Pfaff–Saalschütz balanced sum

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21: 15.15 Sums

§15.15 Sums

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15.15.1 $$𝐅(\genfrac{}{}{0pt}{}{a,b}{c};\frac{1}{z})={\left(1-\frac{z_0}{z}\right)}^{-a}\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(a\right)}_s}{s!}𝐅(\genfrac{}{}{0pt}{}{-s,b}{c};\frac{1}{z_0}){\left(1-\frac{z}{z_0}\right)}^{-s}.$$

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s 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:

§15.19(i)

Permalink:

http://dlmf.nist.gov/15.15.E1

Encodings:

pMML, png, TeX

Correction (effective with 1.1.2):

The Pochhammer symbol now links to its definition.

See also:

Annotations for §15.15 and Ch.15

… ►For compendia of finite sums and infinite series involving hypergeometric functions see Prudnikov et al. (1990, §§5.3 and 6.7) and Hansen (1975).

22: 6.15 Sums

§6.15 Sums

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6.15.1 $$\sum _{n=1}^{\mathrm{\infty }}\mathrm{Ci}\left(\pi n\right)=\frac{1}{2}(\ln 2-\gamma ),$$

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Symbols:

$\gamma$: Euler’s constant, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{Ci}\left(z\right)$: cosine integral, $\ln z$: principal branch of logarithm function and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/6.15.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §6.15 and Ch.6

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6.15.2 $$\sum _{n=1}^{\mathrm{\infty }}\frac{\mathrm{si}\left(\pi n\right)}{n}=\frac{1}{2}\pi (\ln \pi -1),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\ln z$: principal branch of logarithm function, $\mathrm{si}\left(z\right)$: sine integral and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/6.15.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §6.15 and Ch.6

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6.15.3 $$\sum _{n=1}^{\mathrm{\infty }}{(-1)}^n\mathrm{Ci}\left(2\pi n\right)=1-\ln 2-\frac{1}{2}\gamma ,$$

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Symbols:

$\gamma$: Euler’s constant, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{Ci}\left(z\right)$: cosine integral, $\ln z$: principal branch of logarithm function and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/6.15.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §6.15 and Ch.6

… ►For further sums see Fempl (1960), Hansen (1975, pp. 423–424), Harris (2000), Prudnikov et al. (1986b, vol. 2, pp. 649–650), and Slavić (1974).

23: 1.7 Inequalities

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§1.7(i) Finite Sums

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Cauchy–Schwarz Inequality

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Minkowski’s Inequality

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Cauchy–Schwarz Inequality

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Minkowski’s Inequality

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24: 27.6 Divisor Sums

§27.6 Divisor Sums

►Sums of number-theoretic functions extended over divisors are of special interest. … ►

27.6.1

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Symbols:

$\lambda \left(n\right)$: Liouville’s function, $d$: positive integer and $n$: positive integer

Permalink:

http://dlmf.nist.gov/27.6.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §27.6 and Ch.27

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

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

…

25: 24.6 Explicit Formulas

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24.6.1 $$B_{2n}=\sum _{k=2}^{2n+1}\frac{{(-1)}^{k-1}}{k}\left(\genfrac{}{}{0pt}{}{2n+1}{k}\right)\sum _{j=1}^{k-1}{j}^{2n},$$

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Symbols:

$B_n$: Bernoulli 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.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §24.6 and Ch.24

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24.6.2 $$B_n=\frac{1}{n+1}\sum _{k=1}^n\sum _{j=1}^k{(-1)}^j{j}^n\left(\genfrac{}{}{0.0pt}{}{n+1}{k-j}\right)/\left(\genfrac{}{}{0.0pt}{}{n}{k}\right),$$

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Symbols:

$B_n$: Bernoulli 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.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §24.6 and Ch.24

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24.6.3 $$B_{2n}=\sum _{k=1}^n\frac{(k-1)!k!}{(2k+1)!}\sum _{j=1}^k{(-1)}^{j-1}\left(\genfrac{}{}{0pt}{}{2k}{k+j}\right)j^{2n}.$$

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Symbols:

