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31: Frank Garvan

… ►His research is in the areas of $q$-series and modular forms, and he enjoys using MAPLE in his research. …

32: Christopher J. Howls

… ►In 2008 he was appointed to the editorial board of the Proceedings of the Royal Society of London Series A. …

33: 27.5 Inversion Formulas

… ►If a Dirichlet series $F(s)$ generates $f(n)$, and $G(s)$ generates $g(n)$, then the product $F(s)G(s)$ generates … ►

27.5.6 $$G(x)=\sum _{n\le x}F\left(\frac{x}{n}\right)⟺F(x)=\sum _{n\le x}\mu \left(n\right)G\left(\frac{x}{n}\right),$$

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Symbols:

$\mu \left(n\right)$: Möbius function, $n$: positive integer, $x$: real number, $F(s)$: Dirichlet series and $G(s)$: Dirichlet series

A&S Ref:

24.3.1 II.C

Permalink:

http://dlmf.nist.gov/27.5.E6

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Annotations for §27.5 and Ch.27

►

27.5.7 $$G(x)=\sum _{m=1}^{\mathrm{\infty }}\frac{F(mx)}{m^s}⟺F(x)=\sum _{m=1}^{\mathrm{\infty }}\mu \left(m\right)\frac{G(mx)}{m^s},$$

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Symbols:

$\mu \left(n\right)$: Möbius function, $m$: positive integer, $x$: real number, $F(s)$: Dirichlet series and $G(s)$: Dirichlet series

Referenced by:

§27.17, §27.5

Permalink:

http://dlmf.nist.gov/27.5.E7

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Annotations for §27.5 and Ch.27

…

34: 33.23 Methods of Computation

… ►

§33.23(ii) Series Solutions

►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. … ►Thus the regular solutions can be computed from the power-series expansions (§§33.6, 33.19) for small values of the radii and then integrated in the direction of increasing values of the radii. … ►Curtis (1964a, §10) describes the use of series, radial integration, and other methods to generate the tables listed in §33.24. … ►Thompson and Barnett (1985, 1986) and Thompson (2004) use combinations of series, continued fractions, and Padé-accelerated asymptotic expansions (§3.11(iv)) for the analytic continuations of Coulomb functions. …

35: 8.27 Approximations

… ►

Luke (1969b, pp. 25, 40–41) gives Chebyshev-series expansions for $\mathrm{\Gamma }(a,\omega z)$ (by specifying parameters) with , and $\gamma (a,\omega z)$ with $0\le \omega \le 1$; see also Temme (1994b, §3).

… ►

Luke (1975, p. 103) gives Chebyshev-series expansions for $E_1\left(x\right)$ and related functions for $x\ge 5$.

…

36: 11.15 Approximations

… ►

§11.15(i) Expansions in Chebyshev Series

►

Luke (1975, pp. 416–421) gives Chebyshev-series expansions for ${𝐇}_n\left(x\right)$, ${𝐋}_n\left(x\right)$, $0\le \left|x\right|\le 8$, and ${𝐇}_n\left(x\right)-Y_n\left(x\right)$, $x\ge 8$, for $n=0,1$; ${\int }_0^x{t}^{-m}{𝐇}_0\left(t\right)dt$, ${\int }_0^x{t}^{-m}{𝐋}_0\left(t\right)dt$, $0\le \left|x\right|\le 8$, $m=0,1$ and ${\int }_0^x({𝐇}_0\left(t\right)-Y_0\left(t\right))dt$, ${\int }_x^{\mathrm{\infty }}{t}^{-1}({𝐇}_0\left(t\right)-Y_0\left(t\right))dt$, $x\ge 8$; the coefficients are to 20D.

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MacLeod (1993) gives Chebyshev-series expansions for ${𝐋}_0\left(x\right)$, ${𝐋}_1\left(x\right)$, $0\le x\le 16$, and $I_0\left(x\right)-{𝐋}_0\left(x\right)$, $I_1\left(x\right)-{𝐋}_1\left(x\right)$, $x\ge 16$; the coefficients are to 20D.

