representations as sums of powers

SearchAdvancedHelp

(0.003 seconds)

21—26 of 26 matching pages

21: Bibliography R

… ►

A non-negative representation of the linearization coefficients of the product of Jacobi polynomials. Canad. J. Math. 33 (4), pp. 915–928.

ⓘ

External Links:

ISSN 0008-414X, Document, Link, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§18.18(v).

Encodings:

BibTeX

… ►

Integral representations for products of Airy functions. Z. Angew. Math. Phys. 46 (2), pp. 159–170.

ⓘ

External Links:

ISSN 0044-2275, Document, MathReview (P. K. Banerji), ZentralBlatt

Referenced by:

§9.11(iii), §9.11(v), §9.11(v), (9.5.6), §9.5(ii).

Encodings:

BibTeX

►

Integral representations for products of Airy functions. II. Cubic products. Z. Angew. Math. Phys. 48 (4), pp. 646–655.

ⓘ

External Links:

ISSN 0044-2275, Document, MathReview (E. R. Love), ZentralBlatt

Referenced by:

(9.11.16), (9.11.17), §9.11(iii), §9.11(v).

Encodings:

BibTeX

►

Integral representations for products of Airy functions. III. Quartic products. Z. Angew. Math. Phys. 48 (4), pp. 656–664.

ⓘ

External Links:

ISSN 0044-2275, Document, MathReview (E. R. Love), ZentralBlatt

Referenced by:

§9.11(iii), §9.11(v).

Encodings:

BibTeX

… ►

Research Institute for Symbolic Computation, Hagenberg im Mühlkreis, Austria.

ⓘ

Notes:

Software packages for hypergeometric and $q$-hypergeometric summation, sequences and power series, permutation groups, partition analysis, and difference equations.

External Links:

RISC Combinatorics Group

Referenced by:

6th item.

Encodings:

BibTeX

…

22: 32.8 Rational Solutions

… ►where the $Q_n(z)$ are monic polynomials (coefficient of highest power of $z$ is $1$) satisfying … ►

32.8.8 $$\sum _{m=0}^{\mathrm{\infty }}{p}_m(z){\lambda }^m=\exp \left(z\lambda -\frac{4}{3}{\lambda }^3\right).$$

ⓘ

Symbols:

$\exp z$: exponential function, $m$: integer, $z$: real and $p_m(z)$: polynomials

Permalink:

http://dlmf.nist.gov/32.8.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §32.8(ii), §32.8 and Ch.32

… ►For determinantal representations see Kajiwara and Masuda (1999). … ►For determinantal representations see Kajiwara and Ohta (1998) and Noumi and Yamada (1999). … ►For determinantal representations see Masuda et al. (2002). …

23: Errata

… ►For some classical polynomials we give some positive sums and discriminants. … ►

A sentence indicating that explicit representations for the coefficients $c_k$ are given in Volkmer (2023) was inserted just below (16.11.5).

… ►

Equations: (3.3.3_1), (3.3.3_2), (5.15.9) (suggested by Calvin Khor on 2021-09-04), (8.15.2), Pochhammer symbol representation in (10.17.1) for $a_k(\nu )$ coefficient, Pochhammer symbol representation in (11.9.4) for $a_k(\mu ,\nu )$ coefficient, and (12.14.4_5).

… ►

Pochhammer symbol representations for the functions $F_k(\nu )$ and $G_k(\nu )$ were inserted.

… ►

Clarifications regarding $t$-powers and asymptotics were added, along with extra citations.

…

24: 11.10 Anger–Weber Functions

… ►

§11.10(i) Definitions

… ►

§11.10(iii) Maclaurin Series

… ►

§11.10(x) Integrals and Sums

►For collections of integral representations and integrals see Erdélyi et al. (1954a, §§4.19 and 5.17), Marichev (1983, pp. 194–195 and 214–215), Oberhettinger (1972, p. 128), Oberhettinger (1974, §§1.12 and 2.7), Oberhettinger (1990, pp. 105 and 189–190), Prudnikov et al. (1990, §§1.5 and 2.8), Prudnikov et al. (1992a, §3.18), Prudnikov et al. (1992b, §3.18), and Zanovello (1977). ►For sums see Hansen (1975, pp. 456–457) and Prudnikov et al. (1990, §§6.4.2–6.4.3).

