delta sequence

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1: 1.17 Integral and Series Representations of the Dirac Delta

… ►

§1.17(i) Delta Sequences

… ►for a suitably chosen sequence of functions ${\delta }_n\left(x\right)$, $n=1,2,\mathrm{\dots }$. Such a sequence is called a delta sequence and we write, symbolically, ►

1.17.4 $$\underset{n\to \mathrm{\infty }}{lim}{\delta }_n\left(x\right)=\delta \left(x\right),$$ $x\in ℝ$.

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Symbols:

$\delta \left(x-a\right)$: Dirac delta (or Dirac delta function), ${\delta }_n\left(x\right)$: Dirac delta sequence, $\in$: element of, $ℝ$: real line and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/1.17.E4

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See also:

Annotations for §1.17(i), §1.17 and Ch.1

►An example of a delta sequence is provided by …

2: 3.9 Acceleration of Convergence

… ►

§3.9(iii) Aitken’s ${\mathrm{\Delta }}^2$-Process

►

3.9.7 $$t_n=s_n-\frac{{(\mathrm{\Delta }{s}_n)}^2}{{\mathrm{\Delta }}^2{s}_n}=s_n-\frac{{(s_{n+1}-s_n)}^2}{s_{n+2}-2s_{n+1}+s_n}.$$

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Symbols:

$s_n$: sequence and $t_n$: sequence

A&S Ref:

3.9.7

Permalink:

http://dlmf.nist.gov/3.9.E7

Encodings:

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Annotations for §3.9(iii), §3.9 and Ch.3

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3.9.9 $$t_{n,2k}=\frac{H_{k+1}(s_n)}{H_k({\mathrm{\Delta }}^2{s}_n)},$$ $n=0,1,2,\mathrm{\dots }$,

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Symbols:

$s_n$: sequence, $t_n$: sequence and $H_m(u)$: Hankel determinant

Referenced by:

§3.9(iv)

Permalink:

http://dlmf.nist.gov/3.9.E9

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Annotations for §3.9(iv), §3.9 and Ch.3

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3: 30.15 Signal Analysis

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30.15.7 $${\int }_{-\tau }^{\tau }{\varphi }_k(t){\varphi }_n(t)dt={\mathrm{\Lambda }}_n{\delta }_{k,n},$$

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Symbols:

${\delta }_{j,k}$: Kronecker delta, $dx$: differential of $x$ , $\int$: integral, $n\ge m$: integer degree, $\tau >0$: parameter, ${\varphi }_n(t)$: function and $\mathrm{\Lambda }$

Permalink:

http://dlmf.nist.gov/30.15.E7

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Annotations for §30.15(iv), §30.15 and Ch.30

►

30.15.8 $${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\varphi }_k(t){\varphi }_n(t)dt={\delta }_{k,n}.$$

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Symbols:

${\delta }_{j,k}$: Kronecker delta, $dx$: differential of $x$ , $\int$: integral, $n\ge m$: integer degree and ${\varphi }_n(t)$: function

Permalink:

http://dlmf.nist.gov/30.15.E8

Encodings:

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See also:

Annotations for §30.15(iv), §30.15 and Ch.30

►The sequence ${\varphi }_n$, $n=0,1,2,\mathrm{\dots }$ forms an orthonormal basis in the space of $\sigma$-bandlimited functions, and, after normalization, an orthonormal basis in $L^2(-\tau ,\tau )$. …

4: 3.6 Linear Difference Equations

… ►

3.6.2 $$a_n{\mathrm{\Delta }}^2{w}_{n-1}+(2a_n-b_n)\mathrm{\Delta }{w}_{n-1}+(a_n-b_n+c_n)w_{n-1}=d_n,$$

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Symbols:

$\mathrm{\Delta }$: forward difference operator, $w_n$: sequence, $a_n$: coefficient, $b_n$: coefficient, $c_n$: coefficient and $d_n$: coefficient

Permalink:

http://dlmf.nist.gov/3.6.E2

Encodings:

