DLMF: Untitled Document

… ►

C\left(n\right)

is the Catalan number. …(Sixty-six equivalent definitions of

C\left(n\right)

are given in Stanley (1999, pp. 219–229).) … ►

26.5.3

C\left(n+1\right)=\sum_{k=0}^{n}C\left(k\right)C\left(n-k\right),

ⓘ

Symbols:: $C\left(\NVar{n}\right)$ : Catalan number, $k$ : nonnegative integer and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/26.5.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §26.5(iii), §26.5 and Ch.26

►

26.5.4

C\left(n+1\right)=\frac{2(2n+1)}{n+2}C\left(n\right),

ⓘ

Symbols:: $C\left(\NVar{n}\right)$ : Catalan number and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/26.5.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §26.5(iii), §26.5 and Ch.26

… ►

26.5.7

\lim_{n\to\infty}\frac{C\left(n+1\right)}{C\left(n\right)}=4.

ⓘ

Symbols:: $C\left(\NVar{n}\right)$ : Catalan number and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/26.5.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §26.5(iv), §26.5 and Ch.26

… ►

P. I. Hadži (1975a) Certain integrals that contain a probability function. Bul. Akad. Štiince RSS Moldoven. 1975 (2), pp. 86–88, 95 (Russian).

ⓘ

External Links:: MathReview (C. Tanaka)
Referenced by:: §7.14(iii).
Encodings:: BibTeX

… ►

P. I. Hadži (1976a) Expansions for the probability function in series of Čebyšev polynomials and Bessel functions. Bul. Akad. Štiince RSS Moldoven. 1976 (1), pp. 77–80, 96 (Russian).

ⓘ

External Links:: MathReview (V. L. Makarov)
Referenced by:: §7.14(iii).
Encodings:: BibTeX

►

P. I. Hadži (1976b) Integrals that contain a probability function of complicated arguments. Bul. Akad. Štiince RSS Moldoven. 1976 (1), pp. 80–84, 96 (Russian).

ⓘ

External Links:: MathReview (V. L. Makarov)
Referenced by:: §7.14(iii).
Encodings:: BibTeX

►

P. I. Hadži (1978) Sums with cylindrical functions that reduce to the probability function and to related functions. Bul. Akad. Shtiintse RSS Moldoven. 1978 (3), pp. 80–84, 95 (Russian).

ⓘ

External Links:: MathReview (M. Lawrence Glasser)
Referenced by:: §7.14(iii).
Encodings:: BibTeX

… ►

J. Hammack, D. McCallister, N. Scheffner, and H. Segur (1995) Two-dimensional periodic waves in shallow water. II. Asymmetric waves. J. Fluid Mech. 285, pp. 95–122.

ⓘ

External Links:: ISSN 0022-1120, Document, MathReview
Referenced by:: Figure 21.9.1, Figure 21.9.1, Figure 21.9.2, Figure 21.9.2, §21.9, Sidebar 21.SB1, Sidebar 21.SB2.
Encodings:: BibTeX

…

… ►

I. C. Tang (1969) Some definite integrals and Fourier series for Jacobian elliptic functions. Z. Angew. Math. Mech. 49, pp. 95–96.

ⓘ

External Links:: ISSN 0044-2267, Document, MathReview (B. C. Carlson), ZentralBlatt
Referenced by:: §22.11.
Encodings:: BibTeX

… ►

N. M. Temme (1994a) A set of algorithms for the incomplete gamma functions. Probab. Engrg. Inform. Sci. 8, pp. 291–307.

ⓘ

Notes:: Includes Pascal program
External Links:: FullText
Referenced by:: §5.24(ii), §7.25(ii), §8.28(ii).
Encodings:: BibTeX

… ►

N. M. Temme (1995b) Bernoulli polynomials old and new: Generalizations and asymptotics. CWI Quarterly 8 (1), pp. 47–66.

ⓘ

External Links:: ISSN 0922-5366, FullText, MathReview (Michael Hauss), ZentralBlatt
Referenced by:: §24.11, §24.16(i).
Encodings:: BibTeX

… ►

P. Terwilliger (2013) The universal Askey-Wilson algebra and DAHA of type

(C^{\vee}_{1},C_{1})

. SIGMA 9, pp. Paper 047, 40 pp..

