

U\left(-\tfrac{1}{2}\mu^{2},\mu t\sqrt{2}\right)

, which is related to the Hermite polynomials via …

32: Bibliography V

… ►

G. Valent (1986) An integral transform involving Heun functions and a related eigenvalue problem. SIAM J. Math. Anal. 17 (3), pp. 688–703.

ⓘ

External Links:: ISSN 0036-1410, Document, MathReview (William L. Perry), ZentralBlatt
Referenced by:: §31.10(i).
Encodings:: BibTeX

… ►

J. F. Van Diejen and V. P. Spiridonov (2001) Modular hypergeometric residue sums of elliptic Selberg integrals. Lett. Math. Phys. 58 (3), pp. 223–238.

ⓘ

External Links:: ISSN 0377-9017, Document, MathReview (Laurent Habsieger), ZentralBlatt
Referenced by:: §19.35(i).
Encodings:: BibTeX

… ►

R. Vidūnas and N. M. Temme (2002) Symbolic evaluation of coefficients in Airy-type asymptotic expansions. J. Math. Anal. Appl. 269 (1), pp. 317–331.

ⓘ

External Links:: ISSN 0022-247X, Document, MathReview, ZentralBlatt
Referenced by:: §2.4(v).
Encodings:: BibTeX

… ►

H. Volkmer (1982) Integral relations for Lamé functions. SIAM J. Math. Anal. 13 (6), pp. 978–987.

ⓘ

External Links:: ISSN 0036-1410, Document, MathReview (F. M. Arscott), ZentralBlatt
Referenced by:: §29.8, §29.8.
Encodings:: BibTeX

… ►

H. Volkmer (1984) Integral representations for products of Lamé functions by use of fundamental solutions. SIAM J. Math. Anal. 15 (3), pp. 559–569.

ⓘ

External Links:: ISSN 0036-1410, Document, MathReview (F. M. Arscott), ZentralBlatt
Referenced by:: §29.8.
Encodings:: BibTeX

…

33: 18.34 Bessel Polynomials

… ►

§18.34(i) Definitions and Recurrence Relation

… ►With the notation of Koekoek et al. (2010, (9.13.1)) the left-hand side of (18.34.1) has to be replaced by

y_{n}\left(x;a-2\right)

. …where

\mathsf{k}_{n}

is a modified spherical Bessel function (10.49.9), and … … ►where primes denote derivatives with respect to

x

. …

34: 19.19 Taylor and Related Series

§19.19 Taylor and Related Series

… ►The following two multivariate hypergeometric series apply to each of the integrals (19.16.14)–(19.16.18) and (19.16.20)–(19.16.23): … ►where

M=\sum_{j=1}^{n}m_{j}

and the summation extends over all nonnegative integers

m_{1},\dots,m_{n}

such that

\sum_{j=1}^{n}jm_{j}=N

. … ►The number of terms in

T_{N}

can be greatly reduced by using variables

\mathbf{Z}=\boldsymbol{{1}}-(\mathbf{z}/A)

with

A

chosen to make

E_{1}(\mathbf{Z})=0

. …For

R_{J}

and

R_{D}

T_{N}

has at most one term if

N\leq 3

, and two terms if

N=4

or 5. …

35: Bibliography D

… ►

A. Deaño, J. Segura, and N. M. Temme (2010) Computational properties of three-term recurrence relations for Kummer functions. J. Comput. Appl. Math. 233 (6), pp. 1505–1510.

ⓘ

External Links:: ISSN 0377-0427, Document, FullText, MathReview (Richard B. Paris), ZentralBlatt
Referenced by:: §13.29(iv), §13.29(iv).
Encodings:: BibTeX

… ►

B. I. Dunlap and B. R. Judd (1975) Novel identities for simple

n

j

symbols. J. Mathematical Phys. 16, pp. 318–319.

ⓘ

External Links:: Document, MathReview (M. J. Westwater)
Referenced by:: §34.5(vi).
Encodings:: BibTeX

… ►

T. M. Dunster (1999) Asymptotic approximations for the Jacobi and ultraspherical polynomials, and related functions. Methods Appl. Anal. 6 (3), pp. 21–56.

ⓘ

External Links:: ISSN 1073-2772, Pdf, MathReview (Walter Van Assche), ZentralBlatt
Referenced by:: §15.12(iii), §18.15(i), §18.15(vi).
Encodings:: BibTeX

… ►

T. M. Dunster (2001c) Uniform asymptotic expansions for the reverse generalized Bessel polynomials, and related functions. SIAM J. Math. Anal. 32 (5), pp. 987–1013.

ⓘ

External Links:: ISSN 0036-1410, Document, MathReview (Hasan Taşeli), ZentralBlatt
Referenced by:: §10.49(ii), §18.34(iii).
Encodings:: BibTeX

… ►

P. L. Duren (1991) The Legendre Relation for Elliptic Integrals. In Paul Halmos: Celebrating 50 Years of Mathematics, J. H. Ewing and F. W. Gehring (Eds.), pp. 305–315.

