Gauss–Christoffel quadrature

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21: 16.7 Relations to Other Functions

… ►Further representations of special functions in terms of

{{}_{p}F_{q}}

functions are given in Luke (1969a, §§6.2–6.3), and an extensive list of

{{}_{q+1}F_{q}}

functions with rational numbers as parameters is given in Krupnikov and Kölbig (1997).

22: 16.10 Expansions in Series of ${{}_{p}F_{q}}$ Functions

§16.10 Expansions in Series of ${{}_{p}F_{q}}$ Functions

… ►

16.10.1

{{}_{p+r}F_{q+s}}\left({a_{1},\dots,a_{p},c_{1},\dots,c_{r}\atop b_{1},\dots,b% _{q},d_{1},\dots,d_{s}};z\zeta\right)=\sum_{k=0}^{\infty}\frac{{\left(\mathbf{% a}\right)_{k}}{\left(\alpha\right)_{k}}{\left(\beta\right)_{k}}(-z)^{k}}{{% \left(\mathbf{b}\right)_{k}}{\left(\gamma+k\right)_{k}}k!}{{}_{p+2}F_{q+1}}% \left({\alpha+k,\beta+k,a_{1}+k,\dots,a_{p}+k\atop\gamma+2k+1,b_{1}+k,\dots,b_% {q}+k};z\right){{}_{r+2}F_{s+2}}\left({-k,\gamma+k,c_{1},\dots,c_{r}\atop% \alpha,\beta,d_{1},\dots,d_{s}};\zeta\right).

ⓘ

Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $p$ : nonnegative integer, $q$ : nonnegative integer, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : 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

… ►

16.10.2

{{}_{p+1}F_{p}}\left({a_{1},\dots,a_{p+1}\atop b_{1},\dots,b_{p}};z\zeta\right% )=(1-z)^{-a_{1}}\sum_{k=0}^{\infty}\frac{{\left(a_{1}\right)_{k}}}{k!}{{}_{p+1% }F_{p}}\left({-k,a_{2},\dots,a_{p+1}\atop b_{1},\dots,b_{p}};\zeta\right)\left% (\frac{z}{z-1}\right)^{k}.

ⓘ

Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $p$ : nonnegative integer, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : 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

… ►Expansions of the form

\sum_{n=1}^{\infty}(\pm 1)^{n}{{}_{p}F_{p+1}}\left(\mathbf{a};\mathbf{b};-n^{2% }z^{2}\right)

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).

23: 27.10 Periodic Number-Theoretic Functions

… ►Another generalization of Ramanujan’s sum is the Gauss sum

G\left(n,\chi\right)

associated with a Dirichlet character

\chi\pmod{k}

. …In particular,

G\left(n,\chi_{1}\right)=c_{k}\left(n\right)

. ►

G\left(n,\chi\right)

is separable for some

n

if … ►For a primitive character

\chi\pmod{k}

G\left(n,\chi\right)

is separable for every

n

, and … ►Conversely, if

G\left(n,\chi\right)

is separable for every

n

, then

\chi

is primitive (mod

k

). …

24: 16.2 Definition and Analytic Properties

… ►

§16.2(i) Generalized Hypergeometric Series

… ►Equivalently, the function is denoted by

{{}_{p}F_{q}}\left({\mathbf{a}\atop\mathbf{b}};z\right)

{{}_{p}F_{q}}\left(\mathbf{a};\mathbf{b};z\right)

, and sometimes, for brevity, by

{{}_{p}F_{q}}\left(z\right)

. … ►The branch obtained by introducing a cut from

1

+\infty

on the real axis, that is, the branch in the sector

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

, is the principal branch (or principal value) of

{{}_{q+1}F_{q}}\left(\mathbf{a};\mathbf{b};z\right)

; compare §4.2(i). … ►

16.2.3

{{}_{p+1}F_{q}}\left({-m,\mathbf{a}\atop\mathbf{b}};z\right)=\frac{{\left(% \mathbf{a}\right)_{m}}(-z)^{m}}{{\left(\mathbf{b}\right)_{m}}}{{}_{q+1}F_{p}}% \left({-m,1-m-\mathbf{b}\atop 1-m-\mathbf{a}};\frac{(-1)^{p+q}}{z}\right)

ⓘ

Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $p$ : nonnegative integer, $q$ : nonnegative integer and $z$ : complex variable
Referenced by:: §13.6(v)
Permalink:: http://dlmf.nist.gov/16.2.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §16.2(iv), §16.2(iv), §16.2 and Ch.16

