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1: 26.14 Permutations: Order Notation

… ►

26.14.2

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Symbols:

$j$: nonnegative integer, $n$: nonnegative integer, $\sigma$: permutation and $maj(\sigma )$: major index

Permalink:

http://dlmf.nist.gov/26.14.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §26.14(i), §26.14 and Ch.26

… ►The major index is also called the greater index of the permutation. …

2: 25.11 Hurwitz Zeta Function

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25.11.15 $$\zeta (s,ka)=k^{-s}\sum _{n=0}^{k-1}\zeta (s,a+\frac{n}{k}),$$ $s\ne 1$, $k=1,2,3,\mathrm{\dots }$.

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Symbols:

$\zeta (s,a)$: Hurwitz zeta function, $k$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer, $a$: real or complex parameter and $s$: complex variable

Keywords:

special value

Proof sketch:

Derivable by starting with (25.11.1), replacing $a\mapsto ka$, pulling out a factor of $k^{-s}$ and performing a change of sum index $n=km+l$ with $m\ge 0$ and $0\le l\le k-1$, which through Euclidean division converts the single sum into a double sum of the correct form.

Referenced by:

(25.11.24)

Permalink:

http://dlmf.nist.gov/25.11.E15

Encodings:

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See also:

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

…

3: 1.2 Elementary Algebra

… ►The last two equations require $a_j>0$ for all $j$. … ►The full index form of an $m\times n$ matrix $𝐀$ is … ► $𝐀$ is an upper or lower triangular matrix if all $a_{ij}$ vanish for $i>j$ or , respectively. ►Equation (3.2.7) displays a tridiagonal matrix in index form; (3.2.4) does the same for a lower triangular matrix. … ►The matrix $𝐀$ has a determinant, $det(𝐀)$, explored further in §1.3, denoted, in full index form, as …

4: 3.1 Arithmetics and Error Measures

… ►

§3.1(iv) Level-Index Arithmetic

… ►with $x\ge 0$ and $\mathrm{\ell }$ the unique nonnegative integer such that $a\equiv {\ln }_{\mathrm{\ell }}(x)\in [0,1)$. In level-index arithmetic $x$ is represented by $\mathrm{\ell }+a$ (or $-(\mathrm{\ell }+a)$ for negative numbers). … ►For further references on level-index arithmetic (and also other arithmetics) see Anuta et al. (1996). … ►where $x{x}^{\ast }>0$ for real variables, and $x{x}^{\ast }\ne 0$ for complex variables (with the principal value of the logarithm). …

5: 25.15 Dirichlet $L$-functions

… ►This implies that $L(s,\chi )\ne 0$ if $\mathrm{\Re }s>1$. … ►

25.15.3 $$L(s,\chi )=k^{-s}\sum _{r=1}^{k-1}\chi (r)\zeta (s,\frac{r}{k}),$$

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Symbols:

$L(s,\chi )$: Dirichlet $L$-function, $\zeta (s,a)$: Hurwitz zeta function, $k$: nonnegative integer, $s$: complex variable and $\chi (n)$: Dirichlet character

Keywords:

finite sum

Source:

Apostol (1976, (8), p. 255)

Notes:

In the original source the upper-index of the finite sum is $k$. We use $k-1$ since $\chi (k)=0$.

Referenced by:

(25.11.36), (25.11.36), §25.11(x), §25.15(i), Erratum (V1.0.23) for Equation (25.11.36)

Permalink:

http://dlmf.nist.gov/25.15.E3

Encodings:

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See also:

Annotations for §25.15(i), §25.15 and Ch.25

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25.15.6 $$G(\chi )\equiv \sum _{r=1}^{k-1}\chi (r){\mathrm{e}}^{2\pi \mathrm{i}r/k}.$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\equiv$: equals by definition, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $k$: nonnegative integer and $\chi (n)$: Dirichlet character

Keywords:

definition

Source:

Apostol (1976, p. 262)

Referenced by:

Erratum (V1.1.4) for Equation (25.15.6)

Permalink:

http://dlmf.nist.gov/25.15.E6

Encodings:

pMML, png, TeX

Clarification (effective with 1.1.4):

The upper-index of the finite sum which originally was $k$, was replaced with $k-1$ since $\chi (k)=0$.

