Appell%0Afunctions

SearchAdvancedHelp

(0.001 seconds)

11—20 of 701 matching pages

11: 16.1 Special Notation

… ►The main functions treated in this chapter are the generalized hypergeometric function ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$, the Appell (two-variable hypergeometric) functions $F_1(\alpha ;\beta ,{\beta }^{\prime };\gamma ;x,y)$, $F_2(\alpha ;\beta ,{\beta }^{\prime };\gamma ,{\gamma }^{\prime };x,y)$, $F_3(\alpha ,{\alpha }^{\prime };\beta ,{\beta }^{\prime };\gamma ;x,y)$, $F_4(\alpha ,\beta ;\gamma ,{\gamma }^{\prime };x,y)$, and the Meijer $G$-function $G_{p,q}^{m,n}(z;\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q})$. Alternative notations are ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$, ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$, and ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ for the generalized hypergeometric function, $F_1(\alpha ,\beta ,{\beta }^{\prime };\gamma ;x,y)$, $F_2(\alpha ,\beta ,{\beta }^{\prime };\gamma ,{\gamma }^{\prime };x,y)$, $F_3(\alpha ,{\alpha }^{\prime },\beta ,{\beta }^{\prime };\gamma ;x,y)$, $F_4(\alpha ,\beta ;\gamma ,{\gamma }^{\prime };x,y)$, for the Appell functions, and $G_{p,q}^{m,n}(z;𝐚;𝐛)$ for the Meijer $G$-function.

12: Bibliography F

… ►

Tablicy značeniĭ funkcii $w(z)=e^{-z^2}(1+\frac{2i}{\sqrt{\pi }}{\int }_0^z{e}^{t^2}𝑑t)$ ot kompleksnogo argumenta. Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow (Russian).

ⓘ

Notes:

In Russian. English translation, see Faddeyeva and Terent’ev (1961)

External Links:

MathReview (John Todd), ZentralBlatt

Referenced by:

§7.21.

Encodings:

BibTeX

►

Tables of Values of the Function $w(z)=e^{-z^2}(1+2i{\pi }^{-1/2}{\int }_0^z{e}^{t^2}𝑑t)$ for Complex Argument. Edited by V. A. Fok; translated from the Russian by D. G. Fry. Mathematical Tables Series, Vol. 11, Pergamon Press, Oxford.

ⓘ

External Links:

MathReview, ZentralBlatt

Referenced by:

V. N. Faddeeva and N. M. Terent’ev (1954).

Encodings:

BibTeX

… ►

The third Appell function for one large variable. J. Approx. Theory 165, pp. 60–69.

ⓘ

External Links:

ISSN 0021-9045, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§16.13, §16.13.

Encodings:

BibTeX

… ►

The second Appell function for one large variable. Mediterr. J. Math. 10 (4), pp. 1853–1865.

ⓘ

External Links:

ISSN 1660-5446, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§16.13, §16.13.

Encodings:

BibTeX

…

13: 19.25 Relations to Other Functions

… ►with $\alpha \ne 0$. … ►

§19.25(vii) Hypergeometric Function

… ►For these results and extensions to the Appell function $F_1$ (§16.13) and Lauricella’s function $F_D$ see Carlson (1963). ($F_1$ and $F_D$ are equivalent to the $R$-function of 3 and $n$ variables, respectively, but lack full symmetry.)

14: Bibliography

… ►

Recurrence relations for the Fresnel integral ${\int }_0^{\mathrm{\infty }}\frac{\exp (-ct)dt}{\sqrt{t}(1+t^2)}$ and similar integrals. Comm. ACM 17 (8), pp. 480–481.

ⓘ

External Links:

ISSN 0001-0782, Document, MathReview, ZentralBlatt

Referenced by:

§6.18(ii).

Encodings:

BibTeX

… ►

Algorithm 511: CDC 6600 subroutines IBESS and JBESS for Bessel functions $I_{\nu }(x)$ and $J_{\nu }(x)$, $x\ge 0$, $\nu \ge 0$ . ACM Trans. Math. Software 3 (1), pp. 93–95.

ⓘ

Notes:

Single-precision Fortran, maximum accuracy 16S.

External Links:

ISSN 0098-3500, Document, Companion paper. Erratum. GAMS (module 511; package TOMS)

Referenced by:

1st item, 1st item.

Encodings:

BibTeX

… ►

Summations and transformations for basic Appell series. J. London Math. Soc. (2) 4, pp. 618–622.

ⓘ

External Links:

MathReview (R. N. Jain), ZentralBlatt

Referenced by:

§17.11.

Encodings:

BibTeX

… ►

Fonctions hypergéométriques et hypersphériques. Polynomes d’Hermite. Gauthier-Villars, Paris.

ⓘ

External Links:

ZentralBlatt

Referenced by:

§16.13, §16.15, Ch.16.

Encodings:

BibTeX

…

15: Bibliography L

… ►

Ratios of Bessel functions and roots of $\alpha J_{\nu }(x)+x{J}_{\nu }^{\prime }(x)=0$ . J. Math. Anal. Appl. 240 (1), pp. 174–204.

ⓘ

External Links:

ISSN 0022-247X, Document, MathReview (Boro Döring), ZentralBlatt

Referenced by:

§10.14, §10.15, §10.21(xii).

Encodings:

BibTeX

… ►

Note sur la fonction $𝔎(w,x,s)={\sum }_{k=0}^{\mathrm{\infty }}\frac{e^{2k\pi ix}}{{(w+k)}^s}$ . Acta Math. 11 (1-4), pp. 19–24 (French).