ⓘ

External Links:: ISSN 1815-0659, Link, MathReview (Renato Álvarez-Nodarse), ZentralBlatt
Referenced by:: §18.38(iii).
Encodings:: BibTeX

… ►

J. Todd (1975) The lemniscate constants. Comm. ACM 18 (1), pp. 14–19.

ⓘ

Notes:: Collection of articles honoring Alston S. Householder. For corrigendum see same journal v. 18 (1975), no. 8, p. 462
External Links:: ISSN 0001-0782, Document, MathReview (L. Fox), ZentralBlatt
Referenced by:: §19.20(i), §19.20(iv), §19.36(iii).
Encodings:: BibTeX

…

… ►where ►

33.20.4

{\sf F}_{k}(\ell;r)=\sum_{p=2k}^{3k}(2r)^{(p+1)/2}C_{k,p}J_{2\ell+1+p}\left(% \sqrt{8r}\right),

r>0

,

ⓘ

Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $k$ : nonnegative integer, $\ell$ : nonnegative integer, $r$ : real variable, $\mathsf{F}_{k}(\ell;r)$ : coefficient and $C_{k,p}$ : coefficients
Permalink:: http://dlmf.nist.gov/33.20.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §33.20(ii), §33.20 and Ch.33

►

33.20.5

{\sf F}_{k}(\ell;r)=\sum_{p=2k}^{3k}(-1)^{\ell+1+p}(2|r|)^{(p+1)/2}C_{k,p}I_{2% \ell+1+p}\left(\sqrt{8|r|}\right),

r<0

.

ⓘ

Symbols:: $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $k$ : nonnegative integer, $\ell$ : nonnegative integer, $r$ : real variable, $\mathsf{F}_{k}(\ell;r)$ : coefficient and $C_{k,p}$ : coefficients
Permalink:: http://dlmf.nist.gov/33.20.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §33.20(ii), §33.20 and Ch.33

►The functions

J

and

I

are as in §§10.2(ii), 10.25(ii), and the coefficients

C_{k,p}

are given by

C_{0,0}=1

,

C_{1,0}=0

, and … ►The functions

Y

and

K

are as in §§10.2(ii), 10.25(ii), and the coefficients

C_{k,p}

are given by (33.20.6). …

… ►

M. Abramowitz and P. Rabinowitz (1954) Evaluation of Coulomb wave functions along the transition line. Physical Rev. (2) 96, pp. 77–79.

ⓘ

External Links:: Document, MathReview (A. Erdélyi), ZentralBlatt
Referenced by:: §33.12(i).
Encodings:: BibTeX

… ►

H. Airault, H. P. McKean, and J. Moser (1977) Rational and elliptic solutions of the Korteweg-de Vries equation and a related many-body problem. Comm. Pure Appl. Math. 30 (1), pp. 95–148.

ⓘ

External Links:: MathReview (Alexander A. Pankov), ZentralBlatt
Referenced by:: §23.21(ii).
Encodings:: BibTeX

… ►

G. Almkvist and B. Berndt (1988) Gauss, Landen, Ramanujan, the arithmetic-geometric mean, ellipses,

\pi

, and the Ladies Diary. Amer. Math. Monthly 95 (7), pp. 585–608.

ⓘ

External Links:: ISSN 0002-9890, FullText, Document, MathReview (R. A. Askey), ZentralBlatt
Referenced by:: §19.9(i).
Encodings:: BibTeX

… ►

G. E. Andrews and D. Foata (1980) Congruences for the

q

-secant numbers. European J. Combin. 1 (4), pp. 283–287.

ⓘ

External Links:: ISSN 0195-6698, MathReview (J. E. Cigler), ZentralBlatt
Referenced by:: §24.16(iii).
Encodings:: BibTeX

►

G. E. Andrews, I. P. Goulden, and D. M. Jackson (1986) Shanks’ convergence acceleration transform, Padé approximants and partitions. J. Combin. Theory Ser. A 43 (1), pp. 70–84.

ⓘ

External Links:: ISSN 0097-3165, Document, MathReview (Kenneth A. Jukes), ZentralBlatt
Referenced by:: §17.18.
Encodings:: BibTeX

…

… ►

6.14.1

\int_{0}^{\infty}e^{-at}E_{1}\left(t\right)\,\mathrm{d}t=\frac{1}{a}\ln\left(1% +a\right),

\Re a>-1

,

ⓘ

Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $E_{1}\left(\NVar{z}\right)$ : exponential integral, $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function and $\Re$ : real part
A&S Ref:: 5.1.34 (special case of)
Permalink:: http://dlmf.nist.gov/6.14.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §6.14(i), §6.14 and Ch.6

… ►

6.14.4

\int_{0}^{\infty}{E_{1}}^{2}\left(t\right)\,\mathrm{d}t=2\ln 2,

ⓘ

Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $E_{1}\left(\NVar{z}\right)$ : exponential integral, $\int$ : integral and $\ln\NVar{z}$ : principal branch of logarithm function
A&S Ref:: 5.1.33
Permalink:: http://dlmf.nist.gov/6.14.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §6.14(ii), §6.14 and Ch.6

