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1: Bibliography K

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Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library. ACM Trans. Math. Software 20 (4), pp. 447–459.

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External Links:

ISSN 0098-3500, Document, GAMS (module 737; package TOMS), ZentralBlatt

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1st item.

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Methods of computing the Riemann zeta-function and some generalizations of it. USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.

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External Links:

Document, MathReview (Matti Jutila), ZentralBlatt

Referenced by:

§25.18(i).

Encodings:

BibTeX

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Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation. I. Inverse Problems 20 (4), pp. 1165–1206.

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External Links:

ISSN 0266-5611, Document, MathReview (Shun Shimomura), ZentralBlatt

Referenced by:

§32.11(iv).

Encodings:

BibTeX

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The Askey scheme as a four-manifold with corners. Ramanujan J. 20 (3), pp. 409–439.

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ISSN 1382-4090, Document, FullText, MathReview (José Luis López), ZentralBlatt

Referenced by:

§18.26(v), §18.26(v).

Encodings:

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HYP and HYPQ. Mathematica packages for the manipulation of binomial sums and hypergeometric series respectively $q$-binomial sums and basic hypergeometric series. Séminaire Lotharingien de Combinatoire 30, pp. 61–76.

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Notes:

Also published in J. Symbolic Comput. (1995), v. 20, no. 5–6, pp. 737–744.

External Links:

Code, Document, MathReview (G. Gasper), ZentralBlatt

Referenced by:

1st item.

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2: Bibliography S

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Transformations of the Jacobian amplitude function and its calculation via the arithmetic-geometric mean. SIAM J. Math. Anal. 20 (6), pp. 1514–1528.

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External Links:

ISSN 0036-1410, Document, MathReview (J. M. H. Peters), ZentralBlatt

Referenced by:

§22.19(i), §22.20(vi).

Encodings:

BibTeX

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Uniform asymptotic forms of modified Mathieu functions. Quart. J. Mech. Appl. Math. 20 (3), pp. 365–380.

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External Links:

Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§28.8(iv).

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BibTeX

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A Maple package for symmetric functions. J. Symbolic Comput. 20 (5-6), pp. 755–768.

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Notes:

For software see John Stembridge’s home page. The SF package contains Maple software for zonal, Jack, and Macdonald polynomials.

External Links:

Home Page, ISSN 0747-7171, Document, MathReview, ZentralBlatt

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3rd item.

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Error bounds for asymptotic solutions of differential equations. I. The distinct eigenvalue case. J. Res. Nat. Bur. Standards Sect. B 70B, pp. 167–186.

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External Links:

MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§2.7(ii).

Encodings:

BibTeX

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Error bounds for asymptotic solutions of differential equations. II. The general case. J. Res. Nat. Bur. Standards Sect. B 70B, pp. 187–210.

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External Links:

MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§2.7(ii).

Encodings:

BibTeX

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3: Bibliography N

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On an integral transform involving a class of Mathieu functions. SIAM J. Math. Anal. 20 (6), pp. 1500–1513.

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External Links:

ISSN 0036-1410, Document, MathReview (K. C. Gupta), ZentralBlatt

Referenced by:

§28.26(ii).

Encodings:

BibTeX

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Reduction and evaluation of elliptic integrals. Math. Comp. 20 (94), pp. 223–231.

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External Links:

FullText, MathReview (I. A. Stegun), ZentralBlatt

Referenced by:

§19.37(iv).

Encodings:

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Géza Freud, orthogonal polynomials and Christoffel functions. A case study. J. Approx. Theory 48 (1), pp. 3–167.

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ISSN 0021-9045, Document, MathReview (T. S. Chihara), ZentralBlatt

Referenced by:

§18.32.

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A table of integrals of the error functions. J. Res. Nat. Bur. Standards Sect B. 73B, pp. 1–20.

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Notes:

See for additions and corrections same journal, Sect. B 75, 149–163 (1975)

External Links:

MathReview, ZentralBlatt

Referenced by:

§7.14(iii), §7.21, §7.7(iii).

Encodings:

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Askey-Wilson polynomials: an affine Hecke algebra approach. In Laredo Lectures on Orthogonal Polynomials and Special Functions, Adv. Theory Spec. Funct. Orthogonal Polynomials, pp. 111–144.

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External Links:

MathReview (V. L. Deshpande), ZentralBlatt

Referenced by:

§18.38(iii).

Encodings:

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4: 18.5 Explicit Representations

§18.5 Explicit Representations

… ►For this reason, and also in the interest of simplicity, in the case of the Jacobi polynomials $P_n^{(\alpha ,\beta )}\left(x\right)$ we assume throughout this chapter that $\alpha >-1$ and $\beta >-1$, unless stated otherwise. Similarly in the cases of the ultraspherical polynomials $C_n^{(\lambda )}\left(x\right)$ and the Laguerre polynomials $L_n^{(\alpha )}\left(x\right)$ we assume that $\lambda >-\frac{1}{2},\lambda \ne 0$, and $\alpha >-1$, unless stated otherwise. … ►

5: Bibliography M

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Rational approximations, software and test methods for sine and cosine integrals. Numer. Algorithms 12 (3-4), pp. 259–272.

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Notes:

Includes Fortran code, maximum accuracy 20S.

External Links:

ISSN 1017-1398, Document, MathReview, ZentralBlatt

Referenced by:

§6.13, 4th item, §6.21(ii).

Encodings:

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Exact misclassification probabilities for plug-in normal quadratic discriminant functions. I. The equal-means case. J. Multivariate Anal. 77 (1), pp. 21–53.

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External Links:

ISSN 0047-259X, Document, MathReview (Peter A. Lachenbruch), ZentralBlatt

Referenced by:

§35.9.