Suggested 2021-08-23 by Gergő Nemes

See also:

Annotations for §25.15(i), §25.15 and Ch.25

… ►Since $L(s,\chi )\ne 0$ if $\mathrm{\Re }s>1$, (25.15.5) shows that for a primitive character $\chi$ the only zeros of $L(s,\chi )$ for (the so-called trivial zeros) are as follows: … ►

25.15.10

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Symbols:

$L(s,\chi )$: Dirichlet $L$-function, $k$: nonnegative integer and $\chi (n)$: Dirichlet character

Keywords:

zeros

Source:

Apostol (1976, Theorem 12.20, p. 268)

Referenced by:

Erratum (V1.1.4) for Equation (25.15.10)

Permalink:

http://dlmf.nist.gov/25.15.E10

Encodings:

pMML, png, TeX

Clarification (effective with 1.1.4):

The upper-index of the finite sum which originally was $k$, was replaced with $k-1$ since $\chi (k)=0$.

Suggested 2021-08-23 by Gergő Nemes

See also:

Annotations for §25.15(ii), §25.15 and Ch.25

6: 9.7 Asymptotic Expansions

… ►In (9.7.5) and (9.7.6) the $n$th error term, that is, the error on truncating the expansion at $n$ terms, is bounded in magnitude by the first neglected term and has the same sign, provided that the following term is of opposite sign, that is, if $n\ge 0$ for (9.7.5) and $n\ge 1$ for (9.7.6). …

7: 18.30 Associated OP’s

… ►Assuming equation (18.2.8) with its initialization defines a set of OP’s, $p_n(x)$, the corresponding associated orthogonal polynomials of order $c$ are the $p_n(x;c)$ as defined by shifting the index $n$ in the recurrence coefficients by adding a constant $c$, functions of $n$, say $f(n)$, being replaced by $f(n+c)$. The inequality $A_n{A}_{n+1}{C}_{n+1}>0$, for $n\ge 0$ is replaced by ►

18.30.1 $$A_{n+c}{A}_{n+c+1}{C}_{n+c+1}>0,$$ $n=0,1,\mathrm{\dots }$.

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Symbols:

$n$: nonnegative integer, $A_n$: coefficient and $C_n$: coefficient

Referenced by:

§18.30

Permalink:

http://dlmf.nist.gov/18.30.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §18.30 and Ch.18

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18.30.10 $${\int }_0^{\mathrm{\infty }}{L}_n^{\lambda }(x;c)L_m^{\lambda }(x;c)w^{\lambda }(x,c)dx=\frac{\mathrm{\Gamma }\left(n+c+\lambda +1\right)\mathrm{\Gamma }\left(c+1\right)}{{\left(c+1\right)}_n}{\delta }_{n,m},$$ $c+\lambda >-1$, $c\ge 0$, or $c+\lambda \ge 0$, $c>-1$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\delta }_{j,k}$: Kronecker delta, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $L_n^{\lambda }(x;c)$: associated Laguerre polynomial, $dx$: differential of $x$ , $\int$: integral, $w(x)$: weight function, $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.30(iii)

Permalink:

http://dlmf.nist.gov/18.30.E10

Encodings:

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See also:

Annotations for §18.30(iii), §18.30 and Ch.18

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18.30.17 $${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{𝒫}_n^{\lambda }(x;\varphi ,c){𝒫}_m^{\lambda }(x;\varphi ,c)w^{(\lambda )}(x,\varphi ,c)dx=\frac{\mathrm{\Gamma }\left(n+c+2\lambda \right)\mathrm{\Gamma }\left(c+1\right)}{{\left(c+1\right)}_n}{\delta }_{n,m},$$ or ,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\delta }_{j,k}$: Kronecker delta, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), ${𝒫}_n^{\lambda }(x;\varphi ,c)$: associated Meixner–Pollaczek polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $w(x)$: weight function, $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Sources:

Askey and Wimp (1984, (1.9)); Pollaczek (1950, formula following (16))

Permalink:

http://dlmf.nist.gov/18.30.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §18.30(v), §18.30 and Ch.18

…

8: Errata

… ►

Meta.Numerics (website) was added to the Software Index.

… ►

Entries for the Sage computational system have been updated in the Software Index.

… ►

Other minor changes were made in the bibliography and index.

… ►Several minor improvements were made affecting display of math and graphics on the website; the software index and help files were updated. … ►

Additions and revisions were made in the Cross Index for Computing Special Functions.

