DLMF: Untitled Document

… ►where

ef=abcq

. … ►where

\lambda=-c(ab/q)^{\frac{1}{2}}

. … ►

17.7.10

{{}_{8}\phi_{7}}\left({-c,q(-c)^{\frac{1}{2}},-q(-c)^{\frac{1}{2}},a,q/a,c,-d,% -q/d\atop(-c)^{\frac{1}{2}},-(-c)^{\frac{1}{2}},-cq/a,-ac,-q,cq/d,cd};q,c% \right)=\frac{\left(-c,-cq;q\right)_{\infty}\left(acd,acq/d,cdq/a,cq^{2}/(ad);% q^{2}\right)_{\infty}}{\left(cd,cq/d,-ac,-cq/a;q\right)_{\infty}}.

ⓘ

Symbols:: ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol and $q$ : complex base
Permalink:: http://dlmf.nist.gov/17.7.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §17.7(iii), §17.7(iii), §17.7 and Ch.17

… ►where

a^{2}q=bcdeq^{-n}

. … ►where

qa^{2}=bcdef

. …

… ►And

f(x)

is continuous at

c

when both (1.4.1) and (1.4.3) apply. … ►A removable singularity of

f(x)

at

x=c

occurs when

f(c+)=f(c-)

but

f(c)

is undefined. … ►A simple discontinuity of

f(x)

at

x=c

occurs when

f(c+)

and

f(c-)

exist, but

f(c+)\not=f(c-)

. … ► … ►

c

and

d

constants. …

…

… ►Let

N(a,b,c)

denote the number of zeros of

F\left(a,b;c;z\right)

in the sector

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

. If

a

,

b

,

c

are real,

a

,

b

,

c

,

c-a

,

c-b\neq 0,-1,-2,\dots

, and, without loss of generality,

b\geq a

,

c\geq a+b

(compare (15.8.1)), then ►

15.13.1

N(a,b,c)=\begin{cases}0,&a>0,\\ \left\lfloor-a\right\rfloor+\tfrac{1}{2}(1+S),&a<0,c-a>0,\\ \left\lfloor-a\right\rfloor+\tfrac{1}{2}(1+S)+\left\lfloor a-c+1\right\rfloor S% ,&a<0,c-a<0,\\ \end{cases}

ⓘ

Symbols:: $\left\lfloor\NVar{x}\right\rfloor$ : floor of $x$ , $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter, $N(a,b,c)$ : number of zeros and $S$
Permalink:: http://dlmf.nist.gov/15.13.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §15.13 and Ch.15

►where

S=\operatorname{sign}\left(\Gamma\left(a\right)\Gamma\left(b\right)\Gamma\left% (c-a\right)\Gamma\left(c-b\right)\right)

. … ►If

a

,

b

,

c

,

c-a

, or

c-b\in\{0,-1,-2,\dots\}

, then

F\left(a,b;c;z\right)

is not defined, or reduces to a polynomial, or reduces to

(1-z)^{c-a-b}

times a polynomial. …

… ►

Bille C. Carlson, Iowa State University, Chap. 19

… ►

Leonard C. Maximon, George Washington University, Chaps. 10, 34

… ►

Roderick S. C. Wong, City University of Hong Kong, Chaps. 1, 2, 18

… ►

Leonard C. Maximon, The George Washington University, for Chap. 34 (deceased)

… ►

Roderick S. C. Wong, City University of Hong Kong, for Chaps. 2, 18

…

… ►

S.-L. Qiu and M. K. Vamanamurthy (1996) Sharp estimates for complete elliptic integrals. SIAM J. Math. Anal. 27 (3), pp. 823–834.

ⓘ

External Links:: ISSN 0036-1410, Document, MathReview (G. D. Anderson), ZentralBlatt
Referenced by:: §19.9(i), §19.9(i).
Encodings:: BibTeX

… ►

W. Qiu and R. Wong (2004) Asymptotic expansion of the Krawtchouk polynomials and their zeros. Comput. Methods Funct. Theory 4 (1), pp. 189–226.

ⓘ

External Links:: ISSN 1617-9447, FullText, MathReview (C. M. Joshi), ZentralBlatt (N. M. Temme)
Referenced by:: §18.24.
Encodings:: BibTeX

►

C. K. Qu and R. Wong (1999) “Best possible” upper and lower bounds for the zeros of the Bessel function

J_{\nu}(x)

. Trans. Amer. Math. Soc. 351 (7), pp. 2833–2859.

ⓘ

External Links:: ISSN 0002-9947, FullText, MathReview (Andrea Laforgia), ZentralBlatt
Referenced by:: §10.21(vii).
Encodings:: BibTeX

►

C. Quesne (2011) Higher-Order SUSY, Exactly Solvable Potentials, and Exceptional Orthogonal Polynomials. Modern Physics Letters A 26, pp. 1843–1852.

