Euler sums

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11: 6.6 Power Series

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6.6.1 $$\mathrm{Ei}\left(x\right)=\gamma +\ln x+\sum _{n=1}^{\mathrm{\infty }}\frac{x^n}{n!n},$$ $x>0$.

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Symbols:

$\gamma$: Euler’s constant, $\mathrm{Ei}\left(x\right)$: exponential integral, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $x$: real variable and $n$: nonnegative integer

A&S Ref:

5.1.10

Referenced by:

§6.5

Permalink:

http://dlmf.nist.gov/6.6.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §6.6 and Ch.6

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6.6.2 $$E_1\left(z\right)=-\gamma -\ln z-\sum _{n=1}^{\mathrm{\infty }}\frac{{(-1)}^n{z}^n}{n!n}.$$

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Symbols:

$\gamma$: Euler’s constant, $E_1\left(z\right)$: exponential integral, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $z$: complex variable and $n$: nonnegative integer

A&S Ref:

5.1.11

Referenced by:

§6.4, §6.5

Permalink:

http://dlmf.nist.gov/6.6.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §6.6 and Ch.6

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6.6.6 $$\mathrm{Ci}\left(z\right)=\gamma +\ln z+\sum _{n=1}^{\mathrm{\infty }}\frac{{(-1)}^n{z}^{2n}}{(2n)!(2n)}.$$

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Symbols:

$\gamma$: Euler’s constant, $\mathrm{Ci}\left(z\right)$: cosine integral, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $z$: complex variable and $n$: nonnegative integer

A&S Ref:

5.2.16

Referenced by:

§6.5

Permalink:

http://dlmf.nist.gov/6.6.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §6.6 and Ch.6

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12: 24.4 Basic Properties

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§24.4(iii) Sums of Powers

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24.4.8 $$\sum _{k=1}^m{(-1)}^{m-k}{k}^n=\frac{E_n\left(m+1\right)+{(-1)}^m{E}_n\left(0\right)}{2}.$$

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Symbols:

$E_n\left(x\right)$: Euler polynomials, $k$: integer, $m$: integer and $n$: integer

A&S Ref:

23.1.4

Referenced by:

§24.17(iii)

Permalink:

http://dlmf.nist.gov/24.4.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §24.4(iii), §24.4 and Ch.24

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24.4.13 $$E_n\left(x+h\right)=\sum _{k=0}^n\left(\genfrac{}{}{0pt}{}{n}{k}\right)E_k\left(x\right)h^{n-k},$$

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Symbols:

$E_n\left(x\right)$: Euler polynomials, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $k$: integer, $n$: integer and $x$: real or complex

A&S Ref:

23.1.7

Permalink:

http://dlmf.nist.gov/24.4.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §24.4(iv), §24.4 and Ch.24

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24.4.17 $$E_{2n}=1-\sum _{k=1}^n\left(\genfrac{}{}{0pt}{}{2n}{2k-1}\right)\frac{2^{2k}(2^{2k}-1)B_{2k}}{2k}.$$

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Symbols:

$B_n$: Bernoulli numbers, $E_n$: Euler numbers, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $k$: integer and $n$: integer

Permalink:

http://dlmf.nist.gov/24.4.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §24.4(iv), §24.4 and Ch.24

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24.4.20 $$E_n\left(mx\right)=m^n\sum _{k=0}^{m-1}{(-1)}^k{E}_n\left(x+\frac{k}{m}\right),$$ $m=1,3,5,\mathrm{\dots }$.

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Symbols:

$E_n\left(x\right)$: Euler polynomials, $k$: integer, $m$: integer, $n$: integer and $x$: real or complex

A&S Ref:

23.1.10

Permalink:

http://dlmf.nist.gov/24.4.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §24.4(v), §24.4(v), §24.4 and Ch.24

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13: 13.7 Asymptotic Expansions for Large Argument

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13.7.1 $$𝐌(a,b,x)\sim \frac{{\mathrm{e}}^x{x}^{a-b}}{\mathrm{\Gamma }\left(a\right)}\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(1-a\right)}_s{\left(b-a\right)}_s}{s!}{x}^{-s},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $𝐌(a,b,z)$: Olver’s confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $s$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/13.7.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §13.7(i), §13.7 and Ch.13

