Euler sums

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21: 15.8 Transformations of Variable

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15.8.8 $$𝐅(\genfrac{}{}{0pt}{}{a,a+m}{c};z)=\frac{{(-z)}^{-a}}{\mathrm{\Gamma }\left(a+m\right)}\sum _{k=0}^{m-1}\frac{{\left(a\right)}_k(m-k-1)!}{k!\mathrm{\Gamma }\left(c-a-k\right)}{z}^{-k}+\frac{{(-z)}^{-a}}{\mathrm{\Gamma }\left(a\right)}\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(a+m\right)}_k}{k!(k+m)!\mathrm{\Gamma }\left(c-a-k-m\right)}{(-1)}^k{z}^{-k-m}\left(\ln \left(-z\right)+\psi \left(k+1\right)+\psi \left(k+m+1\right)-\psi \left(a+k+m\right)-\psi \left(c-a-k-m\right)\right),$$ ,

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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, $\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $\mathrm{ph}$: phase, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $z$: complex variable, $a$: real or complex parameter, $c$: real or complex parameter, $k$: integer and $m$: integer

Referenced by:

§15.12(i), §15.19(iv), §15.8(ii), §15.8(ii), §16.11(i)

Permalink:

http://dlmf.nist.gov/15.8.E8

Encodings:

pMML, png, TeX

Correction (effective with 1.1.2):

Pochhammer symbols now link to their definition.

See also:

Annotations for §15.8(ii), §15.8 and Ch.15

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15.8.9 $$𝐅(\genfrac{}{}{0pt}{}{a,a+m}{c};z)=\frac{{(1-z)}^{-a}}{\mathrm{\Gamma }\left(a+m\right)\mathrm{\Gamma }\left(c-a\right)}\sum _{k=0}^{m-1}\frac{{\left(a\right)}_k{\left(c-a-m\right)}_k(m-k-1)!}{k!}{(z-1)}^{-k}+\frac{{(-1)}^m{(1-z)}^{-a-m}}{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(c-a-m\right)}\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(a+m\right)}_k{\left(c-a\right)}_k}{k!(k+m)!}{(1-z)}^{-k}(\ln \left(1-z\right)+\psi \left(k+1\right)+\psi \left(k+m+1\right)-\psi \left(a+k+m\right)-\psi \left(c-a+k\right)),$$ .

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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, $\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $\mathrm{ph}$: phase, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $z$: complex variable, $a$: real or complex parameter, $c$: real or complex parameter, $k$: integer and $m$: integer

Referenced by:

§15.8(ii)

Permalink:

http://dlmf.nist.gov/15.8.E9

Encodings:

pMML, png, TeX

Correction (effective with 1.1.2):

Pochhammer symbols now link to their definition.

See also:

Annotations for §15.8(ii), §15.8 and Ch.15

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15.8.10 $$𝐅(\genfrac{}{}{0pt}{}{a,b}{a+b+m};z)=\frac{1}{\mathrm{\Gamma }\left(a+m\right)\mathrm{\Gamma }\left(b+m\right)}\sum _{k=0}^{m-1}\frac{{\left(a\right)}_k{\left(b\right)}_k(m-k-1)!}{k!}{(z-1)}^k-\frac{{(z-1)}^m}{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(b\right)}\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(a+m\right)}_k{\left(b+m\right)}_k}{k!(k+m)!}{(1-z)}^k\left(\ln \left(1-z\right)-\psi \left(k+1\right)-\psi \left(k+m+1\right)+\psi \left(a+k+m\right)+\psi \left(b+k+m\right)\right),$$ ,

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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, $\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $\mathrm{ph}$: phase, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $k$: integer and $m$: integer

Referenced by:

§15.4(ii), §15.8(ii)

Permalink:

http://dlmf.nist.gov/15.8.E10

Encodings:

pMML, png, TeX

Correction (effective with 1.1.2):

Pochhammer symbols now link to their definition.

