infinite product

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21: 10.43 Integrals

… ►For infinite integrals of triple products of modified and unmodified Bessel functions, see Gervois and Navelet (1984, 1985a, 1985b, 1986a, 1986b). …

22: 10.22 Integrals

… ►Additional infinite integrals over the product of three Bessel functions (including modified Bessel functions) are given in Gervois and Navelet (1984, 1985a, 1985b, 1986a, 1986b). …

23: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►A complex linear vector space

V

is called an inner product space if an inner product

\left\langle u,v\right\rangle\in\mathbb{C}

is defined for all

u,v\in V

with the properties: (i)

\left\langle u,v\right\rangle

is complex linear in

u

; (ii)

\left\langle u,v\right\rangle=\overline{\left\langle v,u\right\rangle}

; (iii)

\left\langle v,v\right\rangle\geq 0

; (iv) if

\left\langle v,v\right\rangle=0

then

v=0

. With norm defined by …Two elements

u

and

v

V

are orthogonal if

\left\langle u,v\right\rangle=0

. … ►where the infinite sum means convergence in norm, … ►thus generalizing the inner product of (1.18.9). …

24: 16.11 Asymptotic Expansions

… ►

16.11.2

H_{p,q}(z)=\sum_{m=1}^{p}\sum_{k=0}^{\infty}\frac{(-1)^{k}}{k!}\Gamma\left(a_{% m}+k\right)\left({\textstyle\ifrac{\prod\limits_{\begin{subarray}{c}\ell=1\\ \ell\neq m\end{subarray}}^{p}\Gamma\left(a_{\ell}-a_{m}-k\right)}{\prod\limits% _{\ell=1}^{q}\Gamma\left(b_{\ell}-a_{m}-k\right)}}\right)z^{-a_{m}-k}.

ⓘ

Defines:: $H_{p,q}(z)$ : formal infinite series (locally)
Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $!$ : factorial (as in $n!$ ), $p$ : nonnegative integer, $q$ : nonnegative integer, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters
Referenced by:: §16.11(ii), §16.11(ii)
Permalink:: http://dlmf.nist.gov/16.11.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §16.11(i), §16.11 and Ch.16

… ►

16.11.6

\left({\textstyle\ifrac{\prod\limits_{\ell=1}^{q+1}\Gamma\left(a_{\ell}\right)% }{\prod\limits_{\ell=1}^{q}\Gamma\left(b_{\ell}\right)}}\right){{}_{q+1}F_{q}}% \left({a_{1},\dots,a_{q+1}\atop b_{1},\dots,b_{q}};z\right)=H_{q+1,q}(-z),

|\operatorname{ph}\left(-z\right)|\leq\pi

;

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{ph}$ : phase, $q$ : nonnegative integer, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters, $b,b_{1},\ldots,b_{q}$ : real or complex parameters and $H_{p,q}(z)$ : formal infinite series
Permalink:: http://dlmf.nist.gov/16.11.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §16.11(ii), §16.11(ii), §16.11 and Ch.16

… ►

16.11.7

\left({\textstyle\ifrac{\prod\limits_{\ell=1}^{q}\Gamma\left(a_{\ell}\right)}{% \prod\limits_{\ell=1}^{q}\Gamma\left(b_{\ell}\right)}}\right){{}_{q}F_{q}}% \left({a_{1},\dots,a_{q}\atop b_{1},\dots,b_{q}};z\right)\sim H_{q,q}(z{% \mathrm{e}}^{\mp\pi\mathrm{i}})+E_{q,q}(z).

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $q$ : nonnegative integer, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters, $b,b_{1},\ldots,b_{q}$ : real or complex parameters, $E_{p,q}(z)$ : formal infinite series and $H_{p,q}(z)$ : formal infinite series
Referenced by:: §16.11(ii)
Permalink:: http://dlmf.nist.gov/16.11.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §16.11(ii), §16.11(ii), §16.11 and Ch.16

… ►

16.11.8

\left({\textstyle\ifrac{\prod\limits_{\ell=1}^{q-1}\Gamma\left(a_{\ell}\right)% }{\prod\limits_{\ell=1}^{q}\Gamma\left(b_{\ell}\right)}}\right){{}_{q-1}F_{q}}% \left({a_{1},\dots,a_{q-1}\atop b_{1},\dots,b_{q}};-z\right)\sim H_{q-1,q}(z)+% E_{q-1,q}(ze^{-\pi\mathrm{i}})+E_{q-1,q}(ze^{\pi\mathrm{i}}),

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{i}$ : imaginary unit, $q$ : nonnegative integer, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters, $b,b_{1},\ldots,b_{q}$ : real or complex parameters, $E_{p,q}(z)$ : formal infinite series, $H_{p,q}(z)$ : formal infinite series and $e_{k,m}$
Permalink:: http://dlmf.nist.gov/16.11.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §16.11(ii), §16.11(ii), §16.11 and Ch.16

… ►

16.11.9

\left({\textstyle\ifrac{\prod\limits_{\ell=1}^{p}\Gamma\left(a_{\ell}\right)}{% \prod\limits_{\ell=1}^{q}\Gamma\left(b_{\ell}\right)}}\right){{}_{p}F_{q}}% \left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};-z\right)\sim E_{p,q}(ze^{-% \pi\mathrm{i}})+E_{p,q}(ze^{\pi\mathrm{i}}),