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13.7.2 $$𝐌(a,b,z)\sim \frac{{\mathrm{e}}^z{z}^{a-b}}{\mathrm{\Gamma }\left(a\right)}\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(1-a\right)}_s{\left(b-a\right)}_s}{s!}{z}^{-s}+\frac{{\mathrm{e}}^{\pm \pi \mathrm{i}a}{z}^{-a}}{\mathrm{\Gamma }\left(b-a\right)}\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(a\right)}_s{\left(a-b+1\right)}_s}{s!}{(-z)}^{-s},$$ $-\frac{1}{2}\pi +\delta \le \pm \mathrm{ph}z\le \frac{3}{2}\pi -\delta$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $𝐌(a,b,z)$: Olver’s confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $\mathrm{ph}$: phase, $s$: nonnegative integer, $z$: complex variable and $\delta$: small positive constant

A&S Ref:

13.5.1 (with corrected region of validity)

Referenced by:

§13.2(iv), §13.7(ii), §13.9(i)

Permalink:

http://dlmf.nist.gov/13.7.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §13.7(i), §13.7 and Ch.13

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13.7.11 $$R_n(a,b,z)=\frac{{(-1)}^{n}2\pi z^{a-b}}{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(a-b+1\right)}\left(\sum _{s=0}^{m-1}\frac{{\left(1-a\right)}_s{\left(b-a\right)}_s}{s!}{(-z)}^{-s}{G}_{n+2a-b-s}(z)+{\left(1-a\right)}_m{\left(b-a\right)}_m{R}_{m,n}(a,b,z)\right),$$

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Defines:

$R_n(a,b,z)$: remainder (locally)

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $m$: integer, $n$: nonnegative integer, $s$: nonnegative integer, $z$: complex variable and $G_p(z)$: expansion

Referenced by:

§13.29(i)

Permalink:

http://dlmf.nist.gov/13.7.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §13.7(iii), §13.7 and Ch.13

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14: 8.18 Asymptotic Expansions of $I_x(a,b)$

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8.18.1 $$I_x(a,b)=\mathrm{\Gamma }\left(a+b\right)x^a{(1-x)}^{b-1}\left(\sum _{k=0}^{n-1}\frac{1}{\mathrm{\Gamma }\left(a+k+1\right)\mathrm{\Gamma }\left(b-k\right)}{\left(\frac{x}{1-x}\right)}^k+O\left(\frac{1}{\mathrm{\Gamma }\left(a+n+1\right)}\right)\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, $I_x(a,b)$: incomplete beta function, $k$: nonnegative integer, $n$: nonnegative integer, $a$: parameter, $b$: parameter and $x$: variable

Referenced by:

§8.18(i)

Permalink:

http://dlmf.nist.gov/8.18.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §8.18(i), §8.18 and Ch.8

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8.18.3 $$I_x(a,b)=\frac{\mathrm{\Gamma }\left(a+b\right)}{\mathrm{\Gamma }\left(a\right)}\left(\sum _{k=0}^{n-1}{d}_k{F}_k+O\left(a^{-n}\right)F_0\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, $I_x(a,b)$: incomplete beta function, $\sim$: Poincaré asymptotic expansion, $k$: nonnegative integer, $n$: nonnegative integer, $a$: parameter, $b$: parameter, $x$: variable, $F_k$: functions and $d_k$: coefficients

Referenced by:

Erratum (V1.0.14) for Equation (8.18.3)

Permalink:

http://dlmf.nist.gov/8.18.E3

Encodings:

pMML, png, TeX

Addition (effective with 1.0.14):

Previously this equation appeared without the order estimate as $I_x(a,b)\sim \frac{\mathrm{\Gamma }\left(a+b\right)}{\mathrm{\Gamma }\left(a\right)}{\sum }_{k=0}^{\mathrm{\infty }}{d}_k{F}_k$. The range of $x$ was extended to include $1$.

