alternative forms

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1: 32.2 Differential Equations

… ►

§32.2(iii) Alternative Forms

…

4: 33.2 Definitions and Basic Properties

… ►The normalizing constant

C_{\ell}\left(\eta\right)

is always positive, and has the alternative form …

5: 18.39 Applications in the Physical Sciences

… ►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)

, …

… ►This section may also include important alternative notations that have appeared in the literature. … ►Other examples are: (a) the notation for the Ferrers functions—also known as associated Legendre functions on the cut—for which existing notations can easily be confused with those for other associated Legendre functions (§14.1); (b) the spherical Bessel functions for which existing notations are unsymmetric and inelegant (§§10.47(i) and 10.47(ii)); and (c) elliptic integrals for which both Legendre’s forms and the more recent symmetric forms are treated fully (Chapter 19). … ►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. … ►For equations or other technical information that appeared previously in AMS 55, the DLMF usually includes the corresponding AMS 55 equation number, or other form of reference, together with corrections, if needed. …

7: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►These are based on the Liouville normal form of (1.13.29). … ► … ►Consider formally self-adjoint operators of the form … ►

Self-adjoint extensions of (1.18.28) and the Weyl alternative

… ►By Weyl’s alternative

n_{1}

equals either 1 (the limit point case) or 2 (the limit circle case), and similarly for

n_{2}

. …

8: 18.38 Mathematical Applications

… ►

18.38.3

\sum_{m=0}^{n}P^{(\alpha,0)}_{m}\left(x\right)=\frac{{\left(\alpha+2\right)_{n% }}}{n!}{{}_{3}F_{2}}\left({-n,n+\alpha+2,\frac{1}{2}(\alpha+1)\atop\alpha+1,% \frac{1}{2}(\alpha+3)};\tfrac{1}{2}(1-x)\right)\geq 0,

x\geq-1

\alpha\geq-2

n=0,1,\dots

ⓘ

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, $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$ : Jacobi polynomial, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $m$ : nonnegative integer, $n$ : nonnegative integer and $x$ : real variable
Proved:: Askey and Gasper (1976, Theorem 3, p.717)(proved)
Referenced by:: §18.14(iv), §18.18(viii), Erratum (V1.2.0) for Equation (18.38.3)
Permalink:: http://dlmf.nist.gov/18.38.E3
Encodings:: pMML, png, TeX
Modification (effective with 1.2.0):: This equation was updated to include the value of the sum in terms of the ${{}_{3}F_{2}}$ function. Also the constraint was previously $\alpha>-1$ .
See also:: Annotations for §18.38(ii), §18.38(ii), §18.38 and Ch.18

… ►The

3j

symbol (34.2.6), with an alternative expression as a terminating

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

of unit argument, can be expressed in terms of Hahn polynomials (18.20.5) or, by (18.21.1), dual Hahn polynomials. … ►The

6j

symbol (34.4.3), with an alternative expression as a terminating balanced

{{}_{4}F_{3}}

of unit argument, can be expressend in terms of Racah polynomials (18.26.3). … ►A symmetric Laurent polynomial is a function of the form … ►Exceptional OP’s (EOP’s) are those which are ‘missing’ a finite number of lower order polynomials, but yet form complete sets with respect to suitable measures. …

9: 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

►with limiting form

a\left(\psi\left(a+n+1\right)-\psi\left(a\right)\right)=\frac{a\frac{\mathrm{d% }}{\mathrm{d}a}{\left(a\right)_{n+1}}}{{\left(a\right)_{n+1}}}

in the case that

c=a+1

. … ►

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

…

10: 2.9 Difference Equations

… ►Alternatively, suppose that

2g_{1}=f_{0}f_{1}

. …Provided that

\alpha_{2}-\alpha_{1}

is not zero or an integer, (2.9.1) has independent solutions

w_{j}(n)

j=1,2

, of the form … ►For analogous results for difference equations of the form …

alternative forms

1—10 of 26 matching pages

1: 32.2 Differential Equations

§32.2(iii) Alternative Forms

2: 22.8 Addition Theorems

§22.8(ii) Alternative Forms for Sum of Two Arguments

3: 1.8 Fourier Series

Alternative Form