ⓘ

External Links:: ISBN 0-387-97509-8, MathReview (J. M. H. Peters), ZentralBlatt
Referenced by:: §19.7(i).
Encodings:: BibTeX

…

36: 31.16 Mathematical Applications

… ►The coefficients

A_{j}

satisfy the relations: ►

31.16.3

A_{0}=\frac{n!}{{\left(\gamma+\delta\right)_{n}}}\mathit{Hp}_{n,m}\left(1% \right),\quad Q_{0}A_{0}+R_{0}A_{1}=0,

ⓘ

Symbols:: $\mathit{Hp}_{\NVar{n},\NVar{m}}\left(\NVar{a},\NVar{q_{n,m}};\NVar{-n},\NVar{% \beta},\NVar{\gamma},\NVar{\delta};\NVar{z}\right)$ : Heun polynomials, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $m$ : nonnegative integer, $n$ : nonnegative integer, $a$ : complex parameter, $q$ : real or complex parameter, $\beta$ : real or complex parameter, $A_{j}$ : coefficients, $Q_{j}$ and $R_{j}$
Referenced by:: §31.16(ii), Erratum (V1.0.21) for Equations (31.16.2) and (31.16.3), Erratum (V1.0.21) for Equations (31.16.2) and (31.16.3)
Permalink:: http://dlmf.nist.gov/31.16.E3
Encodings:: pMML, png, TeX
Correction (effective with 1.0.21):: The initial data $A_{0}$ , previously missing, has now been included.
See also:: Annotations for §31.16(ii), §31.16 and Ch.31

►

31.16.4

P_{j}A_{j-1}+Q_{j}A_{j}+R_{j}A_{j+1}=0,

j=1,2,\dots,n

ⓘ

Symbols:: $j$ : nonnegative integer, $n$ : nonnegative integer, $A_{j}$ : coefficients, $P_{j}$ , $Q_{j}$ and $R_{j}$
Permalink:: http://dlmf.nist.gov/31.16.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §31.16(ii), §31.16 and Ch.31

… ►

31.16.5

P_{j}=\frac{(\epsilon-j+n)j(\beta+j-1)(\gamma+\delta+j-2)}{(\gamma+\delta+2j-3% )(\gamma+\delta+2j-2)},

ⓘ

Defines:: $P_{j}$ (locally)
Symbols:: $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\epsilon$ : real or complex parameter, $j$ : nonnegative integer, $n$ : nonnegative integer and $\beta$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/31.16.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §31.16(ii), §31.16 and Ch.31

… ►

31.16.7

R_{j}=\frac{(n-j)(j+n+\gamma+\delta)(j+\gamma)(j+\delta)}{(\gamma+\delta+2j)(% \gamma+\delta+2j+1)}.

ⓘ

Defines:: $R_{j}$ (locally)
Symbols:: $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $j$ : nonnegative integer and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/31.16.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §31.16(ii), §31.16 and Ch.31

…

37: 26.8 Set Partitions: Stirling Numbers

… ►where

{\left(x\right)_{n}}

is the Pochhammer symbol:

x(x+1)\cdots(x+n-1)

. … ►

§26.8(iv) Recurrence Relations

… ►Let

A

and

B

be the

n\times n

matrices with

(j,k)

th elements

s\left(j,k\right)

, and

S\left(j,k\right)

, respectively. … ►

§26.8(vi) Relations to Bernoulli Numbers

… ►For asymptotic approximations for

s\left(n+1,k+1\right)

and

S\left(n,k\right)

that apply uniformly for

1\leq k\leq n

n\to\infty

see Temme (1993) and Temme (2015, Chapter 34). …

38: 19.29 Reduction of General Elliptic Integrals

… ►The advantages of symmetric integrals for tables of integrals and symbolic integration are illustrated by (19.29.4) and its cubic case, which replace the

8+8+12=28

formulas in Gradshteyn and Ryzhik (2000, 3.147, 3.131, 3.152) after taking

x^{2}

as the variable of integration in 3. … ►where

x>y

h=3

or 4,

n\geq h

, and

m_{j}

is an integer. … ►Partial fractions provide a reduction to integrals in which

\mathbf{m}

has at most one nonzero component, and these are then reduced to basic integrals by the recurrence relations. … ►If

h=3

, then the recurrence relation (Carlson (1999, (3.5))) has the special case … ►The other recurrence relation is …

39: 11.10 Anger–Weber Functions

… ►

§11.10(vi) Relations to Other Functions

… ► ►For

n=1,2,3,\dots

, … ►where the prime on the second summation symbols means that the first term is to be halved. ►

§11.10(ix) Recurrence Relations and Derivatives

…

40: 30.14 Wave Equation in Oblate Spheroidal Coordinates

… ►Oblate spheroidal coordinates

\xi,\eta,\phi

are related to Cartesian coordinates

x, y, z

by …(On the use of the symbol

\theta

in place of

\phi

see §1.5(ii).) … ►The wave equation (30.13.7), transformed to oblate spheroidal coordinates

(\xi,\eta,\phi)

, admits solutions of the form (30.13.8), where

w_{1}

satisfies the differential equation …and

w_{2}

w_{3}

satisfy (30.13.10) and (30.13.11), respectively, with

\gamma^{2}=-\kappa^{2}c^{2}\leq 0

and separation constants

\lambda

and

\mu^{2}

. Equation (30.14.7) can be transformed to equation (30.2.1) by the substitution

z=\pm i\xi

. …