…

37: 25.8 Sums

… ►

25.8.1 $$\sum _{k=2}^{\mathrm{\infty }}\left(\zeta \left(k\right)-1\right)=1.$$

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function and $k$: nonnegative integer

Keywords:

infinite series

Source:

Adamchik and Srivastava (1998, (1.4), p. 132)

Permalink:

http://dlmf.nist.gov/25.8.E1

Encodings:

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See also:

Annotations for §25.8 and Ch.25

►

25.8.2 $$\sum _{k=0}^{\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(s+k\right)}{(k+1)!}\left(\zeta \left(s+k\right)-1\right)=\mathrm{\Gamma }\left(s-1\right),$$ $s\ne 1,0,-1,-2,\mathrm{\dots }$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta \left(s\right)$: Riemann zeta function, $!$: factorial (as in $n!$), $k$: nonnegative integer and $s$: complex variable

Keywords:

infinite series

Source:

Landau (1953, (2), p. 273); with (5.2.4), (5.2.5)

Permalink:

http://dlmf.nist.gov/25.8.E2

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Annotations for §25.8 and Ch.25

… ►

25.8.5 $$\sum _{k=2}^{\mathrm{\infty }}\zeta \left(k\right)z^k=-\gamma z-z\psi \left(1-z\right),$$ .

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Symbols:

$\gamma$: Euler’s constant, $\zeta \left(s\right)$: Riemann zeta function, $\psi \left(z\right)$: psi (or digamma) function, $k$: nonnegative integer and $z$: complex variable

Keywords:

infinite series

Source:

Erdélyi et al. (1953a, (1.17.5), p. 45); with $z\mapsto -z$

Referenced by:

(25.8.7)

Permalink:

http://dlmf.nist.gov/25.8.E5

Encodings:

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See also:

Annotations for §25.8 and Ch.25

… ►

25.8.9 $$\sum _{k=1}^{\mathrm{\infty }}\frac{\zeta \left(2k\right)}{(2k+1)2^{2k}}=\frac{1}{2}-\frac{1}{2}\ln 2.$$

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $\ln z$: principal branch of logarithm function and $k$: nonnegative integer

Keywords:

infinite series

Source:

Srivastava and Choi (2001, (467), p. 212)

Permalink:

http://dlmf.nist.gov/25.8.E9

Encodings:

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See also:

Annotations for §25.8 and Ch.25

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25.8.10 $$\sum _{k=1}^{\mathrm{\infty }}\frac{\zeta \left(2k\right)}{(2k+1)(2k+2)2^{2k}}=\frac{1}{4}-\frac{7}{4{\pi }^2}\zeta \left(3\right).$$

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter and $k$: nonnegative integer

Keywords:

infinite series

Source:

Ewell (1990, (1), p. 219)

Permalink:

http://dlmf.nist.gov/25.8.E10

Encodings:

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Annotations for §25.8 and Ch.25

…

38: 24.8 Series Expansions

§24.8 Series Expansions

►

§24.8(i) Fourier Series

… ►If $n=1,2,\mathrm{\dots }$ and $0\le x\le 1$, then … ►

§24.8(ii) Other Series

… ►

24.8.9 $$E_{2n}={(-1)}^n\sum _{k=1}^{\mathrm{\infty }}\frac{k^{2n}}{\cosh \left(\frac{1}{2}\pi k\right)}-4\sum _{k=0}^{\mathrm{\infty }}\frac{{(-1)}^k{(2k+1)}^{2n}}{{\mathrm{e}}^{2\pi (2k+1)}-1},$$ $n=1,2,\mathrm{\dots }$.

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Symbols:

$E_n$: Euler numbers, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $k$: integer and $n$: integer

Keywords:

Euler polynomials, infinite series expansions, other

Permalink:

http://dlmf.nist.gov/24.8.E9

Encodings:

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See also:

Annotations for §24.8(ii), §24.8 and Ch.24

39: 36.8 Convergent Series Expansions

§36.8 Convergent Series Expansions

… ►For multinomial power series for ${\mathrm{\Psi }}_K\left(𝐱\right)$, see Connor and Curtis (1982). …

40: 28.11 Expansions in Series of Mathieu Functions

§28.11 Expansions in Series of Mathieu Functions

… ►The series (28.11.1) converges absolutely and uniformly on any compact subset of the strip $S$. … ►

28.11.7 $$\sin (2m+2)z=\sum _{n=0}^{\mathrm{\infty }}{B}_{2m+2}^{2n+2}(q){\mathrm{se}}_{2n+2}(z,q).$$

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Symbols:

${\mathrm{se}}_n(z,q)$: Mathieu function, $\sin z$: sine function, $m$: integer, $q=h^2$: parameter, $n$: integer, $z$: complex variable and $B_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.11.E7

Encodings:

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See also:

Annotations for §28.11, §28.11 and Ch.28