boundary conditions and the Weyl alternative

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11: 18.39 Applications in the Physical Sciences

… ►The solutions of (18.39.8) are subject to boundary conditions at

a

and

b

. … ►Namely the

k

th eigenfunction, listed in order of increasing eigenvalues, starting at

k=0

, has precisely

k

nodes, as real zeros of wave-functions away from boundaries are often referred to. … ►An alternative, and often used, form of (18.39.25) is that for the spherical radial function

\psi_{n,l}(r)=rR_{n,l}(r)

, … ►These cases correspond to the two distinct orthogonality conditions of (18.35.6) and (18.35.6_3). … ►The Coulomb–Pollaczek polynomials provide alternate representations of the positive energy Coulomb wave functions of Chapter 33. …

12: 27.13 Functions

… ►If

3^{k}=q2^{k}+r

with

0<r<2^{k}

, then equality holds in (27.13.2) provided

r+q\leq 2^{k}

, a condition that is satisfied with at most a finite number of exceptions. … ►

27.13.4

\vartheta\left(x\right)=1+2\sum_{m=1}^{\infty}x^{m^{2}},

|x|<1

ⓘ

Defines:: $\vartheta\left(\NVar{x}\right)=\theta_{3}\left(0,x\right)$ : alternative notation
Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $m$ : positive integer and $x$ : real number
A&S Ref:: 16.27.3 (with $z=0$ , $q=x$ )
Referenced by:: §27.13(iv)
Permalink:: http://dlmf.nist.gov/27.13.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §27.13(iv), §27.13 and Ch.27

… ►

27.13.5

(\vartheta\left(x\right))^{2}=1+\sum_{n=1}^{\infty}r_{2}\left(n\right)x^{n}.

ⓘ

Symbols:: $\vartheta\left(\NVar{x}\right)=\theta_{3}\left(0,x\right)$ : alternative notation, $r_{\NVar{k}}\left(\NVar{n}\right)$ : number of squares, $n$ : positive integer and $x$ : real number
Referenced by:: §27.13(iv)
Permalink:: http://dlmf.nist.gov/27.13.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §27.13(iv), §27.13 and Ch.27

… ►

27.13.6

(\vartheta\left(x\right))^{2}=1+4\sum_{n=1}^{\infty}\left(\delta_{1}(n)-\delta% _{3}(n)\right)x^{n},

ⓘ

Symbols:: $\vartheta\left(\NVar{x}\right)=\theta_{3}\left(0,x\right)$ : alternative notation, $n$ : positive integer, $x$ : real number and $\delta_{j}(n)$ : number of divisors
Referenced by:: §27.13(iv)
Permalink:: http://dlmf.nist.gov/27.13.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §27.13(iv), §27.13 and Ch.27

…

13: Mathematical Introduction

… ►This section may also include important alternative notations that have appeared in the literature. … ►In the corresponding section for the DLMF some of the alternative notations that appear in the first section of the special function chapters are also included. … ►Lastly, users may notice some lack of smoothness in the color boundaries of some of the 4D-type surfaces; see, for example, Figure 10.3.9. This nonsmoothness arises because the mesh that was used to generate the figure was optimized only for smoothness of the surface, and not for smoothness of the color boundaries. …

14: 18.37 Classical OP’s in Two or More Variables

… ►The following three conditions, taken together, determine

R^{(\alpha)}_{m,n}\left(z\right)

uniquely: … ►Orthogonal polynomials associated with root systems are certain systems of trigonometric polynomials in several variables, symmetric under a certain finite group (Weyl group), and orthogonal on a torus. …

15: Bibliography

… ►

V. I. Arnol’d (1972) Normal forms of functions near degenerate critical points, the Weyl groups

{A}_{k},{D}_{k},{E}_{k}

and Lagrangian singularities. Funkcional. Anal. i Priložen. 6 (4), pp. 3–25 (Russian).

ⓘ

Notes:: In Russian. English translation: Functional Anal. Appl., 6(1973), pp. 254–272.
External Links:: MathReview (G. R. Belickii), ZentralBlatt
Referenced by:: §36.2(i).
Encodings:: BibTeX

… ►

U. M. Ascher, R. M. M. Mattheij, and R. D. Russell (1995) Numerical Solution of Boundary Value Problems for Ordinary Differential Equations. Classics in Applied Mathematics, Vol. 13, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.

