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11: 1.14 Integral Transforms

… ►Sufficient conditions for the integral to converge are that $s$ is a positive real number, and $f(t)=O\left(t^{-\delta }\right)$ as $t\to \mathrm{\infty }$, where $\delta >0$. …

12: 14.16 Zeros

… ►The number of zeros of ${𝖯}_{\nu }^{\mu }\left(x\right)$ in the interval $(-1,1)$ is $\max (\lceil \nu -|\mu |\rceil ,0)$ if any of the following sets of conditions hold: … ►The number of zeros of ${𝖯}_{\nu }^{\mu }\left(x\right)$ in the interval $(-1,1)$ is $\max (\lceil \nu -|\mu |\rceil ,0)+1$ if either of the following sets of conditions holds: …

13: 31.15 Stieltjes Polynomials

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31.15.2 $$\sum _{j=1}^N\frac{{\gamma }_j/2}{z_k-a_j}+\sum _{\begin{array}{c}j=1\\ j\ne k\end{array}}^n\frac{1}{z_k-z_j}=0,$$ $k=1,2,\mathrm{\dots },n$.

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Symbols:

$z$: complex variable, $\gamma$: real or complex parameter, $j$: nonnegative integer, $n$: nonnegative integer, $a$: complex parameter and $N+1$: number of singularities

Referenced by:

§31.15(ii)

Permalink:

http://dlmf.nist.gov/31.15.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §31.15(ii), §31.15 and Ch.31

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31.15.3 $$\sum _{j=1}^N\frac{{\gamma }_j}{t_k-a_j}+\sum _{j=1}^{n-1}\frac{1}{t_k-z_j^{\prime }}=0.$$

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Symbols:

$z$: complex variable, $\gamma$: real or complex parameter, $j$: nonnegative integer, $n$: nonnegative integer, $a$: complex parameter, $N+1$: number of singularities and $t_k$: zero of $V(z)$

Permalink:

http://dlmf.nist.gov/31.15.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §31.15(ii), §31.15 and Ch.31

… ►If the following conditions are satisfied: … ►

31.15.6 $j=1,2,\mathrm{\dots },N-1$,

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Symbols:

$j$: nonnegative integer, $a$: complex parameter and $N+1$: number of singularities

Referenced by:

§31.15(iii)

Permalink:

http://dlmf.nist.gov/31.15.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §31.15(ii), §31.15 and Ch.31

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31.15.7 $$q_j={\gamma }_j\sum _{k=1}^n\frac{1}{z_k-a_j},$$ $j=1,2,\mathrm{\dots },N$.

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Symbols:

$z$: complex variable, $\gamma$: real or complex parameter, $j$: nonnegative integer, $n$: nonnegative integer, $a$: complex parameter, $q$: real or complex parameter and $N+1$: number of singularities

Permalink:

http://dlmf.nist.gov/31.15.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §31.15(ii), §31.15 and Ch.31

…

14: 27.8 Dirichlet Characters

§27.8 Dirichlet Characters

… ►In other words, Dirichlet characters (mod $k$) satisfy the four conditions: … ►An example is the principal character (mod $k$): … ►If $\left(n,k\right)=1$, then the characters satisfy the orthogonality relation … ►A divisor $d$ of $k$ is called an induced modulus for $\chi$ if …

15: 1.13 Differential Equations

… ►For a regular Sturm-Liouville system, equations (1.13.26) and (1.13.29) have: (i) identical eigenvalues, $\lambda$; (ii) the corresponding (real) eigenfunctions, $u(x)$ and $w(t)$, have the same number of zeros, also called nodes, for $t\in (0,c)$ as for $x\in (a,b)$; (iii) the eigenfunctions also satisfy the same type of boundary conditions, un-mixed or periodic, for both forms at the corresponding boundary points. …

16: 18.2 General Orthogonal Polynomials

… ►The moments for an orthogonality measure $d\mu (x)$ are the numbers …It is to be noted that, although formally correct, the results of (18.2.30) are of little utility for numerical work, as Hankel determinants are notoriously ill-conditioned. … ►The OP’s $p_n^{(1)}(x)$ may also be calculated from the original recursion (18.2.11_5), but with independent initial conditions for $p_0,p_1$: … ►are the Christoffel numbers, see also (3.5.18). … ►Under further conditions on the weight function there is an equiconvergence theorem, see Szegő (1975, Theorem 13.1.2). …

17: 3.11 Approximation Techniques

… ►A sufficient condition for $p_n(x)$ to be the minimax polynomial is that $\left|{ϵ}_n(x)\right|$ attains its maximum at $n+2$ distinct points in $[a,b]$ and ${ϵ}_n(x)$ changes sign at these consecutive maxima. … ►The matrix is symmetric and positive definite, but the system is ill-conditioned when $n$ is large because the lower rows of the matrix are approximately proportional to one another. … … ►Then the system (3.11.33) is diagonal and hence well-conditioned. … ►In consequence of this structure the number of operations can be reduced to $nm=n{\log }_{2}n$ operations. …

18: 10.72 Mathematical Applications

… ►In regions in which (10.72.1) has a simple turning point $z_0$, that is, $f(z)$ and $g(z)$ are analytic (or with weaker conditions if $z=x$ is a real variable) and $z_0$ is a simple zero of $f(z)$, asymptotic expansions of the solutions $w$ for large $u$ can be constructed in terms of Airy functions or equivalently Bessel functions or modified Bessel functions of order $\frac{1}{3}$ (§9.6(i)). … ►The number $m$ can also be replaced by any real constant $\lambda$ $(>-2)$ in the sense that ${(z-z_0)}^{-\lambda }$ $f(z)$ is analytic and nonvanishing at $z_0$; moreover, $g(z)$ is permitted to have a single or double pole at $z_0$. …

19: Bibliography L

… ►

An asymptotic estimate for the Bernoulli and Euler numbers. Canad. Math. Bull. 20 (1), pp. 109–111.