ⓘ

External Links:

ISSN 0001-5962, Document, MathReview Entry, ZentralBlatt

Referenced by:

§25.14(i), Notations K.

Encodings:

BibTeX

… ►

Orthogonal polynomials for exponential weights $x^{2\rho }{e}^{-2Q(x)}$ on $[0,d)$ . J. Approx. Theory 134 (2), pp. 199–256.

ⓘ

External Links:

ISSN 0021-9045, Document, Link, MathReview (Leonid B. Golinskiĭ)

Referenced by:

§18.32.

Encodings:

BibTeX

… ►

Asymptotics of the first Appell function $F_1$ with large parameters II. Integral Transforms Spec. Funct. 24 (12), pp. 982–999.

ⓘ

External Links:

ISSN 1065-2469, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§16.13, §16.13.

Encodings:

BibTeX

►

Asymptotics of the first Appell function $F_1$ with large parameters. Integral Transforms Spec. Funct. 24 (9), pp. 715–733.

ⓘ

External Links:

ISSN 1065-2469, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§16.13, §16.13.

Encodings:

BibTeX

…

16: Bibliography B

… ►

Ueber die Entwickelung einer willkürlichen Function nach den Nennern des Kettenbruches für ${\int }_{-\mathrm{\infty }}^0\frac{\phi (\xi )d\xi }{z-\xi }$ .

ⓘ

Notes:

Dissertation, Göttingen

Referenced by:

§18.2(xi).

Encodings:

BibTeX

… ►

Expansions of Appell’s double hypergeometric functions. Quart. J. Math., Oxford Ser. 11, pp. 249–270.

ⓘ

External Links:

Document, MathReview (G. Szegő), ZentralBlatt

Referenced by:

§15.16, (16.15.4), §16.15, §16.16(i).

Encodings:

BibTeX

►

Expansions of Appell’s double hypergeometric functions. II. Quart. J. Math., Oxford Ser. 12, pp. 112–128.

ⓘ

External Links:

Document, MathReview (G. Szegő), ZentralBlatt

Referenced by:

§16.16(i).

Encodings:

BibTeX

…

17: Bibliography C

… ►

Quadratic transformations of Appell functions. SIAM J. Math. Anal. 7 (2), pp. 291–304.

ⓘ

External Links:

Document, MathReview (L. J. Slater), ZentralBlatt

Referenced by:

§16.16(ii), §19.22(i).

Encodings:

BibTeX

… ►

A Fortran subroutine for the Bessel function $J_n(x)$ of order $0$ to $10$ . Comput. Phys. Comm. 21 (1), pp. 109–118.

ⓘ

Notes:

Double-precision Fortran, maximum accuracy 16S.

External Links:

ISSN 0010-4655, Document, Code

Referenced by:

4th item.

Encodings:

BibTeX

…

18: Errata

… ►

17.11.2 $${\mathrm{\Phi }}^{(2)}(a;b,b^{\prime };c,c^{\prime };q;x,y)=\frac{{(b,ax;q)}_{\mathrm{\infty }}}{{(c,x;q)}_{\mathrm{\infty }}}\sum _{n,r\geqq 0}\frac{{(a,b^{\prime };q)}_n{(c/b,x;q)}_r{b}^r{y}^n}{{(q,c^{\prime };q)}_n{(q;q)}_r{(ax;q)}_{n+r}}$$

The factor ${\left(q\right)}_r$ originally used in the denominator has been corrected to be ${(q;q)}_r$.

… ►

The constraint $a\ne 0$ was added.

… ►

The vector at the origin, previously given as $0$, has been clarified to read $\mathrm{𝟎}$.

… ►

The notation used for the $q$-Appell functions in Equations (17.4.5), (17.4.6),(17.4.7), (17.4.8), (17.11.1), (17.11.2) and (17.11.3) was updated to explicitly include the argument $q$, as used in Gasper and Rahman (2004).

… ►

This equation is true only for $n>0$. Previously, $n=0$ was also allowed.

Reported 2012-05-14 by Vladimir Yurovsky.

…

19: 33.24 Tables

… ►

Abramowitz and Stegun (1964, Chapter 14) tabulates $F_0(\eta ,\rho )$, $G_0(\eta ,\rho )$, $F_0^{\prime }(\eta ,\rho )$, and $G_0^{\prime }(\eta ,\rho )$ for $\eta =0.5(.5)20$ and $\rho =1(1)20$, 5S; $C_0\left(\eta \right)$ for $\eta =0(.05)3$, 6S.

►

Curtis (1964a) tabulates $P_{\mathrm{\ell }}(ϵ,r)$, $Q_{\mathrm{\ell }}(ϵ,r)$ (§33.1), and related functions for $\mathrm{\ell }=0,1,2$ and $ϵ=-2(.2)2$, with $x=0(.1)4$ for and $x=0(.1)10$ for $ϵ\ge 0$; 6D.

…

20: 26.15 Permutations: Matrix Notation

… ►The set ${𝔖}_n$ (§26.13) can be identified with the set of $n\times n$ matrices of 0’s and 1’s with exactly one 1 in each row and column. The permutation $\sigma$ corresponds to the matrix in which there is a 1 at the intersection of row $j$ with column $\sigma (j)$, and 0’s in all other positions. … ►Define $r_0(B)=1$. … ►The Ferrers board of shape $(b_1,b_2,\mathrm{\dots },b_n)$, $0\le b_1\le b_2\le \mathrm{\cdots }\le b_n$, is the set $B=\{(j,k)|\mathrm{\hspace{0.25em}1}\le j\le n,1\le k\le b_j\}$. …If $B$ is the Ferrers board of shape $(0,1,2,\mathrm{\dots },n-1)$, then …