ⓘ

Notes:: Corrected reprint of the 1988 original
External Links:: ISBN 0-89871-354-4, MathReview, ZentralBlatt
Referenced by:: §3.7(iii).
Encodings:: BibTeX

…

16: 10.22 Integrals

… ►Sufficient conditions for the validity of (10.22.77) are that

\int_{0}^{\infty}|f(x)|\,\mathrm{d}x<\infty

when

\nu\geq-\tfrac{1}{2}

, or that

\int_{0}^{\infty}|f(x)|\,\mathrm{d}x<\infty

and

\int_{0}^{1}x^{\nu+\frac{1}{2}}|f(x)|\,\mathrm{d}x<\infty

when

-1<\nu<-\tfrac{1}{2}

; see Titchmarsh (1986a, Theorem 135, Chapter 8) and Akhiezer (1988, p. 62). … ►These are examples of the self-adjoint extensions and the Weyl alternatives of §1.18(ix). …A sufficient condition for the validity is

\int_{a}^{\infty}|f(y)|\,\mathrm{d}y<\infty

. …Sufficient conditions for the validity of (10.22.79) are that

\int_{0}^{\infty}|f(x)|\,\mathrm{d}x<\infty

when

0<\nu\leq\tfrac{1}{2}

, or that

\int_{0}^{\infty}|f(x)|\,\mathrm{d}x<\infty

and

\int_{0}^{1}x^{\frac{1}{2}-\nu}|f(x)|\,\mathrm{d}x<\infty

when

\tfrac{1}{2}<\nu<1

; see Titchmarsh (1962a, pp. 88–90). …

17: 16.11 Asymptotic Expansions

… ►(If this condition is violated, then the definition of

H_{p,q}(z)

has to be modified so that the residues are those associated with the multiple poles. … ►

16.11.6

\left({\textstyle\ifrac{\prod\limits_{\ell=1}^{q+1}\Gamma\left(a_{\ell}\right)% }{\prod\limits_{\ell=1}^{q}\Gamma\left(b_{\ell}\right)}}\right){{}_{q+1}F_{q}}% \left({a_{1},\dots,a_{q+1}\atop b_{1},\dots,b_{q}};z\right)=H_{q+1,q}(-z),

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

;

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ 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, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{ph}$ : phase, $q$ : nonnegative integer, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters, $b,b_{1},\ldots,b_{q}$ : real or complex parameters and $H_{p,q}(z)$ : formal infinite series
Permalink:: http://dlmf.nist.gov/16.11.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §16.11(ii), §16.11(ii), §16.11 and Ch.16

… ►

16.11.7

\left({\textstyle\ifrac{\prod\limits_{\ell=1}^{q}\Gamma\left(a_{\ell}\right)}{% \prod\limits_{\ell=1}^{q}\Gamma\left(b_{\ell}\right)}}\right){{}_{q}F_{q}}% \left({a_{1},\dots,a_{q}\atop b_{1},\dots,b_{q}};z\right)\sim H_{q,q}(z{% \mathrm{e}}^{\mp\pi\mathrm{i}})+E_{q,q}(z).

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ 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, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $q$ : nonnegative integer, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters, $b,b_{1},\ldots,b_{q}$ : real or complex parameters, $E_{p,q}(z)$ : formal infinite series and $H_{p,q}(z)$ : formal infinite series
Referenced by:: §16.11(ii)
Permalink:: http://dlmf.nist.gov/16.11.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §16.11(ii), §16.11(ii), §16.11 and Ch.16

… ►

16.11.8

\left({\textstyle\ifrac{\prod\limits_{\ell=1}^{q-1}\Gamma\left(a_{\ell}\right)% }{\prod\limits_{\ell=1}^{q}\Gamma\left(b_{\ell}\right)}}\right){{}_{q-1}F_{q}}% \left({a_{1},\dots,a_{q-1}\atop b_{1},\dots,b_{q}};-z\right)\sim H_{q-1,q}(z)+% E_{q-1,q}(ze^{-\pi\mathrm{i}})+E_{q-1,q}(ze^{\pi\mathrm{i}}),