… ►See §16.5 for the definition of

{{}_{p}F_{q}}\left(\mathbf{a};\mathbf{b};z\right)

as a contour integral when

p>q+1

and none of the

a_{k}

is a nonpositive integer. …

25: 16.3 Derivatives and Contiguous Functions

… ►

16.3.1

\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}{{}_{p}F_{q}}\left({a_{1},\dots,a_{p% }\atop b_{1},\dots,b_{q}};z\right)=\frac{{\left(\mathbf{a}\right)_{n}}}{{\left% (\mathbf{b}\right)_{n}}}{{}_{p}F_{q}}\left({a_{1}+n,\dots,a_{p}+n\atop b_{1}+n% ,\dots,b_{q}+n};z\right),

ⓘ

Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $p$ : nonnegative integer, $q$ : nonnegative integer, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters
Permalink:: http://dlmf.nist.gov/16.3.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §16.3(i), §16.3 and Ch.16

… ►

16.3.3

\left(z\frac{\mathrm{d}}{\mathrm{d}z}z\right)^{n}\left(z^{\gamma-1}{{}_{p+1}F_% {q}}\left({\gamma,a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)\right)={% \left(\gamma\right)_{n}}z^{\gamma+n-1}{{}_{p+1}F_{q}}\left({\gamma+n,a_{1},% \dots,a_{p}\atop b_{1},\dots,b_{q}};z\right),

ⓘ

Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $p$ : nonnegative integer, $q$ : nonnegative integer, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters
Permalink:: http://dlmf.nist.gov/16.3.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §16.3(i), §16.3 and Ch.16

►

16.3.4

\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(z^{\gamma-1}{{}_{p}F_{q+1}}% \left({a_{1},\dots,a_{p}\atop\gamma,b_{1},\dots,b_{q}};z\right)\right)={\left(% \gamma-n\right)_{n}}z^{\gamma-n-1}{{}_{p}F_{q+1}}\left({a_{1},\dots,a_{p}\atop% \gamma-n,b_{1},\dots,b_{q}};z\right).

ⓘ

Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $p$ : nonnegative integer, $q$ : nonnegative integer, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters
Permalink:: http://dlmf.nist.gov/16.3.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §16.3(i), §16.3 and Ch.16

… ►Two generalized hypergeometric functions

{{}_{p}F_{q}}\left(\mathbf{a};\mathbf{b};z\right)

are (generalized) contiguous if they have the same pair of values of

p

and

q

, and corresponding parameters differ by integers. … ►

16.3.6

z{{}_{0}F_{1}}\left(-;b+1;z\right)+b(b-1){{}_{0}F_{1}}\left(-;b;z\right)-b(b-1% ){{}_{0}F_{1}}\left(-;b-1;z\right)=0,

ⓘ

Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, $z$ : complex variable and $b,b_{1},\ldots,b_{q}$ : real or complex parameters
Permalink:: http://dlmf.nist.gov/16.3.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §16.3(ii), §16.3 and Ch.16

…

26: 15.8 Transformations of Variable

… ►

15.8.13

F\left({a,b\atop 2b};z\right)=\left(1-\tfrac{1}{2}z\right)^{-a}F\left({\tfrac{% 1}{2}a,\tfrac{1}{2}a+\tfrac{1}{2}\atop b+\tfrac{1}{2}};\left(\frac{z}{2-z}% \right)^{2}\right),

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

ⓘ

Symbols:: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{ph}$ : phase, $z$ : complex variable, $a$ : real or complex parameter and $b$ : real or complex parameter
Referenced by:: §15.8(iv)
Permalink:: http://dlmf.nist.gov/15.8.E13
Encodings:: pMML, png, TeX
See also:: Annotations for §15.8(iii), §15.8(iii), §15.8 and Ch.15

►

15.8.14

F\left({a,b\atop 2b};z\right)=\left(1-z\right)^{-\ifrac{a}{2}}F\left({\tfrac{1% }{2}a,b-\tfrac{1}{2}a\atop b+\tfrac{1}{2}};\frac{z^{2}}{4z-4}\right),

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

ⓘ

Symbols:: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{ph}$ : phase, $z$ : complex variable, $a$ : real or complex parameter and $b$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/15.8.E14
Encodings:: pMML, png, TeX
See also:: Annotations for §15.8(iii), §15.8(iii), §15.8 and Ch.15