… ►For collections of integrals, see Apelblat (1983, pp. 110–123), Bierens de Haan (1939, pp. 373–374, 409, 479, 571–572, 637, 664–673, 680–682, 685–697), Erdélyi et al. (1954a, vol. 1, pp. 40–42, 96–98, 177–178, 325), Geller and Ng (1969), Gradshteyn and Ryzhik (2000, §§5.2–5.3 and 6.2–6.27), Marichev (1983, pp. 182–184), Nielsen (1906b), Oberhettinger (1974, pp. 139–141), Oberhettinger (1990, pp. 53–55 and 158–160), Oberhettinger and Badii (1973, pp. 172–179), Prudnikov et al. (1986b, vol. 2, pp. 24–29 and 64–92), Prudnikov et al. (1992a, §§3.4–3.6), Prudnikov et al. (1992b, §§3.4–3.6), and Watrasiewicz (1967).

… ►

F. W. Schäfke and D. Schmidt (1966) Ein Verfahren zur Berechnung des charakteristischen Exponenten der Mathieuschen Differentialgleichung III. Numer. Math. 8 (1), pp. 68–71.

ⓘ

Notes:: Includes Algol program
External Links:: ISSN 0029-599X, Document, MathReview (S. C. van Veen), ZentralBlatt
Referenced by:: §28.36(ii).
Encodings:: BibTeX

… ►

B. I. Schneider, X. Guan, and K. Bartschat (2016) Time propagation of partial differential equations using the short iterative Lanczos method and finite-element discrete variable representation. Adv. Quantum Chem. 72, pp. 95–127.

ⓘ

Referenced by:: §18.38(i).
Encodings:: BibTeX

… ►

J. B. Seaborn (1991) Hypergeometric Functions and Their Applications. Texts in Applied Mathematics, Vol. 8, Springer-Verlag, New York.

ⓘ

External Links:: ISBN 0-387-97558-6, MathReview (Wolfgang Bühring), ZentralBlatt
Referenced by:: Erratum (V1.2.0) Section 18.39.
Encodings:: BibTeX

… ►

M. J. Seaton (1982) Coulomb functions analytic in the energy. Comput. Phys. Comm. 25 (1), pp. 87–95.

ⓘ

Notes:: Double-precision Fortran code.
External Links:: Document, Code
Referenced by:: §33.1, §33.14(iv), 9th item.
Encodings:: BibTeX

… ►

R. Spigler, M. Vianello, and F. Locatelli (1999) Liouville-Green-Olver approximations for complex difference equations. J. Approx. Theory 96 (2), pp. 301–322.

ⓘ

External Links:: ISSN 0021-9045, Document, MathReview (Dumitru Acu), ZentralBlatt
Referenced by:: §2.9(iii).
Encodings:: BibTeX

…

… ►

R. C. McCann (1977) Inequalities for the zeros of Bessel functions. SIAM J. Math. Anal. 8 (1), pp. 166–170.

ⓘ

External Links:: ISSN 1095-7154, Document, MathReview (Mary L. Boas), ZentralBlatt
Referenced by:: §10.21(iv).
Encodings:: BibTeX

… ►

Fr. Mechel (1966) Calculation of the modified Bessel functions of the second kind with complex argument. Math. Comp. 20 (95), pp. 407–412.

ⓘ

External Links:: ISSN 0025-5718, FullText, MathReview (H. E. Fettis), ZentralBlatt
Referenced by:: §10.74(iii).
Encodings:: BibTeX

… ►

J. N. Merner (1962) Algorithm 149: Complete elliptic integral. Comm. ACM 5 (12), pp. 605.

ⓘ

Notes:: Includes Algol program, maximum accuracy 10D. For remark see ACM Trans. Math. Software 4 (1978), p. 95
External Links:: ISSN 0001-0782, Document
Referenced by:: §19.39(ii).
Encodings:: BibTeX

… ►

R. Metzler, J. Klafter, and J. Jortner (1999) Hierarchies and logarithmic oscillations in the temporal relaxation patterns of proteins and other complex systems. Proc. Nat. Acad. Sci. U .S. A. 96 (20), pp. 11085–11089.

ⓘ

External Links:: FullText
Referenced by:: §8.24(i).
Encodings:: BibTeX

… ►

E. W. Montroll (1964) Lattice Statistics. In Applied Combinatorial Mathematics, E. F. Beckenbach (Ed.), University of California Engineering and Physical Sciences Extension Series, pp. 96–143.