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External Links:

MathReview (D. H. Lehmer), ZentralBlatt

Referenced by:

§24.11.

Encodings:

BibTeX

… ►

Guide to Tables in the Theory of Numbers. Bulletin of the National Research Council, No. 105, National Research Council, Washington, D.C..

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Notes:

Reprinted in 1961.

External Links:

MathReview (R. D. Carmichael), ZentralBlatt

Referenced by:

§27.20, §27.20, §27.21, §27.21.

Encodings:

BibTeX

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List of Prime Numbers from 1 to 10,006,721. Publ. No. 165, Carnegie Institution of Washington, Washington, D.C..

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External Links:

ZentralBlatt

Referenced by:

§27.21.

Encodings:

BibTeX

… ►

Optimal cylindrical and spherical Bessel transforms satisfying bound state boundary conditions. Comput. Phys. Comm. 99 (2-3), pp. 297–306.

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Notes:

Single-precision Fortran, maximum accuracy 8S.

External Links:

ISSN 0010-4655, Document, Code, ZentralBlatt

Referenced by:

4th item.

Encodings:

BibTeX

…

20: 35.7 Gaussian Hypergeometric Function of Matrix Argument

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35.7.1 $${}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{a,b}{c};𝐓)=\sum _{k=0}^{\mathrm{\infty }}\frac{1}{k!}\sum _{|\kappa |=k}\frac{{\left[a\right]}_{\kappa }{\left[b\right]}_{\kappa }}{{\left[c\right]}_{\kappa }}{Z}_{\kappa }\left(𝐓\right),$$ $-c+\frac{1}{2}(j+1)\notin \mathbb{N}$, $1\le j\le m$; .

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Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};𝐓)$: generalized hypergeometric function of matrix argument, $!$: factorial (as in $n!$), $\notin$: not an element of, $\mathbb{N}$: set of all positive integers, ${\left[a\right]}_{\kappa }$: partitional shifted factorial, $Z_{\kappa }\left(𝐓\right)$: zonal polynomial, $a$: complex variable, $𝐓$: real symmetric $m\times m$ matrix, $b$: complex variable, $\Vert 𝐓\Vert$: spectral norm of $𝐓$ , $c$: complex variable, $j$: nonnegative integer, $k$: nonnegative integer, $m$: positive integer and $\kappa$: vector of nonnegative integers

Permalink:

http://dlmf.nist.gov/35.7.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §35.7(i), §35.7 and Ch.35

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35.7.8 $${}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{a,b}{c};𝐓)=\frac{{\mathrm{\Gamma }}_m\left(c\right){\mathrm{\Gamma }}_m\left(c-a-b\right)}{{\mathrm{\Gamma }}_m\left(c-a\right){\mathrm{\Gamma }}_m\left(c-b\right)}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{a,b}{a+b-c+\frac{1}{2}(m+1)};𝐈-𝐓),$$ ; $\frac{1}{2}(j+1)-a\in \mathbb{N}$ for some $j=1,\mathrm{\dots },m$; $\frac{1}{2}(j+1)-c\notin \mathbb{N}$ and $c-a-b-\frac{1}{2}(m-j)\notin \mathbb{N}$ for all $j=1,\mathrm{\dots },m$.

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Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};𝐓)$: generalized hypergeometric function of matrix argument, $\in$: element of, ${\mathrm{\Gamma }}_m\left(a\right)$: multivariate gamma function, $\notin$: not an element of, $\mathbb{N}$: set of all positive integers, $\mathrm{\Re }$: real part, $a$: complex variable, $𝐈$: $m\times m$ identity matrix, $𝐓$: real symmetric $m\times m$ matrix, $b$: complex variable, $c$: complex variable, $j$: nonnegative integer, $m$: positive integer and $𝟎$: $m\times m$ matrix of zeros

Referenced by:

§35.7(ii), Erratum (V1.0.26) for Equation (35.7.8)

Permalink:

http://dlmf.nist.gov/35.7.E8

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.26):

Originally the constraint was written as $\mathrm{\Re }\left(c\right),\mathrm{\Re }\left(c-a-b\right)>\frac{1}{2}(m-1)$. The constraint has been replaced with ; $\frac{1}{2}(j+1)-a\in \mathbb{N}$ for some $j=1,\mathrm{\dots },m$; $\frac{1}{2}(j+1)-c\notin \mathbb{N}$ and $c-a-b-\frac{1}{2}(m-j)\notin \mathbb{N}$ for all $j=1,\mathrm{\dots },m$; for details see https://arxiv.org/abs/2002.05248.

See also:

Annotations for §35.7(ii), §35.7(ii), §35.7 and Ch.35

… ►Subject to the conditions (a)–(c), the function $f(𝐓)={}_2{}^{}F_1^{}(a,b;c;𝐓)$ is the unique solution of each partial differential equation …