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31: Bibliography J

… ►

C. Jordan (1939) Calculus of Finite Differences. Hungarian Agent Eggenberger Book-Shop, Budapest.

ⓘ

Notes:: Third edition published in 1965 by Chelsea Publishing Co., New York, and American Mathematical Society, AMS Chelsea Book Series, Providence, RI
External Links:: MathReview (W. E. Milne), ZentralBlatt
Referenced by:: §26.1, §26.1, §26.8(vii), §5.16, Notations S.
Encodings:: BibTeX

►

C. Jordan (1965) Calculus of Finite Differences. 3rd edition, AMS Chelsea, Providence, RI.

ⓘ

External Links:: MathReview, ZentralBlatt
Referenced by:: §24.17(i), §24.6.
Encodings:: BibTeX

…

32: 1.7 Inequalities

… ►

§1.7(i) Finite Sums

…

33: 33.9 Expansions in Series of Bessel Functions

… ►The series (33.9.1) converges for all finite values of

\eta

and

\rho

. … ►The series (33.9.3) and (33.9.4) converge for all finite positive values of

\left|\eta\right|

and

\rho

. …

34: 2.1 Definitions and Elementary Properties

… ►If

c

is a finite limit point of

\mathbf{X}

, then … ►For (2.1.14)

\mathbf{X}

can be the positive real axis or any unbounded sector in

\mathbb{C}

of finite angle. … ►Similarly for finite limit point

c

in place of

\infty

. … ►where

c

is a finite, or infinite, limit point of

\mathbf{X}

. …

35: 18.3 Definitions

… ►For finite power series of the Jacobi, ultraspherical, Laguerre, and Hermite polynomials, see §18.5(iii) (in powers of

x-1

for Jacobi polynomials, in powers of

x

for the other cases). … ►For

-1-\beta>\alpha>-1

a finite system of Jacobi polynomials

P^{(\alpha,\beta)}_{n}\left(x\right)

is orthogonal on

(1,\infty)

with weight function

w(x)=(x-1)^{\alpha}(x+1)^{\beta}

. For

\nu\in\mathbb{R}

and

N>-\tfrac{1}{2}

a finite system of Jacobi polynomials

P^{(-N-1+\mathrm{i}\nu,-N-1-\mathrm{i}\nu)}_{n}\left(\mathrm{i}x\right)

(called pseudo Jacobi polynomials or Routh–Romanovski polynomials) is orthogonal on

(-\infty,\infty)

with

w(x)=\left(1+x^{2}\right)^{-N-1}{\mathrm{e}}^{2\nu\operatorname{arctan}x}

. … ►However, in general they are not orthogonal with respect to a positive measure, but a finite system has such an orthogonality. …

36: 33.8 Continued Fractions

… ►The continued fraction (33.8.1) converges for all finite values of

\rho

, and (33.8.2) converges for all

\rho\neq 0

. …

37: 33.23 Methods of Computation

… ►The power-series expansions of §§33.6 and 33.19 converge for all finite values of the radii

\rho

and

r

, respectively, and may be used to compute the regular and irregular solutions. …

38: Bibliography R

… ►

G. Rota, D. Kahaner, and A. Odlyzko (1973) On the foundations of combinatorial theory. VIII. Finite operator calculus. J. Math. Anal. Appl. 42, pp. 684–760.

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External Links:: ISSN 0022-247X, Document, Link, MathReview (P. Saphar), ZentralBlatt
Referenced by:: §18.2(xii).
Encodings:: BibTeX

… ►

K. Rottbrand (2000) Finite-sum rules for Macdonald’s functions and Hankel’s symbols. Integral Transform. Spec. Funct. 10 (2), pp. 115–124.

ⓘ

External Links:: ISSN 1065-2469, Document, MathReview, ZentralBlatt
Referenced by:: §10.60(iii).
Encodings:: BibTeX

…

39: 13.9 Zeros

… ►When

a,b\in\mathbb{R}

the number of real zeros is finite. … ►For fixed

a

and

z

\mathbb{C}

the function

M\left(a,b,z\right)

has only a finite number of

b

-zeros. … ►For fixed

a

and

b

\mathbb{C}

U\left(a,b,z\right)

has a finite number of

z

-zeros in the sector

|\operatorname{ph}z|\leq\tfrac{3}{2}\pi-\delta(<\tfrac{3}{2}\pi)

. …

40: 25.16 Mathematical Applications

… ►

25.16.6

H\left(s\right)=-\zeta'\left(s\right)+\gamma\zeta\left(s\right)+\frac{1}{2}% \zeta\left(s+1\right)+\sum_{r=1}^{k}\zeta\left(1-2r\right)\zeta\left(s+2r% \right)+\sum_{n=1}^{\infty}\frac{1}{n^{s}}\int_{n}^{\infty}\frac{\widetilde{B}% _{2k+1}\left(x\right)}{x^{2k+2}}\,\mathrm{d}x,

ⓘ

Symbols:: $\gamma$ : Euler’s constant, $H\left(\NVar{s}\right)$ : Euler sums, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\widetilde{B}_{\NVar{n}}\left(\NVar{x}\right)$ : periodic Bernoulli functions, $k$ : nonnegative integer, $n$ : nonnegative integer, $x$ : real variable and $s$ : complex variable
Keywords:: Euler sum, finite sum, improper integral, infinite series, integral representation, series representation
Source:: Apostol and Vu (1984, (3), p. 87)
Permalink:: http://dlmf.nist.gov/25.16.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §25.16(ii), §25.16 and Ch.25

►

25.16.7

H\left(s\right)=\frac{1}{2}\zeta\left(s+1\right)+\frac{\zeta\left(s\right)}{s-% 1}-\sum_{r=1}^{k}\genfrac{(}{)}{0.0pt}{}{s+2r-2}{2r-1}\zeta\left(1-2r\right)% \zeta\left(s+2r\right)-\genfrac{(}{)}{0.0pt}{}{s+2k}{2k+1}\sum_{n=1}^{\infty}% \frac{1}{n}\int_{n}^{\infty}\frac{\widetilde{B}_{2k+1}\left(x\right)}{x^{s+2k+% 1}}\,\mathrm{d}x.

ⓘ

Symbols:: $H\left(\NVar{s}\right)$ : Euler sums, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\widetilde{B}_{\NVar{n}}\left(\NVar{x}\right)$ : periodic Bernoulli functions, $k$ : nonnegative integer, $n$ : nonnegative integer, $x$ : real variable and $s$ : complex variable
Keywords:: Euler sum, finite sum, improper integral, infinite series, integral representation, series representation
Source:: Apostol and Vu (1984, (6), p. 88)
Permalink:: http://dlmf.nist.gov/25.16.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §25.16(ii), §25.16 and Ch.25

… ►when both

H\left(s,z\right)

and

H\left(z,s\right)

are finite. …