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

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External Links:

ISSN 0025-5718, FullText, MathReview (H. E. Fettis), ZentralBlatt

Referenced by:

§10.74(iii).

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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.

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External Links:

FullText

Referenced by:

§8.24(i).

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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6: 19.36 Methods of Computation

… ►Polynomials of still higher degree can be obtained from (19.19.5) and (19.19.7). … ►The computation is slowest for complete cases. … ►Complete cases of Legendre’s integrals and symmetric integrals can be computed with quadratic convergence by the AGM method (including Bartky transformations), using the equations in §19.8(i) and §19.22(ii), respectively. … ►Also, see Todd (1975) for a special case of $K\left(k\right)$. For computation of Legendre’s integral of the third kind, see Abramowitz and Stegun (1964, §§17.7 and 17.8, Examples 15, 17, 19, and 20). …

7: 32.8 Rational Solutions

… ►The rational solutions when the parameters satisfy (32.8.22) are special cases of §32.10(iv). … ►Cases (a) and (b) are special cases of §32.10(v). … ►For the case $\delta =0$ see Airault (1979) and Lukaševič (1968). … ►In the general case, ${\text{P}}_{\text{VI}}$ has rational solutions if …These are special cases of §32.10(vi). …

8: 26.9 Integer Partitions: Restricted Number and Part Size

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26.9.4 $${\left[\genfrac{}{}{0.0pt}{}{m}{n}\right]}_q=\prod _{j=1}^n\frac{1-q^{m-n+j}}{1-q^j},$$ $n\ge 0$,

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Symbols:

${\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q$: $q$-binomial coefficient (or Gaussian polynomial), $j$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer and $q$: parameter

Permalink:

http://dlmf.nist.gov/26.9.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §26.9(ii), §26.9 and Ch.26

►is the Gaussian polynomial (or $q$-binomial coefficient); see also §§17.2(i)–17.2(ii). In the present chapter $m\ge n\ge 0$ in all cases. … ►

26.9.6 $$\sum _{n=0}^{\mathrm{\infty }}{p}_k(\le m,n)q^n={\left[\genfrac{}{}{0.0pt}{}{m+k}{k}\right]}_q.$$

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Symbols:

${\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q$: $q$-binomial coefficient (or Gaussian polynomial), $p_k(\le m,n)$: number of partitions of $n$ into at most $k$ parts, each less than or equal to $m$ , $k$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer and $q$: parameter

Permalink:

http://dlmf.nist.gov/26.9.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §26.9(ii), §26.9 and Ch.26

… ►equivalently, partitions into at most $k$ parts either have exactly $k$ parts, in which case we can subtract one from each part, or they have strictly fewer than $k$ parts. …

9: 5.11 Asymptotic Expansions

… ►Wrench (1968) gives exact values of $g_k$ up to $g_{20}$. … ►where $h(\in ℂ)$ is fixed, and $B_k\left(h\right)$ is the Bernoulli polynomial defined in §24.2(i). … ►In the case $K=1$ the factor $1+\zeta \left(K\right)$ is replaced with 4. … ►In terms of generalized Bernoulli polynomials $B_n^{(\mathrm{\ell })}\left(x\right)$ (§24.16(i)), we have for $k=0,1,\mathrm{\dots }$, … ►For the error term in (5.11.19) in the case $z=x(>0)$ and $c=1$, see Olver (1995). …

10: 25.11 Hurwitz Zeta Function

… ►The Riemann zeta function is a special case: … ►

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Figure 25.11.1: Hurwitz zeta function $\zeta (x,a)$, $a$ = 0. …8, 1, $-20\le x\le 10$. … Magnify

… ►For ${\tilde{B}}_n\left(x\right)$ see §24.2(iii). … ►

25.11.21 $${\zeta }^{\prime }(1-2n,\frac{h}{k})=\frac{(\psi \left(2n\right)-\ln \left(2\pi k\right))B_{2n}\left(h/k\right)}{2n}-\frac{(\psi \left(2n\right)-\ln \left(2\pi \right))B_{2n}}{2n{k}^{2n}}+\frac{{(-1)}^{n+1}\pi }{{(2\pi k)}^{2n}}\sum _{r=1}^{k-1}\sin \left(\frac{2\pi rh}{k}\right){\psi }^{(2n-1)}\left(\frac{r}{k}\right)+\frac{{(-1)}^{n+1}2\cdot (2n-1)!}{{(2\pi k)}^{2n}}\sum _{r=1}^{k-1}\cos \left(\frac{2\pi rh}{k}\right){\zeta }^{\prime }(2n,\frac{r}{k})+\frac{{\zeta }^{\prime }\left(1-2n\right)}{k^{2n}},$$

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Symbols:

$B_n$: Bernoulli numbers, $B_n\left(x\right)$: Bernoulli polynomials, $\zeta (s,a)$: Hurwitz zeta function, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $\sin z$: sine function, $n$: nonnegative integer, $h$: integer and $k$: integer

Keywords:

derivative

Source:

Miller and Adamchik (1998, (5), p. 203)

Notes:

According to the source, (25.11.21) is valid only for . Their derivation, however, is based on (25.11.16) which is true when $h=k$ as well. Alternatively, one can substitute $h=k$ into (25.11.21) and simplify via (25.11.24), (25.6.2) and (25.6.15) confirming that (25.11.21) holds also in the case that $h=k$.

Referenced by:

(25.11.21)

Permalink:

http://dlmf.nist.gov/25.11.E21

Encodings:

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See also:

Annotations for §25.11(vi), §25.11(vi), §25.11 and Ch.25

… ►For the more general case ${\zeta }^{\prime }(-m,a)$, $m=1,2,\mathrm{\dots }$, see Elizalde (1986). …