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See also:

Annotations for §3.6(i), §3.6 and Ch.3

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5: 3.8 Nonlinear Equations

… ►Let $z_1,z_2,\mathrm{\dots }$ be a sequence of approximations to a root, or fixed point, $\zeta$. …for all $n$ sufficiently large, where $A$ and $p$ are independent of $n$, then the sequence is said to have convergence of the $p$th order. … ►For real functions $f(x)$ the sequence of approximations to a real zero $\xi$ will always converge (and converge quadratically) if either: … ►Starting this iteration in the neighborhood of one of the four zeros $\pm 1,\pm \mathrm{i}$, sequences $\{z_n\}$ are generated that converge to these zeros. For an arbitrary starting point $z_0\in ℂ$, convergence cannot be predicted, and the boundary of the set of points $z_0$ that generate a sequence converging to a particular zero has a very complicated structure. …

6: 17.12 Bailey Pairs

… ►

17.12.1 $$\sum _{n=0}^{\mathrm{\infty }}{\alpha }_n{\gamma }_n=\sum _{n=0}^{\mathrm{\infty }}{\beta }_n{\delta }_n,$$

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Symbols:

$n$: nonnegative integer, ${\alpha }_n$: part of Bailey pair and ${\beta }_n$: part of Bailey pair

Permalink:

http://dlmf.nist.gov/17.12.E1

Encodings:

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Annotations for §17.12, §17.12 and Ch.17

… ►

${\gamma }_n={\displaystyle \sum _{j=n}^{\mathrm{\infty }}}{\delta }_j{u}_{j-n}{v}_{j+n}.$

… ►A sequence of pairs of rational functions of several variables $({\alpha }_n,{\beta }_n)$, $n=0,1,2,\mathrm{\dots }$, is called a Bailey pair provided that for each $n\geqq 0$ … ►When (17.12.5) is iterated the resulting infinite sequence of Bailey pairs is called a Bailey Chain. …

7: 4.4 Special Values and Limits

… ►

4.4.16 $$\underset{z\to \mathrm{\infty }}{lim}{z}^a{\mathrm{e}}^{-z}=0,$$ $|\mathrm{ph}z|\le \frac{1}{2}\pi -\delta$ (),

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{ph}$: phase, $a$: real or complex constant, $z$: complex variable and $\delta$: constant

A&S Ref:

4.2.20

Permalink:

http://dlmf.nist.gov/4.4.E16

Encodings:

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Annotations for §4.4(iii), §4.4 and Ch.4

►where $a$ ($\in ℂ$) and $\delta$ ($\in (0,\frac{1}{2}\pi ]$) are constants. … ►

4.4.19 $$\underset{n\to \mathrm{\infty }}{lim}\left(\left(\sum _{k=1}^n\frac{1}{k}\right)-\ln n\right)=\gamma =\mathrm{0.57721\; 56649\; 01532\; 86060}\mathrm{\dots },$$

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Symbols:

$\gamma$: Euler’s constant, $\ln z$: principal branch of logarithm function, $k$: integer and $n$: integer

A&S Ref:

4.1.32 (with 10D value)

Notes:

For more digits see OEIS Sequence A001620; see also Sloane (2003).

Permalink:

http://dlmf.nist.gov/4.4.E19

Encodings:

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See also:

Annotations for §4.4(iii), §4.4 and Ch.4

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8: 18.27 $q$-Hahn Class

… ►

18.27.2 $$\sum _{x\in X}{p}_n(x)p_m(x)|x|v_x=h_n{\delta }_{n,m},$$

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Defines:

$v_x$ (locally)

Symbols:

${\delta }_{j,k}$: Kronecker delta, $\in$: element of, $p_n(x)$: polynomial of degree $n$ , $X$: subset of $ℝ$ , $m$: nonnegative integer, $n$: nonnegative integer, $x$: real variable and $h_n$

Referenced by:

§18.27(i), §18.27(i), §18.27(iii)