See also:

Annotations for §15.8(ii), §15.8 and Ch.15

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15.8.11 $$𝐅(\genfrac{}{}{0pt}{}{a,b}{a+b+m};z)=\frac{z^{-a}}{\mathrm{\Gamma }\left(a+m\right)}\sum _{k=0}^{m-1}\frac{{\left(a\right)}_k(m-k-1)!}{k!\mathrm{\Gamma }\left(b+m-k\right)}{\left(1-\frac{1}{z}\right)}^k-\frac{z^{-a}}{\mathrm{\Gamma }\left(a\right)}\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(a+m\right)}_k}{k!(k+m)!\mathrm{\Gamma }\left(b-k\right)}{(-1)}^k{\left(1-\frac{1}{z}\right)}^{k+m}\left(\ln \left(\frac{1-z}{z}\right)-\psi \left(k+1\right)-\psi \left(k+m+1\right)+\psi \left(a+k+m\right)+\psi \left(b-k\right)\right),$$ .

ⓘ

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, $\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $k$: integer and $m$: integer

Referenced by:

§15.8(ii), §15.8(ii)

Permalink:

http://dlmf.nist.gov/15.8.E11

Encodings:

pMML, png, TeX

Correction (effective with 1.1.2):

Pochhammer symbols now link to their definition.

See also:

Annotations for §15.8(ii), §15.8 and Ch.15

…

22: 24.6 Explicit Formulas

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24.6.4 $$E_{2n}=\sum _{k=1}^n\frac{1}{2^{k-1}}\sum _{j=1}^k{(-1)}^j\left(\genfrac{}{}{0pt}{}{2k}{k-j}\right)j^{2n},$$

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Symbols:

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

Referenced by:

§24.6

Permalink:

http://dlmf.nist.gov/24.6.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §24.6 and Ch.24

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24.6.6 $$E_{2n}=\sum _{k=1}^{2n}\frac{{(-1)}^k}{2^{k-1}}\left(\genfrac{}{}{0pt}{}{2n+1}{k+1}\right)\sum _{j=0}^{\lfloor \frac{1}{2}k-\frac{1}{2}\rfloor }\left(\genfrac{}{}{0pt}{}{k}{j}\right){(k-2j)}^{2n}.$$

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Symbols:

$E_n$: Euler numbers, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $\lfloor x\rfloor$: floor of $x$ , $j$: integer, $k$: integer and $n$: integer

Referenced by:

§24.6

Permalink:

http://dlmf.nist.gov/24.6.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §24.6 and Ch.24

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24.6.8 $$E_n\left(x\right)=\frac{1}{2^n}\sum _{k=1}^{n+1}\sum _{j=0}^{k-1}{(-1)}^j\left(\genfrac{}{}{0pt}{}{n+1}{k}\right){(x+j)}^n.$$

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Symbols:

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

Referenced by:

§24.6, §24.6

Permalink:

http://dlmf.nist.gov/24.6.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §24.6 and Ch.24

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24.6.10 $$E_n=\frac{1}{2^n}\sum _{k=1}^{n+1}\left(\genfrac{}{}{0pt}{}{n+1}{k}\right)\sum _{j=0}^{k-1}{(-1)}^j{(2j+1)}^n.$$

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Symbols:

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

Referenced by:

§24.6, §24.6

Permalink:

http://dlmf.nist.gov/24.6.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §24.6 and Ch.24

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24.6.12 $$E_{2n}=\sum _{k=0}^{2n}\frac{1}{2^k}\sum _{j=0}^k{(-1)}^j\left(\genfrac{}{}{0pt}{}{k}{j}\right){(1+2j)}^{2n}.$$

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Symbols:

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

Referenced by:

§24.6, §24.6

Permalink:

http://dlmf.nist.gov/24.6.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §24.6 and Ch.24

23: 27.2 Functions

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27.2.6 $${\varphi }_k\left(n\right)=\sum _{\left(m,n\right)=1}{m}^k,$$

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Defines:

${\varphi }_k\left(n\right)$: sum of powers of integers relatively prime to a number

Symbols:

$\left(m,n\right)$: greatest common divisor (gcd), $k$: positive integer, $m$: positive integer and $n$: positive integer