ⓘ

External Links:: Document, Link
Referenced by:: §18.38(iii).
Encodings:: BibTeX

Bibliography C

… ►

B. C. Carlson and J. M. Keller (1961) Eigenvalues of Density Matrices. Phys. Rev. 121, pp. 659–661.

ⓘ

External Links:: Document, Link, ZentralBlatt
Referenced by:: Profile Bille C. Carlson.
Encodings:: BibTeX

… ►

B. C. Carlson (2004) Symmetry in c, d, n of Jacobian elliptic functions. J. Math. Anal. Appl. 299 (1), pp. 242–253.

ⓘ

External Links:: ISSN 0022-247X, Document, MathReview Entry, ZentralBlatt
Referenced by:: §19.25(v), §19.25(v), §22.13(i), §22.6(ii), §22.6(iii), §22.8(i), §22.8(ii), Profile Bille C. Carlson.
Encodings:: BibTeX

… ►

W. J. Cody, A. J. Strecok, and H. C. Thacher (1973) Chebyshev approximations for the psi function. Math. Comp. 27 (121), pp. 123–127.

ⓘ

External Links:: FullText, MathReview, ZentralBlatt
Referenced by:: §5.23(i).
Encodings:: BibTeX

… ►

CoStLy (free C-XSC library)

ⓘ

Notes:: Complex interval standard function library. For details see Neher (2007)
External Links:: CoStLy
Referenced by:: § ‣ Software Cross Index, M. Neher (2007).
Encodings:: BibTeX

…

… ► ►►►

$x$	real variable.
…
$a, b, c$	real or complex parameters.
…

… ►

15.1.1

{{}_{2}F_{1}}\left(a,b;c;z\right)=F\left(a,b;c;z\right)=F\left({a,b\atop c};z% \right),

ⓘ

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, ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/15.1.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §15.1 and Ch.15

… ►

15.1.2

\frac{F\left(a,b;c;z\right)}{\Gamma\left(c\right)}=\mathbf{F}\left(a,b;c;z% \right)=\mathbf{F}\left({a,b\atop c};z\right)={{}_{2}{\mathbf{F}}_{1}}\left(a,% b;c;z\right),

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ 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, ${{}_{\NVar{p}}{\mathbf{F}}_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}% };\NVar{z}\right)$ or ${{}_{\NVar{p}}{\mathbf{F}}_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{% \mathbf{b}}};\NVar{z}\right)$ : scaled (or Olver’s) generalized hypergeometric function, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ 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
Permalink:: http://dlmf.nist.gov/15.1.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §15.1 and Ch.15

…

… ►

t_{n}=c+n,

►

u_{2n+1}=(a+n)(c-b+n),

►

u_{2n}=(b+n)(c-a+n).

… ►

v_{n}=c+n+(b-a+n+1)z,

►

w_{n}=(b+n)(c-a+n)z.

…

… ►The coefficients satisfy ►

28.14.4

{qc_{2m+2}-\left(a-(\nu+2m)^{2}\right)c_{2m}+qc_{2m-2}=0},

a=\lambda_{\nu}\left(q\right),c_{2m}=c_{2m}^{\nu}(q)

,

ⓘ

Symbols:: $\lambda_{\NVar{\nu+2n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation, $m$ : integer, $q=h^{2}$ : parameter, $\nu$ : complex parameter, $a$ : parameter and $c_{2m}(q)$ : coefficients
Referenced by:: item (d), item (d)
Permalink:: http://dlmf.nist.gov/28.14.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §28.14 and Ch.28

… ►

28.14.7

c_{-2m}^{-\nu}(q)=c_{2m}^{\nu}(q),

ⓘ

Symbols:: $m$ : integer, $q=h^{2}$ : parameter, $\nu$ : complex parameter and $c_{2m}(q)$ : coefficients
Permalink:: http://dlmf.nist.gov/28.14.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §28.14 and Ch.28

►

28.14.8

c_{2m}^{\nu}(-q)=(-1)^{m}c_{2m}^{\nu}(q).

ⓘ

Symbols:: $m$ : integer, $q=h^{2}$ : parameter, $\nu$ : complex parameter and $c_{2m}(q)$ : coefficients
Permalink:: http://dlmf.nist.gov/28.14.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §28.14 and Ch.28

… ►

c_{0}^{\nu}(0)=1,

…

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11—20 of 470 matching pages

11: 17.7 Special Cases of Higher ${{}_{r}\phi_{s}}$ Functions

12: 1.4 Calculus of One Variable

13: 18 Orthogonal Polynomials

14: 15.13 Zeros

15: Staff

16: Bibliography Q

17: Bibliography C

Bibliography C

18: 15.1 Special Notation

19: 15.7 Continued Fractions

20: 28.14 Fourier Series

.c罗世界杯进了几个球了『wn4.com』南非世界杯彩绘.w6n2c9o.2022年11月29日22时7分27秒.tp3lzbdx5.gov.hk

11—20 of 470 matching pages

11: 17.7 Special Cases of Higher ϕ s r Functions

12: 1.4 Calculus of One Variable

13: 18 Orthogonal Polynomials

14: 15.13 Zeros

15: Staff

16: Bibliography Q

17: Bibliography C

Bibliography C

18: 15.1 Special Notation

19: 15.7 Continued Fractions

20: 28.14 Fourier Series

11: 17.7 Special Cases of Higher ${{}_{r}\phi_{s}}$ Functions