… ►

15.8.15

F\left({a,b\atop a-b+1};z\right)=(1+z)^{-a}F\left({\frac{1}{2}a,\frac{1}{2}a+% \frac{1}{2}\atop a-b+1};\frac{4z}{(1+z)^{2}}\right),

|z|<1

ⓘ

Symbols:: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $z$ : complex variable, $a$ : real or complex parameter and $b$ : real or complex parameter
Referenced by:: §15.4(i), §15.8(iv)
Permalink:: http://dlmf.nist.gov/15.8.E15
Encodings:: pMML, png, TeX
See also:: Annotations for §15.8(iii), §15.8(iii), §15.8 and Ch.15

►

15.8.16

F\left({a,b\atop a-b+1};z\right)=(1-z)^{-a}F\left({\frac{1}{2}a,\frac{1}{2}a-b% +\frac{1}{2}\atop a-b+1};\frac{-4z}{(1-z)^{2}}\right),

|z|<1

ⓘ

Symbols:: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $z$ : complex variable, $a$ : real or complex parameter and $b$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/15.8.E16
Encodings:: pMML, png, TeX
See also:: Annotations for §15.8(iii), §15.8(iii), §15.8 and Ch.15

►

15.8.17

F\left({a,b\atop\frac{1}{2}(a+b+1)};z\right)=(1-2z)^{-a}F\left({\frac{1}{2}a,% \frac{1}{2}a+\frac{1}{2}\atop\frac{1}{2}(a+b+1)};\frac{4z(z-1)}{(1-2z)^{2}}% \right),

\Re z<\frac{1}{2}

ⓘ

Symbols:: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $\Re$ : real part, $z$ : complex variable, $a$ : real or complex parameter and $b$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/15.8.E17
Encodings:: pMML, png, TeX
See also:: Annotations for §15.8(iii), §15.8(iii), §15.8 and Ch.15

…

27: Bibliography R

… ►

W. P. Reinhardt (2018) Universality properties of Gaussian quadrature, the derivative rule, and a novel approach to Stieltjes inversion.

ⓘ

Notes:: arXiv:1812.02196
Referenced by:: §18.40(ii).
Encodings:: BibTeX

… ►

R. Roy (2017) Elliptic and modular functions from Gauss to Dedekind to Hecke. Cambridge University Press, Cambridge.

ⓘ

External Links:: ISBN 978-1-107-15938-9, Document, Link, MathReview (Paul M. Jenkins), ZentralBlatt
Referenced by:: Profile Ranjan Roy.
Encodings:: BibTeX

… ►

J. Rys, M. Dupuis, and H. F. King (1983) Computation of electron repulsion integrals using the Rys quadrature method. J. Comput. Chem. 4 (2), pp. 154–175.

ⓘ

External Links:: Link
Referenced by:: §18.32, §18.39(iii).
Encodings:: BibTeX

28: 16.4 Argument Unity

… ►The function

{{}_{q+1}F_{q}}\left(\mathbf{a};\mathbf{b};z\right)

is well-poised if … ►The function

{{}_{q+1}F_{q}}

with argument unity and general values of the parameters is discussed in Bühring (1992). … ►For generalizations involving

{{}_{r+3}F_{r+2}}

functions see Kim et al. (2013). … ►There are two types of three-term identities for

{{}_{3}F_{2}}

’s. … ►Transformations for both balanced

{{}_{4}F_{3}}\left(1\right)

and very well-poised

{{}_{7}F_{6}}\left(1\right)

are included in Bailey (1964, pp. 56–63). …

29: 35.8 Generalized Hypergeometric Functions of Matrix Argument

… ►

§35.8(iii) ${{}_{3}F_{2}}$ Case

►

Kummer Transformation

… ►

Thomae Transformation

… ►Multidimensional Mellin–Barnes integrals are established in Ding et al. (1996) for the functions

{{}_{p}F_{q}}

and

{{}_{p+1}F_{p}}

of matrix argument. A similar result for the

{{}_{0}F_{1}}

function of matrix argument is given in Faraut and Korányi (1994, p. 346). …

30: 16.9 Zeros

… ►Then

{{}_{p}F_{p}}\left(\mathbf{a};\mathbf{b};z\right)

has at most finitely many zeros if and only if the

a_{j}

can be re-indexed for

j=1,\dots,p

in such a way that

a_{j}-b_{j}

is a nonnegative integer. … ►Then

{{}_{p}F_{p}}\left(\mathbf{a};\mathbf{b};z\right)

has at most finitely many real zeros. …