ⓘ

Notes:: Reprinted by Robert E. Krieger Publishing Co. in 1981.
External Links:: ZentralBlatt
Referenced by:: §26.20.
Encodings:: BibTeX

…

… ►

5.13.4

\int_{-\infty}^{\infty}\frac{\,\mathrm{d}t}{\Gamma\left(a+t\right)\Gamma\left(% b+t\right)\Gamma\left(c-t\right)\Gamma\left(d-t\right)}=\frac{\Gamma\left(a+b+% c+d-3\right)}{\Gamma\left(a+c-1\right)\Gamma\left(a+d-1\right)\Gamma\left(b+c-% 1\right)\Gamma\left(b+d-1\right)},

\Re\left(a+b+c+d\right)>3

.

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\Re$ : real part, $a$ : real or complex variable and $b$ : real or complex variable
Referenced by:: §5.13
Permalink:: http://dlmf.nist.gov/5.13.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §5.13, §5.13 and Ch.5

… ►

5.13.5

\frac{1}{4\pi}\int_{-\infty}^{\infty}\frac{\prod_{k=1}^{4}\Gamma\left(a_{k}+it% \right)\Gamma\left(a_{k}-it\right)}{\Gamma\left(2it\right)\Gamma\left(-2it% \right)}\,\mathrm{d}t=\frac{\prod_{1\leq j<k\leq 4}\Gamma\left(a_{j}+a_{k}% \right)}{\Gamma\left(a_{1}+a_{2}+a_{3}+a_{4}\right)},

\Re\left(a_{k}\right)>0

,

k=1,2,3,4

.

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\Re$ : real part, $j$ : nonnegative integer, $k$ : nonnegative integer and $a$ : real or complex variable
Referenced by:: §5.13
Permalink:: http://dlmf.nist.gov/5.13.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §5.13, §5.13 and Ch.5

…

… ►

34.5.1

\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ 0&j_{3}&j_{2}\end{Bmatrix}=\frac{(-1)^{J}}{\left((2j_{2}+1)(2j_{3}+1)\right)^{% \frac{1}{2}}},

ⓘ

Symbols:: $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half. and $J$ : sum
Permalink:: http://dlmf.nist.gov/34.5.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §34.5(i), §34.5 and Ch.34

►

34.5.2

\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ \frac{1}{2}&j_{3}-\frac{1}{2}&j_{2}+\frac{1}{2}\end{Bmatrix}=(-1)^{J}\left(% \frac{(j_{1}+j_{3}-j_{2})(j_{1}+j_{2}-j_{3}+1)}{(2j_{2}+1)(2j_{2}+2)2j_{3}(2j_% {3}+1)}\right)^{\frac{1}{2}},

ⓘ

Symbols:: $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half. and $J$ : sum
Permalink:: http://dlmf.nist.gov/34.5.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §34.5(i), §34.5 and Ch.34

►

34.5.3

\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ \frac{1}{2}&j_{3}-\frac{1}{2}&j_{2}-\frac{1}{2}\end{Bmatrix}=(-1)^{J}\left(% \frac{(j_{2}+j_{3}-j_{1})(j_{1}+j_{2}+j_{3}+1)}{2j_{2}(2j_{2}+1)2j_{3}(2j_{3}+% 1)}\right)^{\frac{1}{2}},

ⓘ

Symbols:: $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half. and $J$ : sum
Permalink:: http://dlmf.nist.gov/34.5.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §34.5(i), §34.5 and Ch.34

►

34.5.4

\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ 1&j_{3}-1&j_{2}-1\end{Bmatrix}=(-1)^{J}\left(\frac{J(J+1)(J-2j_{1})(J-2j_{1}-1% )}{(2j_{2}-1)2j_{2}(2j_{2}+1)(2j_{3}-1)2j_{3}(2j_{3}+1)}\right)^{\frac{1}{2}},

ⓘ

Symbols:: $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half. and $J$ : sum
Permalink:: http://dlmf.nist.gov/34.5.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §34.5(i), §34.5 and Ch.34

… ►For further recursion relations see Varshalovich et al. (1988, §9.6) and Edmonds (1974, pp. 98–99). …

.02%E5%B9%B4%E4%B8%96%E7%95%8C%E6%9D%AF%E6%B2%99%E7%89%B9%E5%B0%8F%E7%BB%84%E8%B5%9B%E3%80%8E%E7%BD%91%E5%9D%80%3Amxsty.cc%E3%80%8F.%E4%B8%96%E7%95%8C%E6%9D%AF%E8%B6%B3%E5%BD%A9%E7%8E%A9%E6%B3%95%E8%A7%84%E5%88%99.m6q3s2-oeoo02w0g

11—20 of 737 matching pages

11: 26.5 Lattice Paths: Catalan Numbers

12: Bibliography H

13: Bibliography T

14: 33.20 Expansions for Small $|\epsilon|$

15: Bibliography

16: 6.14 Integrals

17: Bibliography S

18: Bibliography M

19: 5.13 Integrals

20: 34.5 Basic Properties: $\mathit{6j}$ Symbol