See also:

Annotations for §8.18(ii), §8.18(ii), §8.18 and Ch.8

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8.18.14 $$I_x(a,b)\sim Q(b,a\zeta )-\frac{{(2\pi b)}^{-1/2}}{{\mathrm{\Gamma }}^{\ast }\left(b\right)}{\left(\frac{x}{x_0}\right)}^a{\left(\frac{1-x}{1-x_0}\right)}^b\sum _{k=0}^{\mathrm{\infty }}\frac{h_k(\zeta ,\mu )}{a^k},$$

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Symbols:

$I_x(a,b)$: incomplete beta function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $Q(a,z)$: normalized incomplete gamma function, ${\mathrm{\Gamma }}^{\ast }\left(z\right)$: scaled gamma function, $k$: nonnegative integer, $a$: parameter, $b$: parameter, $x$: variable, $\mu$ , $\zeta$: variable and $h_k(\zeta ,\mu )$: coefficients

Permalink:

http://dlmf.nist.gov/8.18.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §8.18(ii), §8.18(ii), §8.18 and Ch.8

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15: 27.6 Divisor Sums

… ►Generating functions, Euler products, and Möbius inversion are used to evaluate many sums extended over divisors. … ►

27.6.5 $$\sum _{d|n}\frac{|\mu \left(d\right)|}{\varphi \left(d\right)}=\frac{n}{\varphi \left(n\right)},$$

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Symbols:

$\varphi \left(n\right)$: Euler’s totient, $\mu \left(n\right)$: Möbius function, $d$: positive integer and $n$: positive integer

Permalink:

http://dlmf.nist.gov/27.6.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §27.6 and Ch.27

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27.6.6 $$\sum _{d|n}{\varphi }_k\left(d\right){\left(\frac{n}{d}\right)}^k=1^k+2^k+\mathrm{\cdots }+n^k,$$

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Symbols:

${\varphi }_k\left(n\right)$: sum of powers of integers relatively prime to a number, $d$: positive integer, $k$: positive integer and $n$: positive integer

Permalink:

http://dlmf.nist.gov/27.6.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §27.6 and Ch.27

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16: 36.6 Scaling Relations

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$\text{cuspoids: }{\gamma }_K={\displaystyle \sum _{m=1}^K}{\gamma }_{mK}={\displaystyle \frac{K(K+3)}{2(K+2)}},$

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$\text{umbilics: }{\gamma }^{(\mathrm{U})}={\displaystyle \sum _{m=1}^3}{\gamma }_m^{(\mathrm{U})}=\frac{5}{3}.$

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17: 2.10 Sums and Sequences

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§2.10(i) Euler–Maclaurin Formula

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2.10.8 $$\sum _{j=1}^{n-1}\frac{1}{j}\sim \ln n+\gamma -\frac{1}{2n}-\sum _{s=1}^{\mathrm{\infty }}\frac{B_{2s}}{2s}\frac{1}{n^{2s}},$$ $n\to \mathrm{\infty }$.

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Symbols:

$B_n$: Bernoulli numbers, $\gamma$: Euler’s constant, $\sim$: Poincaré asymptotic expansion, $\ln z$: principal branch of logarithm function and $n$: integer

Referenced by:

§2.10(ii)

Permalink:

http://dlmf.nist.gov/2.10.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §2.10(i), §2.10(i), §2.10 and Ch.2

►In both expansions the remainder term is bounded in absolute value by the first neglected term in the sum, and has the same sign, provided that in the case of (2.10.7), truncation takes place at $s=2m-1$, where $m$ is any positive integer satisfying $m\ge \frac{1}{2}(\alpha +1)$. … ► … ►

2.10.20 $$\sum _{j=0}^{n-1}\frac{x^j}{{(j!)}^3}=\frac{1}{2\mathrm{i}}{\int }_{𝒞}\frac{x^t}{{(\mathrm{\Gamma }\left(t+1\right))}^3}\cot \left(\pi t\right)dt,$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cot z$: cotangent function, $dx$: differential of $x$ , $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $\int$: integral and $𝒞$: contour

Permalink:

http://dlmf.nist.gov/2.10.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §2.10(iii), §2.10(iii), §2.10 and Ch.2

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18: 10.23 Sums

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10.23.8 $$\frac{{𝒞}_{\nu }\left(w\right)}{w^{\nu }}=2^{\nu }\mathrm{\Gamma }\left(\nu \right)\sum _{k=0}^{\mathrm{\infty }}(\nu +k)\frac{{𝒞}_{\nu +k}\left(u\right)}{u^{\nu }}\frac{J_{\nu +k}\left(v\right)}{v^{\nu }}{C}_k^{(\nu )}\left(\cos \alpha \right),$$ $\nu \ne 0,-1,\mathrm{\dots }$, ,