Permalink:

http://dlmf.nist.gov/27.2.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §27.2(i), §27.2 and Ch.27

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27.2.7 $$\varphi \left(n\right)={\varphi }_0\left(n\right).$$

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Defines:

$\varphi \left(n\right)$: Euler’s totient

Symbols:

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

Permalink:

http://dlmf.nist.gov/27.2.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §27.2(i), §27.2 and Ch.27

… ►Table 27.2.2 tabulates the Euler totient function $\varphi \left(n\right)$, the divisor function $d\left(n\right)$ ($={\sigma }_0(n)$), and the sum of the divisors $\sigma (n)$ ($={\sigma }_1(n)$), for $n=1(1)52$. …

24: 16.13 Appell Functions

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16.13.1 $$F_1(\alpha ;\beta ,{\beta }^{\prime };\gamma ;x,y)=\sum _{m,n=0}^{\mathrm{\infty }}\frac{{\left(\alpha \right)}_{m+n}{\left(\beta \right)}_m{\left({\beta }^{\prime }\right)}_n}{{\left(\gamma \right)}_{m+n}m!n!}{x}^m{y}^n,$$ ,

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Defines:

$F_1(\alpha ;\beta ,{\beta }^{\prime };\gamma ;x,y)$: first Appell function

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial) and $!$: factorial (as in $n!$)

Permalink:

http://dlmf.nist.gov/16.13.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §16.13 and Ch.16

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16.13.2 $$F_2(\alpha ;\beta ,{\beta }^{\prime };\gamma ,{\gamma }^{\prime };x,y)=\sum _{m,n=0}^{\mathrm{\infty }}\frac{{\left(\alpha \right)}_{m+n}{\left(\beta \right)}_m{\left({\beta }^{\prime }\right)}_n}{{\left(\gamma \right)}_m{\left({\gamma }^{\prime }\right)}_{n}m!n!}{x}^m{y}^n,$$ ,

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Defines:

$F_2(\alpha ;\beta ,{\beta }^{\prime };\gamma ,{\gamma }^{\prime };x,y)$: second Appell function

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial) and $!$: factorial (as in $n!$)

Permalink:

http://dlmf.nist.gov/16.13.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §16.13 and Ch.16

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16.13.3 $$F_3(\alpha ,{\alpha }^{\prime };\beta ,{\beta }^{\prime };\gamma ;x,y)=\sum _{m,n=0}^{\mathrm{\infty }}\frac{{\left(\alpha \right)}_m{\left({\alpha }^{\prime }\right)}_n{\left(\beta \right)}_m{\left({\beta }^{\prime }\right)}_n}{{\left(\gamma \right)}_{m+n}m!n!}{x}^m{y}^n,$$ ,

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Defines:

$F_3(\alpha ,{\alpha }^{\prime };\beta ,{\beta }^{\prime };\gamma ;x,y)$: third Appell function

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial) and $!$: factorial (as in $n!$)

Permalink:

http://dlmf.nist.gov/16.13.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §16.13 and Ch.16

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16.13.4 $$F_4(\alpha ,\beta ;\gamma ,{\gamma }^{\prime };x,y)=\sum _{m,n=0}^{\mathrm{\infty }}\frac{{\left(\alpha \right)}_{m+n}{\left(\beta \right)}_{m+n}}{{\left(\gamma \right)}_m{\left({\gamma }^{\prime }\right)}_{n}m!n!}{x}^m{y}^n,$$ .

