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11: Bibliography G

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Some elementary inequalities relating to the gamma and incomplete gamma function. J. Math. Phys. 38 (1), pp. 77–81.

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External Links:

MathReview (H. O. Pollak), ZentralBlatt

Referenced by:

§5.6(i), §6.8, §7.8, §8.10.

Encodings:

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Computational aspects of three-term recurrence relations. SIAM Rev. 9 (1), pp. 24–82.

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External Links:

Document, MathReview (E. Frank), ZentralBlatt

Referenced by:

§10.74(iv), §14.32, §3.10(iii), §3.6(iii).

Encodings:

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Questions of Numerical Condition Related to Polynomials. In Studies in Numerical Analysis, G. H. Golub (Ed.), pp. 140–177.

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External Links:

MathReview Entry, ZentralBlatt

Referenced by:

§3.8(vi).

Encodings:

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Algorithm 939: computation of the Marcum Q-function. ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.

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Notes:

Includes Fortran program claiming 12S accuracy

External Links:

ISSN 0098-3500, Link, Document, MathReview Entry, ZentralBlatt

Referenced by:

3rd item, §10.77(ix).

Encodings:

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Fourier transforms related to a root system of rank 1. Transform. Groups 12 (1), pp. 77–116.

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External Links:

ISSN 1083-4362, Link, MathReview (Genkai Zhang), ZentralBlatt

Referenced by:

§18.28(xi), §18.38(iii).

Encodings:

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12: 26.5 Lattice Paths: Catalan Numbers

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§26.5(i) Definitions

… ►It counts the number of lattice paths from $(0,0)$ to $(n,n)$ that stay on or above the line $y=x$. … ►

§26.5(iii) Recurrence Relations

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26.5.6 $$C\left(n\right)\sim \frac{4^n}{\sqrt{\pi n^3}},$$ $n\to \mathrm{\infty }$,

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Symbols:

$C\left(n\right)$: Catalan number, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/26.5.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §26.5(iv), §26.5 and Ch.26

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26.5.7 $$\underset{n\to \mathrm{\infty }}{lim}\frac{C\left(n+1\right)}{C\left(n\right)}=4.$$

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Symbols:

$C\left(n\right)$: 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

13: Bibliography M

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Connection between quantum systems involving the fourth Painlevé transcendent and $k$-step rational extensions of the harmonic oscillator related to Hermite exceptional orthogonal polynomial. J. Math. Phys. 57 (5), pp. Paper 052101, 15 pp..

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External Links:

ISSN 0022-2488, Document, Link, MathReview Entry

Referenced by:

§18.38(iii).

Encodings:

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Recursion relations for the $3$-$j$ symbols. Nuclear Physics A 113 (1), pp. 215–220.

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External Links:

Document

Referenced by:

§34.3(iii).

Encodings:

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Evaluation of complex logarithms and related functions. SIAM J. Numer. Anal. 18 (4), pp. 744–750.

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External Links:

ISSN 0036-1429, Document, MathReview (M. Brannigan), ZentralBlatt

Referenced by:

§4.45(ii).

Encodings:

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A new symmetry related to $\mathrm{𝑆𝑈}(n)$ for classical basic hypergeometric series. Adv. in Math. 57 (1), pp. 71–90.

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External Links:

ISSN 0001-8708, Document, MathReview (George E. Andrews), ZentralBlatt

Referenced by:

§17.15.

Encodings:

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The $q$-analogue of the Laguerre polynomials. J. Math. Anal. Appl. 81 (1), pp. 20–47.

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External Links:

ISSN 0022-247X, Document, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§18.27(v).

Encodings:

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14: Bibliography B

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A short table of the functions ${\mathrm{Ki}}_n(x)$, from $n=1$ to $n=16$ . Phil. Mag. Series 7 20, pp. 343–347.

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External Links:

Document, ZentralBlatt

Referenced by:

§10.43(iii), 2nd item.

Encodings:

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Tables Relating to the Radial Mathieu Functions. Vol. 1: Functions of the First Kind. U.S. Government Printing Office, Washington, D.C..

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External Links:

MathReview (C. J. Bouwkamp)

Referenced by:

1st item.

Encodings:

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Tables Relating to the Radial Mathieu Functions. Vol. 2: Functions of the Second Kind. U.S. Government Printing Office, Washington, D.C..

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External Links:

MathReview (C. J. Bouwkamp)

Referenced by:

2nd item, 1st item.

Encodings:

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Bessel functions and modular relations of higher type and hyperbolic differential equations. Comm. Sém. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.] 1952 (Tome Supplementaire), pp. 12–20.

