addition formulas

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31—40 of 54 matching pages

31: 15.6 Integral Representations

… ►

15.6.1

\mathbf{F}\left(a,b;c;z\right)=\frac{1}{\Gamma\left(b\right)\Gamma\left(c-b% \right)}\int_{0}^{1}\frac{t^{b-1}(1-t)^{c-b-1}}{(1-zt)^{a}}\,\mathrm{d}t,

|\operatorname{ph}\left(1-z\right)|<\pi

;

\Re c>\Re b>0

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\operatorname{ph}$ : phase, $\Re$ : real part, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter
Keywords:: Mellin transform
Referenced by:: §15.19(iii), (15.4.34), §15.6, §15.6, Erratum (V1.0.23) for Equations (15.6.1)–(15.6.9), Erratum (V1.0.23) for Equations (15.6.1)–(15.6.9)
Permalink:: http://dlmf.nist.gov/15.6.E1
Encodings:: pMML, png, TeX
Addition (effective with 1.0.23):: The Olver hypergeometric function $\mathbf{F}\left(a,b;c;z\right)$ , previously omitted from the left-hand side to make the formula more concise, has been added. The constraint $|\operatorname{ph}\left(1-z\right)|<\pi$ , originally mentioned in the text, has been directly added to the formula.
See also:: Annotations for §15.6 and Ch.15

►

15.6.2

\mathbf{F}\left(a,b;c;z\right)=\frac{\Gamma\left(1+b-c\right)}{2\pi\mathrm{i}% \Gamma\left(b\right)}\int_{0}^{(1+)}\frac{t^{b-1}(t-1)^{c-b-1}}{(1-zt)^{a}}\,% \mathrm{d}t,

|\operatorname{ph}\left(1-z\right)|<\pi

;

c-b\neq 1,2,3,\dots

\Re b>0

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\operatorname{ph}$ : phase, $\Re$ : real part, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter
Keywords:: Mellin transform
Referenced by:: §15.6
Permalink:: http://dlmf.nist.gov/15.6.E2
Encodings:: pMML, png, TeX
Addition (effective with 1.0.23):: The Olver hypergeometric function $\mathbf{F}\left(a,b;c;z\right)$ , previously omitted from the left-hand side to make the formula more concise, has been added. The constraint $|\operatorname{ph}\left(1-z\right)|<\pi$ , originally mentioned in the text, has been directly added to the formula.
See also:: Annotations for §15.6 and Ch.15

►

15.6.2_5

\mathbf{F}\left(a,b;c;z\right)=\frac{1}{\Gamma\left(b\right)\Gamma\left(c-b% \right)}\int_{0}^{\infty}\frac{t^{b-1}\left(t+1\right)^{a-c}}{\left(t-zt+1% \right)^{a}}\,\mathrm{d}t,

|\operatorname{ph}\left(1-z\right)|<\pi

;

\Re c>\Re b>0

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\operatorname{ph}$ : phase, $\Re$ : real part, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter
Keywords:: Mellin transform
Referenced by:: §15.6, Erratum (V1.1.5) for Additions
Permalink:: http://dlmf.nist.gov/15.6.E2_5
Encodings:: pMML, png, TeX
Addition (effective with 1.1.5):: This equation was added.
See also:: Annotations for §15.6 and Ch.15

… ►

15.6.8

\mathbf{F}\left(a,b;c;z\right)=\frac{1}{\Gamma\left(c-d\right)}\int_{0}^{1}% \mathbf{F}\left(a,b;d;zt\right)t^{d-1}(1-t)^{c-d-1}\,\mathrm{d}t,

|\operatorname{ph}\left(1-z\right)|<\pi

;

\Re c>\Re d>0

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\operatorname{ph}$ : phase, $\Re$ : real part, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter
Keywords:: Mellin transform
Proof sketch:: Follows from (15.6.9) with $\lambda=a+b$ .
Referenced by:: (15.6.9), §15.6, §15.6, §15.6, §18.17(iv), Erratum (V1.0.14) for Equation (15.6.8), Erratum (V1.0.14) for Equation (15.6.8)
Permalink:: http://dlmf.nist.gov/15.6.E8
Encodings:: pMML, png, TeX
Addition (effective with 1.0.23):: The Olver hypergeometric function $\mathbf{F}\left(a,b;c;z\right)$ , previously omitted from the left-hand side to make the formula more concise, has been added. The constraint $|\operatorname{ph}\left(1-z\right)|<\pi$ , originally mentioned in the text, has been directly added to the formula.
See also:: Annotations for §15.6 and Ch.15

… ►In addition: …

32: 23.10 Addition Theorems and Other Identities

§23.10 Addition Theorems and Other Identities

►

§23.10(i) Addition Theorems

… ►For further addition-type identities for the

\sigma

-function see Lawden (1989, §6.4). … ►

§23.10(ii) Duplication Formulas

… ►

§23.10(iii) $n$ -Tuple Formulas

…

33: 3.5 Quadrature

… ► … ►

Gauss–Legendre Formula

… ►

Gauss–Chebyshev Formula

… ►

Gauss–Laguerre Formula

… ►a complex Gauss quadrature formula is available. …

34: 15.2 Definitions and Analytical Properties

… ►Because of the analytic properties with respect to

a

b

, and

c

, it is usually legitimate to take limits in formulas involving functions that are undefined for certain values of the parameters. … ►This formula is also valid when

c=-m-\ell

\ell=0,1,2,\dots

, provided that we use the interpretation … ►Formula (15.4.6) reads

F\left(a,b;a;z\right)=(1-z)^{-b}

. …

36: 18.19 Hahn Class: Definitions

… ►In addition to the limit relations in §18.7(iii) there are limit relations involving the further families in the Askey scheme, see §§18.21(ii) and 18.26(ii). …