ⓘ

Defines:

$F_4(\alpha ,\beta ;\gamma ,{\gamma }^{\prime };x,y)$: fourth Appell function

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial) and $!$: factorial (as in $n!$)

Permalink:

http://dlmf.nist.gov/16.13.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §16.13 and Ch.16

…

25: 31.15 Stieltjes Polynomials

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31.15.1 $$\frac{d^{2}w}{{dz}^2}+\left(\sum _{j=1}^N\frac{{\gamma }_j}{z-a_j}\right)\frac{dw}{dz}+\frac{\mathrm{\Phi }(z)}{{\prod }_{j=1}^N(z-a_j)}w=0,$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $\gamma$: real or complex parameter, $j$: nonnegative integer, $a$: complex parameter, $N+1$: number of singularities and $\mathrm{\Phi }(z)$: polynomial

Referenced by:

§31.15(i)

Permalink:

http://dlmf.nist.gov/31.15.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §31.15(i), §31.15 and Ch.31

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31.15.2 $$\sum _{j=1}^N\frac{{\gamma }_j/2}{z_k-a_j}+\sum _{\begin{array}{c}j=1\\ j\ne k\end{array}}^n\frac{1}{z_k-z_j}=0,$$ $k=1,2,\mathrm{\dots },n$.

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Symbols:

$z$: complex variable, $\gamma$: real or complex parameter, $j$: nonnegative integer, $n$: nonnegative integer, $a$: complex parameter and $N+1$: number of singularities

Referenced by:

§31.15(ii)

Permalink:

http://dlmf.nist.gov/31.15.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §31.15(ii), §31.15 and Ch.31

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31.15.3 $$\sum _{j=1}^N\frac{{\gamma }_j}{t_k-a_j}+\sum _{j=1}^{n-1}\frac{1}{t_k-z_j^{\prime }}=0.$$

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Symbols:

$z$: complex variable, $\gamma$: real or complex parameter, $j$: nonnegative integer, $n$: nonnegative integer, $a$: complex parameter, $N+1$: number of singularities and $t_k$: zero of $V(z)$

Permalink:

http://dlmf.nist.gov/31.15.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §31.15(ii), §31.15 and Ch.31

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31.15.7 $$q_j={\gamma }_j\sum _{k=1}^n\frac{1}{z_k-a_j},$$ $j=1,2,\mathrm{\dots },N$.

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Symbols:

$z$: complex variable, $\gamma$: real or complex parameter, $j$: nonnegative integer, $n$: nonnegative integer, $a$: complex parameter, $q$: real or complex parameter and $N+1$: number of singularities

Permalink:

http://dlmf.nist.gov/31.15.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §31.15(ii), §31.15 and Ch.31

…

26: 30.8 Expansions in Series of Ferrers Functions

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30.8.1 $${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)=\sum _{k=-R}^{\mathrm{\infty }}{(-1)}^k{a}_{n,k}^m({\gamma }^2){𝖯}_{n+2k}^m\left(x\right),$$

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Symbols:

: Ferrers function of the first kind, ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the first kind, $x$: real variable, $m$: nonnegative integer, $n\ge m$: integer degree, ${\gamma }^2$: real parameter, $R=\lfloor (n-m)/2\rfloor$: limit and $a_{n,k}^m({\gamma }^2)$: coefficients

A&S Ref:

21.7.1 (in different form)

Referenced by:

§30.16(ii), §30.16(ii)

Permalink:

http://dlmf.nist.gov/30.8.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §30.8(i), §30.8 and Ch.30

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30.8.5 $$\sum _{k=-R}^{\mathrm{\infty }}{a}_{n,k}^m({\gamma }^2)a_{n,k}^{-m}({\gamma }^2)\frac{1}{2n+4k+1}=\frac{1}{2n+1},$$

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Symbols:

$m$: nonnegative integer, $n\ge m$: integer degree, ${\gamma }^2$: real parameter, $R=\lfloor (n-m)/2\rfloor$: limit and $a_{n,k}^m({\gamma }^2)$: coefficients

Referenced by:

§30.16(ii)

Permalink:

http://dlmf.nist.gov/30.8.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §30.8(i), §30.8 and Ch.30

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30.8.9 $${\mathrm{𝖰𝗌}}_n^m(x,{\gamma }^2)=\sum _{k=-\mathrm{\infty }}^{-N-1}{(-1)}^{k}a_{}^{\prime }{}_{n,k}{}^m({\gamma }^2){𝖯}_{n+2k}^m\left(x\right)+\sum _{k=-N}^{\mathrm{\infty }}{(-1)}^k{a}_{n,k}^m({\gamma }^2){𝖰}_{n+2k}^m\left(x\right),$$