…

9: 9.13 Generalized Airy Functions

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9.13.13 $$\frac{d^{2}w}{{dt}^2}=\frac{1}{4}{m}^2{t}^{m-2}w,$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $w$: function and $m$: index

Source:

Olver (1977a, (2.01))

Referenced by:

§9.13(i), §9.13(i)

Permalink:

http://dlmf.nist.gov/9.13.E13

Encodings:

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See also:

Annotations for §9.13(i), §9.13 and Ch.9

►where $m=3,4,5,\mathrm{\dots }.$ For real variables the solutions of (9.13.13) are denoted by $U_m(t)$, $U_m(-t)$ when $m$ is even, and by $V_m(t)$, ${\overline{V}}_m(t)$ when $m$ is odd. … ►

9.13.18 $$w=U_m(t{\mathrm{e}}^{-2j\pi \mathrm{i}/m}),$$ $j=0,\pm 1,\pm 2,\mathrm{\dots }.$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $U_{\mathrm{i}}$: Smirnov’s generalized Airy function, $w$: function, $m$: index and $j$: index

Notes:

See Olver (1978).

Permalink:

http://dlmf.nist.gov/9.13.E18

Encodings:

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See also:

Annotations for §9.13(i), §9.13 and Ch.9

… ►where $\alpha >-2$ and $x>0$. … ►Further properties of these functions, and also of similar contour integrals containing an additional factor ${(\ln t)}^q$, $q=1,2,\mathrm{\dots }$, in the integrand, are derived in Reid (1972), Drazin and Reid (1981, Appendix), and Baldwin (1985). …

10: 9.10 Integrals

… ►

9.10.10 $$\int z^{n+3}w(z)dz=z^{n+2}{w}^{\prime }(z)-(n+2)z^{n+1}w(z)+(n+1)(n+2)\int z^{n}w(z)dz,$$ $n=0,1,2,\mathrm{\dots }.$

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Defines:

$n$: index (locally)

Symbols:

$dx$: differential of $x$ , $\int$: integral, $z$: complex variable and $w$: ODE solution

Proof sketch:

To verify, differentiate and refer to (9.2.1).

Permalink:

http://dlmf.nist.gov/9.10.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §9.10(iii), §9.10 and Ch.9

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9.10.13 $${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{pt}\mathrm{Ai}\left(t\right)dt={\mathrm{e}}^{p^3/3},$$ $\mathrm{\Re }p>0$.

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part and $p$: parameter

Source:

Widder (1979, (3.1))

Permalink:

http://dlmf.nist.gov/9.10.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §9.10(v), §9.10 and Ch.9

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9.10.15 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-pt}\mathrm{Ai}\left(-t\right)dt=\frac{1}{3}{\mathrm{e}}^{p^3/3}\left(\frac{\mathrm{\Gamma }(\frac{1}{3},\frac{1}{3}{p}^3)}{\mathrm{\Gamma }\left(\frac{1}{3}\right)}+\frac{\mathrm{\Gamma }(\frac{2}{3},\frac{1}{3}{p}^3)}{\mathrm{\Gamma }\left(\frac{2}{3}\right)}\right),$$ $\mathrm{\Re }p>0$,

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $\int$: integral, $\mathrm{\Re }$: real part and $p$: parameter

Keywords:

Laplace transform

Source:

Gibbs (1973, Problem 72–21)

Permalink:

http://dlmf.nist.gov/9.10.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §9.10(v), §9.10 and Ch.9

►

9.10.16 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-pt}\mathrm{Bi}\left(-t\right)dt=\frac{1}{\sqrt{3}}{\mathrm{e}}^{p^3/3}\left(\frac{\mathrm{\Gamma }(\frac{2}{3},\frac{1}{3}{p}^3)}{\mathrm{\Gamma }\left(\frac{2}{3}\right)}-\frac{\mathrm{\Gamma }(\frac{1}{3},\frac{1}{3}{p}^3)}{\mathrm{\Gamma }\left(\frac{1}{3}\right)}\right),$$ $\mathrm{\Re }p>0$.

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Symbols:

$\mathrm{Bi}\left(z\right)$: Airy function, $\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $\int$: integral, $\mathrm{\Re }$: real part and $p$: parameter

Keywords:

Laplace transform

Source:

Gibbs (1973, Problem 72–21)

Permalink:

http://dlmf.nist.gov/9.10.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §9.10(v), §9.10 and Ch.9

… ►

9.10.17 $${\int }_0^{\mathrm{\infty }}{t}^{\alpha -1}\mathrm{Ai}\left(t\right)dt=\frac{\mathrm{\Gamma }\left(\alpha \right)}{3^{(\alpha +2)/3}\mathrm{\Gamma }\left(\frac{1}{3}\alpha +\frac{2}{3}\right)},$$ $\mathrm{\Re }\alpha >0$.

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Defines:

$\alpha$: parameter (locally)

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\int$: integral and $\mathrm{\Re }$: real part

Keywords:

Mellin transform

Source:

Olver (1997b, (6.10), p. 338)

Permalink:

http://dlmf.nist.gov/9.10.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §9.10(vi), §9.10 and Ch.9

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