37: 16.16 Transformations of Variables

… ►

§16.16(i) Reduction Formulas

… ►

16.16.5_5

{F_{4}}\left(\alpha,\beta;\gamma,\beta;x(1-y),y(1-x)\right)=(1-x)^{-\alpha}(1-% y)^{-\alpha}{F_{1}}\left(\alpha;\gamma-\beta,\alpha-\gamma+1;\gamma;\frac{x}{x% -1},\frac{xy}{(1-x)(1-y)}\right),

ⓘ

Symbols:: ${F_{1}}\left(\NVar{\alpha};\NVar{\beta},\NVar{\beta^{\prime}};\NVar{\gamma};% \NVar{x},\NVar{y}\right)$ : first Appell function and ${F_{4}}\left(\NVar{\alpha},\NVar{\beta};\NVar{\gamma},\NVar{\gamma^{\prime}};% \NVar{x},\NVar{y}\right)$ : fourth Appell function
Referenced by:: §16.16(i), Erratum (V1.1.12) for Additions
Permalink:: http://dlmf.nist.gov/16.16.E5_5
Encodings:: pMML, png, TeX
Addition (effective with 1.1.12):: This equation was added.
See also:: Annotations for §16.16(i), §16.16 and Ch.16

… ►See Erdélyi et al. (1953a, §5.10) for these and further reduction formulas. …

… ►The Digital Library of Mathematical Functions (DLMF) Project was initiated to perform a complete revision of Abramowitz and Stegun’s Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, published in 1964 by the National Bureau of Standards. … ►They will be called upon to help deal with reports of suspected errors and suggestions for additions or other modifications to the chapters. …

39: DLMF Project News

error generating summary

40: 17.8 Special Cases of ${{}_{r}\psi_{r}}$ Functions

… ►

17.8.8

{{}_{2}\psi_{2}}\left({b^{2},\ifrac{b^{2}}{c}\atop q,cq};q^{2},\ifrac{cq^{2}}{% b^{2}}\right)=\frac{1}{2}\frac{\left(q^{2},qb^{2},\ifrac{q}{b^{2}},\ifrac{cq}{% b^{2}};q^{2}\right)_{\infty}}{\left(cq,\ifrac{cq^{2}}{b^{2}},\ifrac{q^{2}}{b^{% 2}},\ifrac{c}{b^{2}};q^{2}\right)_{\infty}}\left(\frac{\left(\ifrac{c\sqrt{q}}% {b};q\right)_{\infty}}{\left(b\sqrt{q};q\right)_{\infty}}+\frac{\left(\ifrac{-% c\sqrt{q}}{b};q\right)_{\infty}}{\left(-b\sqrt{q};q\right)_{\infty}}\right),

\left|cq^{2}\right|<\left|b^{2}\right|

ⓘ

Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left(\NVar{a_{1},\dots,a_{r}};\NVar{b_{1},\dots% ,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left({\NVar{a_{1},\dots,a_{r}}\atop\NVar{b_{1},% \dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : bilateral basic hypergeometric (or bilateral $q$ -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol and $q$ : complex base
Source:: Verma and Jain (1983, (3.9))
Referenced by:: §17.6(i), §17.8, Erratum (V1.1.0) for Additions
Permalink:: http://dlmf.nist.gov/17.8.E8
Encodings:: pMML, png, TeX
Addition (effective with 1.1.0):: This equation was added.
Suggested 2019-10-19 by Slobodan Damjanovic
See also:: Annotations for §17.8, §17.8 and Ch.17

►For similar formulas see Verma and Jain (1983).

addition formulas

31—40 of 54 matching pages

31: 15.6 Integral Representations

32: 23.10 Addition Theorems and Other Identities

§23.10 Addition Theorems and Other Identities

§23.10(i) Addition Theorems

§23.10(ii) Duplication Formulas

§23.10(iii) $n$ -Tuple Formulas

33: 3.5 Quadrature

Gauss–Legendre Formula

Gauss–Chebyshev Formula

Gauss–Laguerre Formula

34: 15.2 Definitions and Analytical Properties

35: 10.60 Sums

§10.60(i) Addition Theorems

§10.60(ii) Duplication Formulas

36: 18.19 Hahn Class: Definitions

37: 16.16 Transformations of Variables

§16.16(i) Reduction Formulas

38: About the Project

39: DLMF Project News

40: 17.8 Special Cases of ${{}_{r}\psi_{r}}$ Functions

addition formulas

31—40 of 54 matching pages

31: 15.6 Integral Representations

32: 23.10 Addition Theorems and Other Identities

§23.10 Addition Theorems and Other Identities

§23.10(i) Addition Theorems

§23.10(ii) Duplication Formulas

§23.10(iii) n -Tuple Formulas

33: 3.5 Quadrature

Gauss–Legendre Formula

Gauss–Chebyshev Formula

Gauss–Laguerre Formula

34: 15.2 Definitions and Analytical Properties

35: 10.60 Sums

§10.60(i) Addition Theorems

§10.60(ii) Duplication Formulas

36: 18.19 Hahn Class: Definitions

37: 16.16 Transformations of Variables

§16.16(i) Reduction Formulas

38: About the Project

39: DLMF Project News

40: 17.8 Special Cases of ψ r r Functions

§23.10(iii) $n$ -Tuple Formulas

40: 17.8 Special Cases of ${{}_{r}\psi_{r}}$ Functions