ⓘ

Symbols:

: Ferrers function of the first kind, : Ferrers function of the second kind, ${\mathrm{𝖰𝗌}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the second kind, $x$: real variable, $m$: nonnegative integer, $n\ge m$: integer degree, ${\gamma }^2$: real parameter, $a_{n,k}^m({\gamma }^2)$: coefficients and $N$: upper limit

A&S Ref:

21.7.21 (in different form)

Permalink:

http://dlmf.nist.gov/30.8.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §30.8(ii), §30.8 and Ch.30

…

27: 31.14 General Fuchsian Equation

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31.14.1 $$\frac{d^{2}w}{{dz}^2}+\left(\sum _{j=1}^N\frac{{\gamma }_j}{z-a_j}\right)\frac{dw}{dz}+\left(\sum _{j=1}^N\frac{q_j}{z-a_j}\right)w=0,$$ ${\sum }_{j=1}^N{q}_j=0$.

ⓘ

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $\gamma$: real or complex parameter, $j$: nonnegative integer, $a$: complex parameter, $q$: real or complex parameter and $N+1$: number of singularities

Referenced by:

§31.15(i)

Permalink:

http://dlmf.nist.gov/31.14.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §31.14(i), §31.14 and Ch.31

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$\alpha +\beta +1={\displaystyle \sum _{j=1}^N}{\gamma }_j,$

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31.14.4 $$\frac{d^{2}W}{{dz}^2}=\sum _{j=1}^N\left(\frac{{\tilde{\gamma }}_j}{{(z-a_j)}^2}+\frac{{\tilde{q}}_j}{z-a_j}\right)W,$$ ${\sum }_{j=1}^N{\tilde{q}}_j=0$,

ⓘ

Defines:

$W(z)$: solution (locally)

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $\gamma$: real or complex parameter, $j$: nonnegative integer, $a$: complex parameter, $q$: real or complex parameter and $N+1$: number of singularities

Permalink:

http://dlmf.nist.gov/31.14.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §31.14(i), §31.14(i), §31.14 and Ch.31

►

${\tilde{q}}_j={\displaystyle \frac{1}{2}}{\displaystyle \sum _{\begin{array}{c}k=1\\ k\ne j\end{array}}^N}{\displaystyle \frac{{\gamma }_j{\gamma }_k}{a_j-a_k}}-q_j,$

…

28: 13.2 Definitions and Basic Properties

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13.2.3 $$𝐌(a,b,z)=\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(a\right)}_s}{\mathrm{\Gamma }\left(b+s\right)s!}{z}^s,$$

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Defines:

$𝐌(a,b,z)$: Olver’s confluent hypergeometric function

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $s$: nonnegative integer and $z$: complex variable

Referenced by:

§13.2(i)

Permalink:

http://dlmf.nist.gov/13.2.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §13.2(i), §13.2(i), §13.2 and Ch.13

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13.2.9 $$U(a,n+1,z)=\frac{{(-1)}^{n+1}}{n!\mathrm{\Gamma }\left(a-n\right)}\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(a\right)}_k}{{\left(n+1\right)}_{k}k!}{z}^k\left(\ln z+\psi \left(a+k\right)-\psi \left(1+k\right)-\psi \left(n+k+1\right)\right)+\frac{1}{\mathrm{\Gamma }\left(a\right)}\sum _{k=1}^n\frac{(k-1)!{\left(1-a+k\right)}_{n-k}}{(n-k)!}{z}^{-k}.$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $U(a,b,z)$: Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $n$: nonnegative integer and $z$: complex variable

Referenced by:

§13.2(i), §33.6, §6.6, Erratum (V1.0.12) for Equations (13.2.9), (13.2.10)

Permalink:

http://dlmf.nist.gov/13.2.E9

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.12):

The condition $a\ne 0,-1,-2,\mathrm{\dots }$ that appeared below this equation now appears ahead of the equation.

