Bailey 2F1(-1) sum

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11: 16.10 Expansions in Series of ${}_p{}^{}F_q^{}$ Functions

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16.10.1 $${}_{p+r}{}^{}F_{q+s}^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p,c_1,\mathrm{\dots },c_r}{b_1,\mathrm{\dots },b_q,d_1,\mathrm{\dots },d_s};z\zeta )=\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(𝐚\right)}_k{\left(\alpha \right)}_k{\left(\beta \right)}_k{(-z)}^k}{{\left(𝐛\right)}_k{\left(\gamma +k\right)}_{k}k!}{}_{p+2}{}^{}F_{q+1}^{}(\genfrac{}{}{0pt}{}{\alpha +k,\beta +k,a_1+k,\mathrm{\dots },a_p+k}{\gamma +2k+1,b_1+k,\mathrm{\dots },b_q+k};z){}_{r+2}{}^{}F_{s+2}^{}(\genfrac{}{}{0pt}{}{-k,\gamma +k,c_1,\mathrm{\dots },c_r}{\alpha ,\beta ,d_1,\mathrm{\dots },d_s};\zeta ).$$

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Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $p$: nonnegative integer, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters and $b,b_1,\mathrm{\dots },b_q$: real or complex parameters

Permalink:

http://dlmf.nist.gov/16.10.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §16.10 and Ch.16

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16.10.2 $${}_{p+1}{}^{}F_p^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_{p+1}}{b_1,\mathrm{\dots },b_p};z\zeta )={(1-z)}^{-a_1}\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(a_1\right)}_k}{k!}{}_{p+1}{}^{}F_p^{}(\genfrac{}{}{0pt}{}{-k,a_2,\mathrm{\dots },a_{p+1}}{b_1,\mathrm{\dots },b_p};\zeta ){\left(\frac{z}{z-1}\right)}^k.$$

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Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $p$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters and $b,b_1,\mathrm{\dots },b_q$: real or complex parameters

Permalink:

http://dlmf.nist.gov/16.10.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §16.10 and Ch.16

►When the series on the right-hand side converges in the half-plane . ►Expansions of the form ${\sum }_{n=1}^{\mathrm{\infty }}{(\pm 1)}^n{}_p{}^{}F_{p+1}^{}(𝐚;𝐛;-n^2{z}^2)$ are discussed in Miller (1997), and further series of generalized hypergeometric functions are given in Luke (1969b, Chapter 9), Luke (1975, §§5.10.2 and 5.11), and Prudnikov et al. (1990, §§5.3, 6.8–6.9).

12: 26.21 Tables

… ►Andrews (1976) contains tables of the number of unrestricted partitions, partitions into odd parts, partitions into parts $\cancel{\equiv }\pm 2(mod5)$, partitions into parts $\cancel{\equiv }\pm 1(mod5)$, and unrestricted plane partitions up to 100. …

13: 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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14: 17 q-Hypergeometric and Related Functions

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15: 5.7 Series Expansions

… ►where $c_1=1$, $c_2=\gamma$, and … ►

5.7.5 $$\psi \left(1+z\right)=\frac{1}{2z}-\frac{\pi }{2}\cot \left(\pi z\right)+\frac{1}{z^2-1}+1-\gamma -\sum _{k=1}^{\mathrm{\infty }}(\zeta \left(2k+1\right)-1)z^{2k},$$ , $z\ne 0,\pm 1$.

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Symbols:

$\gamma$: Euler’s constant, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cot z$: cotangent function, $\psi \left(z\right)$: psi (or digamma) function, $k$: nonnegative integer and $z$: complex variable

A&S Ref:

6.3.15

Referenced by:

§5.7(i)

Permalink:

http://dlmf.nist.gov/5.7.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §5.7(i), §5.7 and Ch.5

… ►When $z\ne 0,-1,-2,\mathrm{\dots }$, … ►

5.7.7 $$\psi \left(\frac{z+1}{2}\right)-\psi \left(\frac{z}{2}\right)=2\sum _{k=0}^{\mathrm{\infty }}\frac{{(-1)}^k}{k+z}.$$

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Symbols:

$\psi \left(z\right)$: psi (or digamma) function, $k$: nonnegative integer and $z$: complex variable