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ 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, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{i}$ : imaginary unit, $q$ : nonnegative integer, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters, $b,b_{1},\ldots,b_{q}$ : real or complex parameters, $E_{p,q}(z)$ : formal infinite series, $H_{p,q}(z)$ : formal infinite series and $e_{k,m}$
Permalink:: http://dlmf.nist.gov/16.11.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §16.11(ii), §16.11(ii), §16.11 and Ch.16

… ►

16.11.11

{{}_{p}F_{q}}\left({a_{1}+r,\dots,a_{p}+r\atop b_{1}+r,\dots,b_{q}+r};z\right)% =\sum_{n=0}^{m-1}\frac{{\left(a_{1}+r\right)_{n}}\cdots{\left(a_{p}+r\right)_{% n}}}{{\left(b_{1}+r\right)_{n}}\cdots{\left(b_{q}+r\right)_{n}}}\frac{z^{n}}{n% !}+O\left(\frac{1}{r^{(q-p)m}}\right),

ⓘ

Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ 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, $b,b_{1},\ldots,b_{q}$ : real or complex parameters and $r$ : large parameter
Permalink:: http://dlmf.nist.gov/16.11.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §16.11(iii), §16.11 and Ch.16

…

18: 16.5 Integral Representations and Integrals

… ►

16.5.1

\left({\textstyle\ifrac{\prod\limits_{k=1}^{p}\Gamma\left(a_{k}\right)}{\prod% \limits_{k=1}^{q}\Gamma\left(b_{k}\right)}}\right){{}_{p}F_{q}}\left({a_{1},% \dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)=\frac{1}{2\pi\mathrm{i}}\int_{L}% \left({\textstyle\ifrac{\prod\limits_{k=1}^{p}\Gamma\left(a_{k}+s\right)}{% \prod\limits_{k=1}^{q}\Gamma\left(b_{k}+s\right)}}\right)\Gamma\left(-s\right)% (-z)^{s}\,\mathrm{d}s,

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ 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, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{i}$ : imaginary unit, $\int$ : integral, $p$ : 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
Referenced by:: §13.6(vi), §16.11(i), §16.5, §16.5
Permalink:: http://dlmf.nist.gov/16.5.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §16.5 and Ch.16

… ►Then the integral converges when

p<q+1

provided that

z\neq 0

, or when

p=q+1

provided that

0<|z|<1

, and provides an integral representation of the left-hand side with these conditions. … ►

16.5.2

{{}_{p+1}F_{q+1}}\left({a_{0},\dots,a_{p}\atop b_{0},\dots,b_{q}};z\right)=% \frac{\Gamma\left(b_{0}\right)}{\Gamma\left(a_{0}\right)\Gamma\left(b_{0}-a_{0% }\right)}\int_{0}^{1}t^{a_{0}-1}(1-t)^{b_{0}-a_{0}-1}{{}_{p}F_{q}}\left({a_{1}% ,\dots,a_{p}\atop b_{1},\dots,b_{q}};zt\right)\,\mathrm{d}t,

\Re b_{0}>\Re a_{0}>0

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ 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, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\Re$ : real part, $p$ : 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
Keywords:: Mellin transform
Referenced by:: §16.5, §16.5
Permalink:: http://dlmf.nist.gov/16.5.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §16.5 and Ch.16

►

16.5.3

{{}_{p+1}F_{q}}\left({a_{0},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)=\frac% {1}{\Gamma\left(a_{0}\right)}\int_{0}^{\infty}{\mathrm{e}}^{-t}t^{a_{0}-1}{{}_% {p}F_{q}}\left({a_{1},\dots,a_{p}\atop b_{1},\cdots,b_{q}};zt\right)\,\mathrm{% d}t,

\Re z<1

\Re a_{0}>0

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ 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, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\Re$ : real part, $p$ : 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
Keywords:: Laplace transform, Mellin transform
Referenced by:: §16.5
Permalink:: http://dlmf.nist.gov/16.5.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §16.5 and Ch.16