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\cos z$: cosine function, ${𝒞}_{\nu }\left(z\right)$: cylinder function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $k$: nonnegative integer and $\nu$: complex parameter

A&S Ref:

9.1.80

Referenced by:

§10.23(ii), §10.23(ii), §10.23(ii), §10.44(ii), §10.60(i), §10.60(ii)

Permalink:

http://dlmf.nist.gov/10.23.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §10.23(ii), §10.23(ii), §10.23 and Ch.10

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10.23.9 $${\mathrm{e}}^{\mathrm{i}v\cos \alpha }=\frac{\mathrm{\Gamma }\left(\nu \right)}{{(\frac{1}{2}v)}^{\nu }}\sum _{k=0}^{\mathrm{\infty }}(\nu +k){\mathrm{i}}^k{J}_{\nu +k}\left(v\right)C_k^{(\nu )}\left(\cos \alpha \right),$$ $\nu \ne 0,-1,\mathrm{\dots }$.

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $k$: nonnegative integer and $\nu$: complex parameter

A&S Ref:

9.1.81

Referenced by:

§10.23(ii)

Permalink:

http://dlmf.nist.gov/10.23.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §10.23(ii), §10.23(ii), §10.23 and Ch.10

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10.23.15 $${(\frac{1}{2}z)}^{\nu }=\sum _{k=0}^{\mathrm{\infty }}\frac{(\nu +2k)\mathrm{\Gamma }\left(\nu +k\right)}{k!}{J}_{\nu +2k}\left(z\right),$$ $\nu \ne 0,-1,-2,\mathrm{\dots }$,

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $!$: factorial (as in $n!$), $k$: nonnegative integer, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.87

Referenced by:

§10.22(ii), §10.23(iii), §10.44(iii), §10.74(iv)

Permalink:

http://dlmf.nist.gov/10.23.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §10.23(iii), §10.23(iii), §10.23 and Ch.10

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10.23.16 $$Y_0\left(z\right)=\frac{2}{\pi }\left(\ln \left(\frac{1}{2}z\right)+\gamma \right)J_0\left(z\right)-\frac{4}{\pi }\sum _{k=1}^{\mathrm{\infty }}{(-1)}^k\frac{J_{2k}\left(z\right)}{k},$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\gamma$: Euler’s constant, $\pi$: the ratio of the circumference of a circle to its diameter, $\ln z$: principal branch of logarithm function, $k$: nonnegative integer and $z$: complex variable

A&S Ref:

9.1.89

Permalink:

http://dlmf.nist.gov/10.23.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §10.23(iii), §10.23(iii), §10.23 and Ch.10

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19: 15.2 Definitions and Analytical Properties

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15.2.1 $$F(a,b;c;z)=\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(a\right)}_s{\left(b\right)}_s}{{\left(c\right)}_{s}s!}{z}^s=1+\frac{ab}{c}z+\frac{a(a+1)b(b+1)}{c(c+1)2!}{z}^2+\mathrm{\cdots }=\frac{\mathrm{\Gamma }\left(c\right)}{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(b\right)}\sum _{s=0}^{\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(a+s\right)\mathrm{\Gamma }\left(b+s\right)}{\mathrm{\Gamma }\left(c+s\right)s!}{z}^s,$$

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Defines:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $s$: nonnegative integer, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

A&S Ref:

15.1.1

Referenced by:

§15.19(i), §15.4(iii), §18.30(v), §18.35(i), §18.35(ii), §19.19

Permalink:

http://dlmf.nist.gov/15.2.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §15.2(i), §15.2 and Ch.15

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15.2.2 $$𝐅(a,b;c;z)=\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(a\right)}_s{\left(b\right)}_s}{\mathrm{\Gamma }\left(c+s\right)s!}{z}^s,$$ ,

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Defines:

$𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), ${}_p{}^{}𝐅_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}𝐅_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$: scaled (or Olver’s) generalized hypergeometric function, $s$: nonnegative integer, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Referenced by:

§15.12(i), §16.2(v)

Permalink:

http://dlmf.nist.gov/15.2.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §15.2(i), §15.2 and Ch.15

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20: 17.12 Bailey Pairs

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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:

pMML, png, TeX

See also:

Annotations for §17.12, §17.12 and Ch.17

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

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