Referenced by:

§15.4(iii)

Permalink:

http://dlmf.nist.gov/5.7.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §5.7(ii), §5.7 and Ch.5

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5.7.8 $$\mathrm{\Im }\psi \left(1+\mathrm{i}y\right)=\sum _{k=1}^{\mathrm{\infty }}\frac{y}{k^2+y^2}.$$

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Symbols:

$\psi \left(z\right)$: psi (or digamma) function, $\mathrm{\Im }$: imaginary part, $\mathrm{i}$: imaginary unit, $k$: nonnegative integer and $y$: real variable

A&S Ref:

6.3.13

Permalink:

http://dlmf.nist.gov/5.7.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §5.7(ii), §5.7 and Ch.5

16: 18.33 Polynomials Orthogonal on the Unit Circle

… ►Instead of (18.33.9) one might take monic OP’s $\{q_n(x)\}$ with weight function $(1+x)w_1(x)$, and then express $q_n\left(\frac{1}{2}(z+z^{-1})\right)$ in terms of ${\varphi }_{2n}(z^{\pm 1})$ or ${\varphi }_{2n+1}(z^{\pm 1})$. …See Zhedanov (1998, §2). … ►For the hypergeometric function ${}_2{}^{}F_1^{}$ see §§15.1 and 15.2(i). … ►For the notation, including the basic hypergeometric function ${}_2{}^{}\varphi _1^{}$, see §§17.2 and 17.4(i). … ►See Simon (2005a, p. 2, item (2)). …

17: 10.36 Other Differential Equations

… ►The quantity ${\lambda }^2$ in (10.13.1)–(10.13.6) and (10.13.8) can be replaced by $-{\lambda }^2$ if at the same time the symbol $𝒞$ in the given solutions is replaced by $𝒵$. … ►

10.36.1 $$z^2(z^2+{\nu }^2)w^{\prime \prime }+z(z^2+3{\nu }^2)w^{\prime }-\left({(z^2+{\nu }^2)}^2+z^2-{\nu }^2\right)w=0,$$ $w={𝒵}_{\nu }^{\prime }\left(z\right)$,

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Symbols:

${𝒵}_{\nu }\left(z\right)$: modified cylinder function, $z$: complex variable and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/10.36.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.36 and Ch.10

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10.36.2 $$z^2{w}^{\prime \prime }+z(1\pm 2z)w^{\prime }+(\pm z-{\nu }^2)w=0,$$ $w={\mathrm{e}}^{\mp z}{𝒵}_{\nu }\left(z\right)$.

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Symbols:

$\mathrm{e}$: base of natural logarithm, ${𝒵}_{\nu }\left(z\right)$: modified cylinder function, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.41

Permalink:

http://dlmf.nist.gov/10.36.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.36 and Ch.10

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18: 19.21 Connection Formulas

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19.21.1 $$R_F(0,z+1,z)R_D(0,z+1,1)+R_D(0,z+1,z)R_F(0,z+1,1)=3\pi /(2z),$$ $z\in ℂ\setminus (-\mathrm{\infty },0]$.

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Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables, $R_F(x,y,z)$: symmetric elliptic integral of first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $ℂ$: complex plane, $\in$: element of, $(a,b]$: half-closed interval and $\setminus$: set subtraction

Referenced by:

§19.21(i), §19.21(i), §19.7(i)

Permalink:

http://dlmf.nist.gov/19.21.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §19.21(i), §19.21 and Ch.19

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19.21.4 $$R_F(0,z-1,z)=R_F(0,1-z,1)\mp \mathrm{i}{R}_F(0,z,1),$$

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Symbols:

$R_F(x,y,z)$: symmetric elliptic integral of first kind and $\mathrm{i}$: imaginary unit

Permalink:

http://dlmf.nist.gov/19.21.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §19.21(i), §19.21 and Ch.19

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19.21.5 $$2R_G(0,z-1,z)=2R_G(0,1-z,1)\pm \mathrm{i}2R_G(0,z,1)+(z-1)R_F(0,1-z,1)\mp \mathrm{i}z{R}_F(0,z,1).$$

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Symbols:

$R_F(x,y,z)$: symmetric elliptic integral of first kind, $R_G(x,y,z)$: symmetric elliptic integral of second kind and $\mathrm{i}$: imaginary unit

Permalink:

http://dlmf.nist.gov/19.21.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §19.21(i), §19.21 and Ch.19

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19.21.8 $$R_D(y,z,x)+R_D(z,x,y)+R_D(x,y,z)=3x^{-1/2}{y}^{-1/2}{z}^{-1/2},$$

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Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables

Referenced by:

§19.21(ii), §19.33(iii), Erratum (V1.1.3) for Chapter 19

Permalink:

http://dlmf.nist.gov/19.21.E8

Encodings:

pMML, png, TeX

Correction (effective with 1.1.3):

The factors inside the square root on the right-hand side were written as products to ensure the correct multivalued behavior.

Suggested 2021-06-07 by Luc Maisonobe

See also:

Annotations for §19.21(ii), §19.21 and Ch.19

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19.21.11 $$6R_G(x,y,z)=3(x+y+z)R_F(x,y,z)-\sum x^2{R}_D(y,z,x)=\sum x(y+z)R_D(y,z,x),$$

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Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables, $R_F(x,y,z)$: symmetric elliptic integral of first kind and $R_G(x,y,z)$: symmetric elliptic integral of second kind

Referenced by:

§19.21(i), §19.21(ii)

Permalink:

http://dlmf.nist.gov/19.21.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §19.21(ii), §19.21 and Ch.19

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19: 7.5 Interrelations

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7.5.6 $${\mathrm{e}}^{\pm \frac{1}{2}\pi \mathrm{i}{z}^2}(\mathrm{g}\left(z\right)\pm \mathrm{i}\mathrm{f}\left(z\right))=\frac{1}{2}(1\pm \mathrm{i})-(C\left(z\right)\pm \mathrm{i}S\left(z\right)).$$

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Symbols:

$\mathrm{f}\left(z\right)$: auxiliary function for Fresnel integrals, $\mathrm{g}\left(z\right)$: auxiliary function for Fresnel integrals, $C\left(z\right)$: Fresnel integral, $S\left(z\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $z$: complex variable

Referenced by:

§7.5

Permalink:

http://dlmf.nist.gov/7.5.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §7.5 and Ch.7

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7.5.7 $$\zeta =\frac{1}{2}\sqrt{\pi }(1\mp \mathrm{i})z,$$

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Defines:

$\zeta$: change of variable (locally)

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit and $z$: complex variable

Referenced by:

§7.12(ii), §7.5

Permalink:

http://dlmf.nist.gov/7.5.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §7.5 and Ch.7

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7.5.8 $$C\left(z\right)\pm \mathrm{i}S\left(z\right)=\frac{1}{2}(1\pm \mathrm{i})\mathrm{erf}\zeta .$$

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Symbols:

$C\left(z\right)$: Fresnel integral, $S\left(z\right)$: Fresnel integral, $\mathrm{erf}z$: error function, $\mathrm{i}$: imaginary unit, $z$: complex variable and $\zeta$: change of variable

Referenced by:

§7.5, §7.5, §7.6(i)

Permalink:

http://dlmf.nist.gov/7.5.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §7.5 and Ch.7

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7.5.9 $$C\left(z\right)\pm \mathrm{i}S\left(z\right)=\frac{1}{2}(1\pm \mathrm{i})\left(1-{\mathrm{e}}^{\pm \frac{1}{2}\pi \mathrm{i}{z}^2}w\left(\mathrm{i}\zeta \right)\right).$$

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Symbols:

$C\left(z\right)$: Fresnel integral, $S\left(z\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, $w\left(z\right)$: Faddeeva (or Faddeyeva) function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable and $\zeta$: change of variable

Referenced by:

§7.5

Permalink:

http://dlmf.nist.gov/7.5.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §7.5 and Ch.7

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7.5.10 $$\mathrm{g}\left(z\right)\pm \mathrm{i}\mathrm{f}\left(z\right)=\frac{1}{2}(1\pm \mathrm{i}){\mathrm{e}}^{{\zeta }^2}\mathrm{erfc}\zeta .$$

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Symbols:

$\mathrm{f}\left(z\right)$: auxiliary function for Fresnel integrals, $\mathrm{g}\left(z\right)$: auxiliary function for Fresnel integrals, $\mathrm{erfc}z$: complementary error function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable and $\zeta$: change of variable

A&S Ref:

7.3.22 (in different form)

Referenced by:

§7.12(ii), §7.13(iv), §7.5, §7.5

Permalink:

http://dlmf.nist.gov/7.5.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §7.5 and Ch.7

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20: 10.15 Derivatives with Respect to Order

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10.15.1 $$\frac{\partial J_{\pm \nu }\left(z\right)}{\partial \nu }=\pm J_{\pm \nu }\left(z\right)\ln \left(\frac{1}{2}z\right)\mp {(\frac{1}{2}z)}^{\pm \nu }\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{\psi \left(k+1\pm \nu \right)}{\mathrm{\Gamma }\left(k+1\pm \nu \right)}\frac{{(\frac{1}{4}{z}^2)}^k}{k!},$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $k$: nonnegative integer, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.64

Referenced by:

§10.15, §10.38, Erratum (V1.0.17) for Equations (10.15.1), (10.38.1)

Permalink:

http://dlmf.nist.gov/10.15.E1

Encodings:

pMML, png, TeX

Addition (effective with 1.0.17):

This equation has been generalized to include the additional case of $\partial J_{-\nu }\left(z\right)/\partial \nu$ because it will help the user who wants to combine it with (10.15.2).

See also:

Annotations for §10.15, §10.15 and Ch.10

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10.15.3 $${\frac{\partial J_{\nu }\left(z\right)}{\partial \nu }|}_{\nu =n}=\frac{\pi }{2}{Y}_n\left(z\right)+\frac{n!}{2{(\frac{1}{2}z)}^n}\sum _{k=0}^{n-1}\frac{{(\frac{1}{2}z)}^k{J}_k\left(z\right)}{k!(n-k)}.$$

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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, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $n$: integer, $k$: nonnegative integer, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.66

Referenced by:

§10.15, §10.15, §10.38

Permalink:

http://dlmf.nist.gov/10.15.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.15, §10.15 and Ch.10

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10.15.4 $${\frac{\partial Y_{\nu }\left(z\right)}{\partial \nu }|}_{\nu =n}=-\frac{\pi }{2}{J}_n\left(z\right)+\frac{n!}{2{(\frac{1}{2}z)}^n}\sum _{k=0}^{n-1}\frac{{(\frac{1}{2}z)}^k{Y}_k\left(z\right)}{k!(n-k)},$$

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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, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $n$: integer, $k$: nonnegative integer, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.67

Permalink:

http://dlmf.nist.gov/10.15.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.15, §10.15 and Ch.10

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10.15.6 $${\frac{\partial J_{\nu }\left(x\right)}{\partial \nu }|}_{\nu =\frac{1}{2}}=\sqrt{\frac{2}{\pi x}}\left(\mathrm{Ci}\left(2x\right)\sin x-\mathrm{Si}\left(2x\right)\cos x\right),$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{Ci}\left(z\right)$: cosine integral, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $\sin z$: sine function, $\mathrm{Si}\left(z\right)$: sine integral, $x$: real variable and $\nu$: complex parameter

A&S Ref:

10.1.41 (corrected)

Referenced by:

§10.15, §10.38, Ch.10

Permalink:

http://dlmf.nist.gov/10.15.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §10.15, §10.15 and Ch.10

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10.15.7 $${\frac{\partial J_{\nu }\left(x\right)}{\partial \nu }|}_{\nu =-\frac{1}{2}}=\sqrt{\frac{2}{\pi x}}\left(\mathrm{Ci}\left(2x\right)\cos x+\mathrm{Si}\left(2x\right)\sin x\right),$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{Ci}\left(z\right)$: cosine integral, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $\sin z$: sine function, $\mathrm{Si}\left(z\right)$: sine integral, $x$: real variable and $\nu$: complex parameter

A&S Ref:

10.1.42 (corrected)

Referenced by:

§10.15, §10.38

Permalink:

http://dlmf.nist.gov/10.15.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §10.15, §10.15 and Ch.10

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