infinite sequences

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11: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►A (finite or countably infinite, generalizing the definition of (1.2.40)) set $\{v_n\}$ is an orthonormal set if the $v_n$ are normalized and pairwise orthogonal. … ►An inner product space $V$ is called a Hilbert space if every Cauchy sequence $\{v_n\}$ in $V$ (i. … ►A Hilbert space $V$ is separable if there is an (at most countably infinite) orthonormal set $\{v_n\}$ in $V$ such that for every $v\in V$ …where the infinite sum means convergence in norm, … ►Let $X=(a,b)$ be a finite or infinite open interval in $ℝ$. …

12: 18.33 Polynomials Orthogonal on the Unit Circle

… ►This states that for any sequence ${\{{\alpha }_n\}}_{n=0}^{\mathrm{\infty }}$ with ${\alpha }_n\in ℂ$ and the polynomials ${\mathrm{\Phi }}_n(z)$ generated by the recurrence relations (18.33.23), (18.33.24) with ${\mathrm{\Phi }}_0(z)=1$ satisfy the orthogonality relation (18.33.17) for a unique probability measure $\mu$ with infinite support on the unit circle. …

13: 25.11 Hurwitz Zeta Function

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25.11.8 $$\zeta (s,\frac{1}{2}a)=\zeta (s,\frac{1}{2}a+\frac{1}{2})+2^s\sum _{n=0}^{\mathrm{\infty }}\frac{{(-1)}^n}{{(n+a)}^s},$$ $\mathrm{\Re }s>0$, $s\ne 1$, .

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Symbols:

$\zeta (s,a)$: Hurwitz zeta function, $\mathrm{\Re }$: real part, $n$: nonnegative integer, $a$: real or complex parameter and $s$: complex variable

Keywords:

infinite series, series representation

Source:

Srivastava and Choi (2001, (5), p. 89)

Referenced by:

(25.11.35), §25.11(x)

Permalink:

http://dlmf.nist.gov/25.11.E8

Encodings:

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See also:

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

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25.11.35 $$\sum _{n=0}^{\mathrm{\infty }}\frac{{(-1)}^n}{{(n+a)}^s}=\frac{1}{\mathrm{\Gamma }\left(s\right)}{\int }_0^{\mathrm{\infty }}\frac{x^{s-1}{\mathrm{e}}^{-ax}}{1+{\mathrm{e}}^{-x}}dx=2^{-s}\left(\zeta (s,\frac{1}{2}a)-\zeta (s,\frac{1}{2}(1+a))\right),$$ $\mathrm{\Re }a>0$, $\mathrm{\Re }s>0$; or $\mathrm{\Re }a=0$, $\mathrm{\Im }a\ne 0$, .

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta (s,a)$: Hurwitz zeta function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{\Im }$: imaginary part, $\int$: integral, $\mathrm{\Re }$: real part, $n$: nonnegative integer, $x$: real variable, $a$: real or complex parameter and $s$: complex variable

Keywords:

improper integral, infinite series, integral representation, series representation

Proof sketch:

The infinite series connection to the difference of Hurwitz zeta functions is a restatement of (25.11.8). The integral representation is derivable from the difference of Hurwitz zeta functions by substituting (25.11.25) in each, making a change of variables $x=2t$, factoring, and expressing $1-{\mathrm{e}}^{-2t}=(1+{\mathrm{e}}^{-t})(1-{\mathrm{e}}^{-t})$.

Referenced by:

(25.11.31), §25.11(x), §25.11(x)

Permalink:

http://dlmf.nist.gov/25.11.E35

Encodings:

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See also:

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

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25.11.38 $$\sum _{k=1}^{\mathrm{\infty }}\left(\genfrac{}{}{0.0pt}{}{n+k}{k}\right)\zeta (n+k+1,a)z^k=\frac{{(-1)}^n}{n!}\left({\psi }^{(n)}\left(a\right)-{\psi }^{(n)}\left(a-z\right)\right),$$ $n=1,2,3,\mathrm{\dots }$, $\mathrm{\Re }a>0$, .

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Symbols:

$\zeta (s,a)$: Hurwitz zeta function, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $\mathrm{\Re }$: real part, $k$: nonnegative integer, $n$: nonnegative integer, $a$: real or complex parameter and $z$: complex variable

Keywords:

infinite series, series representation

Source:

Adamchik and Srivastava (1998, (2.29), p. 138)

Permalink:

http://dlmf.nist.gov/25.11.E38

Encodings:

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See also:

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

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25.11.39 $$\sum _{k=2}^{\mathrm{\infty }}\frac{k}{2^k}\zeta (k+1,\frac{3}{4})=8G,$$

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Symbols:

$\zeta (s,a)$: Hurwitz zeta function, $k$: nonnegative integer and $G$: Catalan’s constant

Keywords:

Catalan’s constant, infinite series

Source:

Adamchik and Srivastava (1998, (2.30), p. 138)

Permalink:

http://dlmf.nist.gov/25.11.E39

Encodings:

pMML, png, TeX

See also:

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

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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:

pMML, png, TeX

See also:

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

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14: Bibliography S

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On integral representation of Weber’s parabolic cylinder function and its expansion into an infinite series. J. Indian Math. Soc. (N. S.) 4, pp. 34–38.