$B_n$: Bernoulli numbers, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $!$: factorial (as in $n!$), $j$: integer, $k$: integer and $n$: integer

Referenced by:

§24.6

Permalink:

http://dlmf.nist.gov/24.6.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §24.6 and Ch.24

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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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24.6.9 $$B_n=\sum _{k=0}^n\frac{1}{k+1}\sum _{j=0}^k{(-1)}^j\left(\genfrac{}{}{0pt}{}{k}{j}\right)j^n,$$

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Symbols:

$B_n$: Bernoulli 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.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §24.6 and Ch.24

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26: 16.20 Integrals and Series

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27: 27.1 Special Notation

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$d,k,m,n$	positive integers (unless otherwise indicated).
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${\sum }_{d|n}$, ${\prod }_{d|n}$	sum, product taken over divisors of $n$.
${\sum }_{\left(m,n\right)=1}$	sum taken over $m$, $1\le m\le n$ and $m$ relatively prime to $n$.
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${\sum }_p$, ${\prod }_p$	sum, product extended over all primes.
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${\sum }_{n\le x}$	${\sum }_{n=1}^{\lfloor x\rfloor }$.
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28: 27.7 Lambert Series as Generating Functions

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27.7.1 $$\sum _{n=1}^{\mathrm{\infty }}f(n)\frac{x^n}{1-x^n}.$$

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Symbols:

$n$: positive integer and $x$: real number

Referenced by:

§27.7

Permalink:

http://dlmf.nist.gov/27.7.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §27.7 and Ch.27

►If , then the quotient $x^n/(1-x^n)$ is the sum of a geometric series, and when the series (27.7.1) converges absolutely it can be rearranged as a power series: ►

27.7.2 $$\sum _{n=1}^{\mathrm{\infty }}f(n)\frac{x^n}{1-x^n}=\sum _{n=1}^{\mathrm{\infty }}\sum _{d|n}f(d)x^n.$$

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Symbols:

$d$: positive integer, $n$: positive integer and $x$: real number

Referenced by:

§27.7

Permalink:

http://dlmf.nist.gov/27.7.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §27.7 and Ch.27

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27.7.5 $$\sum _{n=1}^{\mathrm{\infty }}{n}^{\alpha }\frac{x^n}{1-x^n}=\sum _{n=1}^{\mathrm{\infty }}{\sigma }_{\alpha }\left(n\right)x^n,$$

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Symbols:

${\sigma }_{\alpha }\left(n\right)$: sum of powers of divisors of a number, $n$: positive integer and $x$: real number

A&S Ref:

24.3.3 I.B

Permalink:

http://dlmf.nist.gov/27.7.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §27.7 and Ch.27

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27.7.6 $$\sum _{n=1}^{\mathrm{\infty }}\lambda \left(n\right)\frac{x^n}{1-x^n}=\sum _{n=1}^{\mathrm{\infty }}{x}^{n^2}.$$

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Symbols:

$\lambda \left(n\right)$: Liouville’s function, $n$: positive integer and $x$: real number

Permalink:

http://dlmf.nist.gov/27.7.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §27.7 and Ch.27

29: 34.13 Methods of Computation

… ►Methods of computation for $\mathit{3}j$ and $\mathit{6}j$ symbols include recursion relations, see Schulten and Gordon (1975a), Luscombe and Luban (1998), and Edmonds (1974, pp. 42–45, 48–51, 97–99); summation of single-sum expressions for these symbols, see Varshalovich et al. (1988, §§8.2.6, 9.2.1) and Fang and Shriner (1992); evaluation of the generalized hypergeometric functions of unit argument that represent these symbols, see Srinivasa Rao and Venkatesh (1978) and Srinivasa Rao (1981). ►For $\mathit{9}j$ symbols, methods include evaluation of the single-sum series (34.6.2), see Fang and Shriner (1992); evaluation of triple-sum series, see Varshalovich et al. (1988, §10.2.1) and Srinivasa Rao et al. (1989). …

30: 10.44 Sums

§10.44 Sums

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§10.44(i) Multiplication Theorem

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§10.44(ii) Addition Theorems

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§10.44(iii) Neumann-Type Expansions

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§10.44(iv) Compendia

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