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External Links:

MathReview (S. C. van Veen), ZentralBlatt

Referenced by:

§35.5(i).

Encodings:

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Concerning the zeros of some functions related to Bessel functions. J. Mathematical Phys. 10 (9), pp. 1729–1744.

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External Links:

Document, MathReview (C. V. Coffman), ZentralBlatt

Referenced by:

§10.21(xi).

Encodings:

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15: 26.9 Integer Partitions: Restricted Number and Part Size

… ►Conjugation establishes a one-to-one correspondence between partitions of $n$ into at most $k$ parts and partitions of $n$ into parts with largest part less than or equal to $k$. It follows that $p_k\left(n\right)$ also equals the number of partitions of $n$ into parts that are less than or equal to $k$. … ►

§26.9(iii) Recurrence Relations

… ►where the inner sum is taken over all positive divisors of $t$ that are less than or equal to $k$. … ►As $n\to \mathrm{\infty }$ with $k$ fixed, …

16: 26.3 Lattice Paths: Binomial Coefficients

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§26.3(i) Definitions

► $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$ is the number of ways of choosing $n$ objects from a collection of $m$ distinct objects without regard to order. $\left(\genfrac{}{}{0.0pt}{}{m+n}{n}\right)$ is the number of lattice paths from $(0,0)$ to $(m,n)$. …The number of lattice paths from $(0,0)$ to $(m,n)$, $m\le n$, that stay on or above the line $y=x$ is $\left(\genfrac{}{}{0.0pt}{}{m+n}{m}\right)-\left(\genfrac{}{}{0.0pt}{}{m+n}{m-1}\right).$ … ►

§26.3(iii) Recurrence Relations

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17: Tom M. Apostol

… ►Apostol was born on August 20, 1923. … ►He was also a coauthor of three textbooks written to accompany the physics telecourse The Mechanical Universe …and Beyond. … ►In 1998, the Mathematical Association of America (MAA) awarded him the annual Trevor Evans Award, presented to authors of an exceptional article that is accessible to undergraduates, for his piece entitled “What Is the Most Surprising Result in Mathematics?” (Answer: the prime number theorem). … Ford Award, given to recognize authors of articles of expository excellence. … ►

25 Zeta and Related Functions

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18: 1.12 Continued Fractions

… ► $C_n$ is called the $n$th approximant or convergent to $C$ . … ►

Recurrence Relations

… ►A continued fraction converges if the convergents $C_n$ tend to a finite limit as $n\to \mathrm{\infty }$. … ►and the even and odd parts of the continued fraction converge to finite values. … ►For analytical and numerical applications of continued fractions to special functions see §3.10. …

19: 26.12 Plane Partitions

… ►It is useful to be able to visualize a plane partition as a pile of blocks, one block at each lattice point $(h,j,k)\in \pi$. … ►A plane partition is self-complementary if it is equal to its complement. … ►

§26.12(iii) Recurrence Relation

… ►As $n\to \mathrm{\infty }$ … ►Addendum: To ten decimal places, …

20: Bibliography W

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Asymptotics of orthogonal polynomials via recurrence relations. Anal. Appl. (Singap.) 10 (2), pp. 215–235.

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External Links:

ISSN 0219-5305, Document, Link, MathReview (Li Zhong Peng), ZentralBlatt

Referenced by:

§2.9(iii), §2.9(iii).

Encodings:

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The Nahm equations, finite-gap potentials and Lamé functions. J. Phys. A 20 (10), pp. 2679–2683.

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External Links:

ISSN 0305-4470, Document, MathReview (James M. D’Archangelo), ZentralBlatt

Referenced by:

§29.19(ii).

Encodings:

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Some useful integrals of $\mathrm{Si}(x),$ $\mathrm{Ci}(x)$ and related integrals. Optica Acta 14 (3), pp. 317–322.

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External Links:

Document, MathReview (S. D. Bajpai)

Referenced by:

§6.14(iii), §6.7(iv).

Encodings:

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Hypergeometric Series, Recurrence Relations and Some New Orthogonal Polynomials. Ph.D. Thesis, University of Wisconsin, Madison, WI.

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Referenced by:

§16.4(iii), §16.4(iii), §16.4(iii).

Encodings:

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Computation with Recurrence Relations. Pitman, Boston, MA.

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External Links:

ISBN 0-273-08508-5, MathReview (S. Conde), ZentralBlatt

Referenced by:

§13.29(iv), §16.25, §2.9(iii), §3.10(iii), §3.6(iii), §3.6(v), §3.6(vii).

Encodings:

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