►

16.5.4

{{}_{p}F_{q+1}}\left({a_{1},\dots,a_{p}\atop b_{0},\dots,b_{q}};z\right)=\frac% {\Gamma\left(b_{0}\right)}{2\pi\mathrm{i}}\int_{c-\mathrm{i}\infty}^{c+\mathrm% {i}\infty}{\mathrm{e}}^{t}t^{-b_{0}}{{}_{p}F_{q}}\left({a_{1},\dots,a_{p}\atop b% _{1},\dots,b_{q}};\frac{z}{t}\right)\,\mathrm{d}t,

c>0

\Re b_{0}>0

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ 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, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\Re$ : real part, $p$ : 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
Keywords:: Mellin transform
Referenced by:: §16.5, §16.5
Permalink:: http://dlmf.nist.gov/16.5.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §16.5 and Ch.16

…

19: 16.4 Argument Unity

… ►

16.4.2_5

{{}_{3}F_{2}}\left({-n,a,1\atop-n,c};1\right)=\sum_{k=0}^{n}\frac{{\left(a% \right)_{k}}}{{\left(c\right)_{k}}}=\frac{c-1}{c-a-1}\left(1-\frac{{\left(a% \right)_{n+1}}}{{\left(c-1\right)_{n+1}}}\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) and $a,a_{1},\ldots,a_{p}$ : real or complex parameters
Source:: Lerch (1905, (3))
Referenced by:: §16.4(ii)
Permalink:: http://dlmf.nist.gov/16.4.E2_5
Encodings:: pMML, png, TeX
Addition (effective with 1.1.0):: This equation was added.
See also:: Annotations for §16.4(ii), §16.4(ii), §16.4 and Ch.16

… ►

16.4.3

{{}_{3}F_{2}}\left({-n,a,b\atop c,d};1\right)=\frac{{\left(c-a\right)_{n}}{% \left(c-b\right)_{n}}}{{\left(c\right)_{n}}{\left(c-a-b\right)_{n}}},

ⓘ

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), $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters
Referenced by:: §18.17(vii)
Permalink:: http://dlmf.nist.gov/16.4.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §16.4(ii), §16.4(ii), §16.4 and Ch.16

… ►

16.4.5

{{}_{3}F_{2}}\left({-n,b,c\atop 1-b-n,1-c-n};1\right)=\begin{cases}0,&n=2k+1,% \\ \dfrac{(2k)!\Gamma\left(b+k\right)\Gamma\left(c+k\right)\Gamma\left(b+c+2k% \right)}{k!\Gamma\left(b+2k\right)\Gamma\left(c+2k\right)\Gamma\left(b+c+k% \right)},&n=2k,\\ \end{cases}

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ 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, $!$ : factorial (as in $n!$ ) and $b,b_{1},\ldots,b_{q}$ : real or complex parameters
Permalink:: http://dlmf.nist.gov/16.4.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §16.4(ii), §16.4(ii), §16.4 and Ch.16

… ►

16.4.7

{{}_{3}F_{2}}\left({a,1-a,c\atop d,2c-d+1};1\right)=\frac{\pi\Gamma\left(d% \right)\Gamma\left(2c-d+1\right)2^{1-2c}}{\Gamma\left(c+\frac{1}{2}(a-d+1)% \right)\Gamma\left(c+1-\frac{1}{2}(a+d)\right)\Gamma\left(\frac{1}{2}(a+d)% \right)\Gamma\left(\frac{1}{2}(d-a+1)\right)},

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ 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, $\pi$ : the ratio of the circumference of a circle to its diameter and $a,a_{1},\ldots,a_{p}$ : real or complex parameters
Referenced by:: §16.4(ii), §17.7(iii), Erratum (V1.1.11) for Subsection 17.7(iii)
Permalink:: http://dlmf.nist.gov/16.4.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §16.4(ii), §16.4(ii), §16.4 and Ch.16

… ►The last condition is equivalent to the sum of the top parameters plus

2

equals the sum of the bottom parameters, that is, the series is 2-balanced. …

20: 2.10 Sums and Sequences

… ►Sufficient conditions for the validity of this second result are: … ►First, the conditions can be weakened. …For example, Condition (b) can be replaced by: … ►Furthermore, (2.10.31) remains valid with the weaker condition … ►In Condition (c) we have …