Suggested 2016-07-05 by Adri Olde Daalhuis

See also:

Annotations for §13.2(i), §13.2(i), §13.2 and Ch.13

…

29: 27.14 Unrestricted Partitions

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27.14.3 $$\frac{1}{f\left(x\right)}=\sum _{n=0}^{\mathrm{\infty }}p\left(n\right)x^n,$$

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Symbols:

$f\left(x\right)$: Euler’s reciprocal function, $p\left(n\right)$: total number of partitions of $n$ , $n$: positive integer and $x$: real number

A&S Ref:

24.2.1 I.B

Permalink:

http://dlmf.nist.gov/27.14.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §27.14(ii), §27.14 and Ch.27

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27.14.4 $$f\left(x\right)=1-x-x^2+x^5+x^7-x^{12}-x^{15}+\mathrm{\cdots }=1+\sum _{k=1}^{\mathrm{\infty }}{(-1)}^k\left(x^{\omega (k)}+x^{\omega (-k)}\right),$$

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Symbols:

$f\left(x\right)$: Euler’s reciprocal function, $k$: positive integer, $x$: real number and $\omega (k)$: pentagonal numbers

Permalink:

http://dlmf.nist.gov/27.14.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §27.14(ii), §27.14 and Ch.27

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27.14.15 $$5\frac{{(f\left(x^5\right))}^5}{{(f\left(x\right))}^6}=\sum _{n=0}^{\mathrm{\infty }}p\left(5n+4\right)x^n$$

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Symbols:

$f\left(x\right)$: Euler’s reciprocal function, $p\left(n\right)$: total number of partitions of $n$ , $n$: positive integer and $x$: real number

Permalink:

http://dlmf.nist.gov/27.14.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §27.14(v), §27.14 and Ch.27

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30: 8.15 Sums

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8.15.1 $$\gamma (a,\lambda x)={\lambda }^a\sum _{k=0}^{\mathrm{\infty }}\gamma (a+k,x)\frac{{(1-\lambda )}^k}{k!}.$$

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Symbols:

$!$: factorial (as in $n!$), $\gamma (a,z)$: incomplete gamma function, $x$: real variable, $a$: parameter and $k$: nonnegative integer

Permalink:

http://dlmf.nist.gov/8.15.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §8.15 and Ch.8

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8.15.2 $$a\sum _{k=1}^{\mathrm{\infty }}\left(\frac{{\mathrm{e}}^{2\pi \mathrm{i}k(z+h)}}{{\left(2\pi \mathrm{i}k\right)}^{a+1}}\mathrm{\Gamma }(a,2\pi \mathrm{i}kz)+\frac{{\mathrm{e}}^{-2\pi \mathrm{i}k(z+h)}}{{\left(-2\pi \mathrm{i}k\right)}^{a+1}}\mathrm{\Gamma }(a,-2\pi \mathrm{i}kz)\right)=\zeta (-a,z+h)+\frac{z^{a+1}}{a+1}+\left(h-\frac{1}{2}\right)z^a,$$ $h\in [0,1]$.

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Symbols:

$\zeta (s,a)$: Hurwitz zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $[a,b]$: closed interval, $\in$: element of, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $z$: complex variable, $a$: parameter and $k$: nonnegative integer

Proof sketch:

Start with and $z>0$. For $\mathrm{\Gamma }(a,\pm 2\pi \mathrm{i}kz)$ take integral representation (8.2.2) and use the substitution $t=\pm 2\pi \mathrm{i}kz(1+\tau )$. The sum and the integral can be interchanged, and the sum can be evaluated via (4.6.1). Use integration by parts. This will result in $\left(h-\frac{1}{2}\right)z^a$ plus two integrals with infinitely many poles. The residue theorem (§1.10(iv)) will give us an infinite series which can be identified via (25.11.1). For other values of $a$ and $z$ use analytic continuation.

Referenced by:

§25.11(x), §25.11(x), §8.15, Erratum (V1.1.3) for Additions

Permalink:

http://dlmf.nist.gov/8.15.E2

Encodings:

pMML, png, TeX

Addition (effective with 1.1.3):

This equation was added.

See also:

Annotations for §8.15 and Ch.8

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