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External Links:

MathReview (M. C. Gray), ZentralBlatt

Referenced by:

§12.13(ii).

Encodings:

BibTeX

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Non-linear transformations of divergent and slowly convergent sequences. J. Math. Phys. 34, pp. 1–42.

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External Links:

MathReview (J. Kuntzmann), ZentralBlatt

Referenced by:

§17.18.

Encodings:

BibTeX

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Computation of infinite integrals involving Bessel functions of arbitrary order by the $\overline{D}$-transformation. J. Comput. Appl. Math. 78 (1), pp. 125–130.

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External Links:

ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

BibTeX

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The On-Line Encyclopedia of Integer Sequences. Notices Amer. Math. Soc. 50 (8), pp. 912–915.

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External Links:

ISSN 0002-9920, FullText, Electronic Database, MathReview (Christian Krattenthaler), ZentralBlatt

Referenced by:

(19.20.2), (19.20.22), (19.30.13), (25.11.40), (3.12.1), (3.12.3), (3.12.4), §3.12, (4.2.11), (4.2.17), (4.2.18), (4.4.19), (5.17.6), (5.2.3), (5.4.10), (5.4.6), (5.4.7), (5.4.8), (5.4.9), (6.13.1).

Encodings:

BibTeX

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15: 5.17 Barnes’ $G$-Function (Double Gamma Function)

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5.17.2 $$G\left(n\right)=(n-2)!(n-3)!\mathrm{\cdots }1!,$$ $n=2,3,\mathrm{\dots }$.

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Symbols:

$G\left(z\right)$: Barnes’ $G$-function (or double gamma function), $!$: factorial (as in $n!$) and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/5.17.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §5.17 and Ch.5

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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:

pMML, png, TeX

See also:

Annotations for §5.17 and Ch.5

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16: Bibliography R

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Algorithm 935: IIPBF, a MATLAB toolbox for infinite integral of products of two Bessel functions. ACM Trans. Math. Softw. 40 (2), pp. 14:1–14:12.

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External Links:

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

Referenced by:

§10.74(vii), §10.74(vii), 7th item, §10.77(ix).

Encodings:

BibTeX

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Infinite Sum of the Incomplete Gamma Function Expressed in Terms of the Hurwitz Zeta Function. Mathematics 9 (16).

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External Links:

Link, ISSN 2227-7390, Document

Referenced by:

§8.15.

Encodings:

BibTeX

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Research Institute for Symbolic Computation, Hagenberg im Mühlkreis, Austria.

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Notes:

Software packages for hypergeometric and $q$-hypergeometric summation, sequences and power series, permutation groups, partition analysis, and difference equations.

External Links:

RISC Combinatorics Group

Referenced by:

6th item.

Encodings:

BibTeX

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The problem of an infinite plate under an inclined loading, with tables of the integrals of $\mathrm{Ai}(\pm \mathrm{x})$ and $\mathrm{Bi}(\pm \mathrm{x})$ . Quart. J. Mech. Appl. Math. 7 (1), pp. 1–7.

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External Links:

Document, MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§9.16, 1st item.

Encodings:

BibTeX

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Sources in the development of mathematics. Cambridge University Press, Cambridge.

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Notes:

Infinite Series and Products from the Fifteenth to the Twenty-first Century

External Links:

ISBN 978-0-521-11470-7, Document, Link, MathReview (Mehdi Hassani), ZentralBlatt

Referenced by:

Profile Ranjan Roy.

Encodings:

BibTeX

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17: 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. …

18: Bibliography L

… ►

On the theory of diffraction by an aperture in an infinite plane screen. I. Phys. Rev. 74 (8), pp. 958–974.

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External Links:

Document, MathReview (C. J. Bouwkamp), ZentralBlatt

Referenced by:

§11.12.

Encodings:

BibTeX

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Note on a method for computing infinite integrals of oscillatory functions. Proc. Cambridge Philos. Soc. 52 (4), pp. 764–768.

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External Links:

Document, MathReview (D. J. Hofsommer), ZentralBlatt

Referenced by:

§3.5(vii).

Encodings:

BibTeX

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Monotonic sequences related to zeros of Bessel functions. Numer. Algorithms 49 (1-4), pp. 221–233.

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External Links:

ISSN 1017-1398, Document, FullText, MathReview (Javier Segura), ZentralBlatt

Referenced by:

§10.21(iv), §10.21(iv).

Encodings:

BibTeX

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Evaluating infinite integrals involving Bessel functions of arbitrary order. J. Comput. Appl. Math. 64 (3), pp. 217–231.

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External Links:

ISSN 0377-0427, Document, MathReview Entry, ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

BibTeX

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Integrating some infinite oscillating tails. J. Comput. Appl. Math. 12/13, pp. 109–117.

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External Links:

ISSN 0377-0427, Document, MathReview Entry, ZentralBlatt

Referenced by:

§3.5(vii).

Encodings:

BibTeX