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{i}$ : imaginary unit, $p$ : nonnegative integer, $q$ : nonnegative integer, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters, $b,b_{1},\ldots,b_{q}$ : real or complex parameters, $E_{p,q}(z)$ : formal infinite series and $e_{k,m}$
Referenced by:: §16.11(ii)
Permalink:: http://dlmf.nist.gov/16.11.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §16.11(ii), §16.11(ii), §16.11 and Ch.16

…

25: 5.19 Mathematical Applications

… ►Many special functions

f(z)

can be represented as a Mellin–Barnes integral, that is, an integral of a product of gamma functions, reciprocals of gamma functions, and a power of

z

, the integration contour being doubly-infinite and eventually parallel to the imaginary axis at both ends. …

26: 1.3 Determinants, Linear Operators, and Spectral Expansions

… ►

§1.3(iii) Infinite Determinants

… ► ►Of importance for special functions are infinite determinants of Hill’s type. These have the property that the double series … ►The adjoint of a matrix

\mathbf{A}

is the matrix

{\mathbf{A}}^{*}

such that

\left\langle\mathbf{A}\mathbf{a},\mathbf{b}\right\rangle=\left\langle\mathbf{a% },{\mathbf{A}}^{*}\mathbf{b}\right\rangle

for all

\mathbf{a},\mathbf{b}\in\mathbf{E}_{n}

. …

27: 25.8 Sums

… ►

25.8.4

\sum_{k=1}^{\infty}\frac{(-1)^{k}}{k}(\zeta\left(nk\right)-1)=\ln\left(\prod_{% j=0}^{n-1}\Gamma\left(2-e^{(2j+1)\pi i/n}\right)\right),

n=2,3,4,\dots

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\ln\NVar{z}$ : principal branch of logarithm function, $k$ : nonnegative integer and $n$ : nonnegative integer
Keywords:: infinite series
Source:: Adamchik and Srivastava (1998, (2.7), p. 135)
Permalink:: http://dlmf.nist.gov/25.8.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §25.8 and Ch.25

…

28: Bibliography S

… ►

L. Z. Salchev and V. B. Popov (1976) A property of the zeros of cross-product Bessel functions of different orders. Z. Angew. Math. Mech. 56 (2), pp. 120–121.

ⓘ

External Links:: Document, MathReview (S. Saran), ZentralBlatt
Referenced by:: §10.21(x).
Encodings:: BibTeX

… ►

H. Shanker (1939) On the expansion of the parabolic cylinder function in a series of the product of two parabolic cylinder functions. J. Indian Math. Soc. (N. S.) 3, pp. 226–230.

ⓘ

External Links:: ZentralBlatt
Referenced by:: §12.13(i).
Encodings:: BibTeX

►

H. Shanker (1940a) 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.

ⓘ

External Links:: MathReview (M. C. Gray), ZentralBlatt
Referenced by:: §12.13(ii).
Encodings:: BibTeX

… ►

N. T. Shawagfeh (1992) The Laplace transforms of products of Airy functions. Dirāsāt Ser. B Pure Appl. Sci. 19 (2), pp. 7–11.

ⓘ

External Links:: ISSN 0253-424X, MathReview
Referenced by:: §9.10(v).
Encodings:: BibTeX

… ►

A. Sidi (1997) Computation of infinite integrals involving Bessel functions of arbitrary order by the

\overline{D}

-transformation. J. Comput. Appl. Math. 78 (1), pp. 125–130.

ⓘ

External Links:: ISSN 0377-0427, Document, MathReview, ZentralBlatt
Referenced by:: §10.74(vii).
Encodings:: BibTeX

…

29: 25.11 Hurwitz Zeta Function

… ►

25.11.37

\sum_{k=1}^{\infty}\frac{(-1)^{k}}{k}\zeta\left(nk,a\right)=-n\ln\Gamma\left(a% \right)+\ln\left(\prod_{j=0}^{n-1}\Gamma\left(a-e^{(2j+1)\pi i/n}\right)\right),

n=2,3,4,\dots

\Re a\geq 1

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\zeta\left(\NVar{s},\NVar{a}\right)$ : Hurwitz zeta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\ln\NVar{z}$ : principal branch of logarithm function, $\operatorname{ph}$ : phase, $\Re$ : real part, $k$ : nonnegative integer, $n$ : nonnegative integer and $a$ : real or complex parameter
Keywords:: infinite series, series representation
Source:: Adamchik and Srivastava (1998, (2.12), p. 136); with the choice $\operatorname{ph}\left(-1\right)=\pi$ , which is reasonable since the gamma function is single valued
Permalink:: http://dlmf.nist.gov/25.11.E37
Encodings:: pMML, png, TeX
See also:: Annotations for §25.11(xi), §25.11 and Ch.25

…

30: 18.2 General Orthogonal Polynomials

… ►Let

(a,b)

be a finite or infinite open interval in

\mathbb{R}

. … ►Let

X

be a finite set of distinct points on

\mathbb{R}

, or a countable infinite set of distinct points on

\mathbb{R}

, and

w_{x}

x\in X

, be a set of positive constants. …when

X

is infinite, or … ►The matrix on the left-hand side is an (infinite tridiagonal) Jacobi matrix. … ►

§18.2(v) Christoffel–Darboux Formula

…