25: 16.8 Differential Equations

… ►

16.8.4 $$z^q{𝐷}^{q+1}w+\sum _{j=1}^q{z}^{j-1}({\alpha }_{j}z+{\beta }_j){𝐷}^{j}w+{\alpha }_{0}w=0,$$ $p\le q$,

ⓘ

Symbols:

$p$: nonnegative integer, $q$: nonnegative integer, $z$: complex variable, $𝐷$: differential operator, $w$: solution, ${\alpha }_j$: constant and ${\beta }_j$: constant

Referenced by:

§16.8(ii)

Permalink:

http://dlmf.nist.gov/16.8.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §16.8(ii), §16.8 and Ch.16

… ►

16.8.5 $$z^q(1-z){𝐷}^{q+1}w+\sum _{j=1}^q{z}^{j-1}({\alpha }_{j}z+{\beta }_j){𝐷}^{j}w+{\alpha }_{0}w=0,$$ $p=q+1$,

ⓘ

Symbols:

$p$: nonnegative integer, $q$: nonnegative integer, $z$: complex variable, $𝐷$: differential operator, $w$: solution, ${\alpha }_j$: constant and ${\beta }_j$: constant

Referenced by:

§16.8(ii)

Permalink:

http://dlmf.nist.gov/16.8.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §16.8(ii), §16.8 and Ch.16

… ►For other values of the $b_j$, series solutions in powers of $z$ (possibly involving also $\ln z$) can be constructed via a limiting process; compare §2.7(i) in the case of second-order differential equations. … ►

16.8.9 $$\left(\prod _{k=1}^{q+1}\mathrm{\Gamma }\left(a_k\right)/\prod _{k=1}^q\mathrm{\Gamma }\left(b_k\right)\right){}_{q+1}{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_{q+1}}{b_1,\mathrm{\dots },b_q};z)=\sum _{j=1}^{q+1}{\left(z_0-z\right)}^{-a_j}\sum _{n=0}^{\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(a_j+n\right)}{n!}\left(\prod _{\begin{array}{c}k=1\\ k\ne j\end{array}}^{q+1}\mathrm{\Gamma }\left(a_k-a_j-n\right)/\prod _{k=1}^q\mathrm{\Gamma }\left(b_k-a_j-n\right)\right){}_{q+1}{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1-a_j-n,\mathrm{\dots },a_{q+1}-a_j-n}{b_1-a_j-n,\mathrm{\dots },b_q-a_j-n};z_0){\left(z-z_0\right)}^{-n}.$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, $!$: factorial (as in $n!$), $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters and $b,b_1,\mathrm{\dots },b_q$: real or complex parameters

Referenced by:

§16.8(ii)

Permalink:

http://dlmf.nist.gov/16.8.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §16.8(ii), §16.8 and Ch.16

… ►In this reference it is also explained that in general when $q>1$ no simple representations in terms of generalized hypergeometric functions are available for the fundamental solutions near $z=1$. …

26: 30.11 Radial Spheroidal Wave Functions

… ►

30.11.3 $$S_n^{m(j)}(z,\gamma )=\frac{{(1-z^{-2})}^{\frac{1}{2}m}}{A_n^{-m}({\gamma }^2)}\sum _{2k\ge m-n}{a}_{n,k}^{-m}({\gamma }^2){\psi }_{n+2k}^{(j)}(\gamma z).$$

ⓘ

Defines:

$S_n^{m(j)}(z,\gamma )$: radial spheroidal wave function

Symbols:

$z$: complex variable, $m$: nonnegative integer, $n\ge m$: integer degree, ${\psi }_k^{(j)}(z)$: spherical function, $A_n^m({\gamma }^2)$ , ${\gamma }^2$: real parameter and $a_{n,k}^m({\gamma }^2)$: coefficients

A&S Ref:

21.9.1 (in different form)

Referenced by:

§30.11(i), §30.16(iii)

Permalink:

http://dlmf.nist.gov/30.11.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §30.11(i), §30.11 and Ch.30

… ►

30.11.4 $$A_n^{\pm m}({\gamma }^2)=\sum _{2k\ge \mp m-n}{(-1)}^k{a}_{n,k}^{\pm m}({\gamma }^2)(\ne 0).$$

ⓘ

Defines:

$A_n^m({\gamma }^2)$ (locally)

Symbols:

$m$: nonnegative integer, $n\ge m$: integer degree, ${\gamma }^2$: real parameter and $a_{n,k}^m({\gamma }^2)$: coefficients

Referenced by:

§30.5, §30.6

Permalink:

http://dlmf.nist.gov/30.11.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §30.11(i), §30.11 and Ch.30

… ►For asymptotic expansions in negative powers of $z$ see Meixner and Schäfke (1954, p. 293). … ►

§30.11(vi) Integral Representations

…