Permalink:

http://dlmf.nist.gov/18.27.E2

Encodings:

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Annotations for §18.27(i), §18.27 and Ch.18

… ►Here $a,b$ are fixed positive real numbers, and $I_{+}$ and $I_{-}$ are sequences of successive integers, finite or unbounded in one direction, or unbounded in both directions. …In case of unbounded sequences (18.27.2) can be rewritten as a $q$-integral, see §17.2(v), and more generally Gasper and Rahman (2004, (1.11.2)). … ►

18.27.4 $$\sum _{y=0}^N{Q}_n(q^{-y})Q_m(q^{-y}){\left[\genfrac{}{}{0.0pt}{}{N}{y}\right]}_q\frac{{(\alpha q;q)}_y{(\beta q;q)}_{N-y}}{{\left(\alpha q\right)}^y}=h_n{\delta }_{n,m},$$ $n,m=0,1,\mathrm{\dots },N$,

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Symbols:

${\delta }_{j,k}$: Kronecker delta, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q$: $q$-binomial coefficient (or Gaussian polynomial), $y$: real variable, $N$: positive integer, $q$: real variable, $m$: nonnegative integer, $n$: nonnegative integer and $h_n$

Notes:

This is Ismail (2009, (18.5.2)) but the presentation has been simplified.

Referenced by:

§18.27(ii), Erratum (V1.2.0) for Equation (18.27.4)

Permalink:

http://dlmf.nist.gov/18.27.E4

Encodings:

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Correction (effective with 1.1.2):

$q$-Pochhammer symbols now link to their definition.

See also:

Annotations for §18.27(ii), §18.27 and Ch.18

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18.27.14 $$\sum _{y=0}^{\mathrm{\infty }}{p}_n(q^y)p_m(q^y)\frac{{(bq;q)}_y{(aq)}^y}{{(q;q)}_y}=h_n{\delta }_{n,m},$$ ,

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Symbols:

${\delta }_{j,k}$: Kronecker delta, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $y$: real variable, $p_n(x)$: polynomial of degree $n$ , $q$: real variable, $m$: nonnegative integer, $n$: nonnegative integer and $h_n$

Permalink:

http://dlmf.nist.gov/18.27.E14

Encodings:

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See also:

Annotations for §18.27(iv), §18.27 and Ch.18

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9: 25.11 Hurwitz Zeta Function

… ►

25.11.40 $$G\equiv \sum _{n=0}^{\mathrm{\infty }}\frac{{(-1)}^n}{{(2n+1)}^2}=\mathrm{0.91596\hspace{0.33em}55941\hspace{0.33em}772}\mathrm{\dots }.$$

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Symbols:

$\equiv$: equals by definition, $n$: nonnegative integer and $G$: Catalan’s constant

Keywords:

Catalan’s constant, definition

Source:

Adamchik and Srivastava (1998, (2.32), p. 139)

Notes:

For more digits see OEIS Sequence A006752; see also Sloane (2003).

Permalink:

http://dlmf.nist.gov/25.11.E40

Encodings:

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

… ►As $a\to \mathrm{\infty }$ in the sector , with $s(\ne 1)$ and $\delta$ fixed, we have the asymptotic expansion … ►Similarly, as $a\to \mathrm{\infty }$ in the sector . …

10: 5.17 Barnes’ $G$-Function (Double Gamma Function)

… ►When $z\to \mathrm{\infty }$ in , … ►

5.17.6 $$A={\mathrm{e}}^C=\mathrm{1.28242\hspace{0.33em}71291\hspace{0.33em}00622\hspace{0.33em}63687}\mathrm{\dots },$$

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Symbols:

$\mathrm{e}$: base of natural logarithm, $A$: Glaisher’s constant and $C$: log of Glaisher’s Constant

Notes:

For more digits see OEIS Sequence A074962; see also Sloane (2003).

Permalink:

http://dlmf.nist.gov/5.17.E6

Encodings:

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See also:

Annotations for §5.17 and Ch.5

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