… ► 1949 in Tasmania) is a professor in the departments of statistics and mathematics at the University of California, Berkeley. …Pitman held a position in the Department of Mathematics and Mathematical Statistics at the University of Cambridge, England. …Pitman holds a B. …in statistics from the Australian National University, Canberra, and a Ph. … ►In November 2015, Pitman was named a Senior Associate Editor of the DLMF.

26: Barry I. Schneider

… ►He is also a General Editor for the DLMF project. A graduate of the NYC Public Schools, he received his B. … in chemistry from Yale University and a Ph. … ►He was a visiting scientist at NIST from 1995 to 2013 and spent a sabbatical year at NIST in 2000-2001. …He also serves as a reviewer for a variety of journals inside and outside the US. …

27: 16.12 Products

… ►

16.12.1

{{}_{0}F_{1}}\left(-;a;z\right){{}_{0}F_{1}}\left(-;b;z\right)={{}_{2}F_{3}}% \left({\frac{1}{2}(a+b),\frac{1}{2}(a+b-1)\atop a,b,a+b-1};4z\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, $z$ : 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:: (10.8.3), §10.8, §10.8, §16.12, Erratum (V1.0.17) for Section 10.8
Permalink:: http://dlmf.nist.gov/16.12.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §16.12 and Ch.16

… ►

16.12.2

\left({{}_{2}F_{1}}\left({a,b\atop a+b+\frac{1}{2}};z\right)\right)^{2}={{}_{3% }F_{2}}\left({2a,2b,a+b\atop a+b+\frac{1}{2},2a+2b};z\right).

ⓘ

Symbols:: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ 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, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters
Referenced by:: §16.12, §17.9(iii), Erratum (V1.1.12) for Subsection 17.9(iii)
Permalink:: http://dlmf.nist.gov/16.12.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §16.12 and Ch.16

… ►

16.12.3

\left({{}_{2}F_{1}}\left({a,b\atop c};z\right)\right)^{2}=\sum_{k=0}^{\infty}% \frac{{\left(2a\right)_{k}}{\left(2b\right)_{k}}{\left(c-\frac{1}{2}\right)_{k% }}}{{\left(c\right)_{k}}{\left(2c-1\right)_{k}}k!}{{}_{4}F_{3}}\left({-\frac{1% }{2}k,\frac{1}{2}(1-k),a+b-c+\frac{1}{2},\frac{1}{2}\atop a+\frac{1}{2},b+% \frac{1}{2},\frac{3}{2}-k-c};1\right)z^{k},

|z|<1

ⓘ

Symbols:: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ 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 $n!$ ), $z$ : 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:: §16.12
Permalink:: http://dlmf.nist.gov/16.12.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §16.12 and Ch.16

…

28: 1.3 Determinants, Linear Operators, and Spectral Expansions

… ►The cofactor

A_{jk}

a_{jk}

is …If

\mu

times a row (column) of a determinant is added to another row (column), then the value of the determinant is unchanged. … ►For real-valued

a_{jk}

, … ►In the case of a real matrix

{\mathbf{A}}^{*}=\mathbf{A}^{\mathrm{T}}

and in the complex case

{\mathbf{A}}^{*}={\mathbf{A}}^{{\rm H}}

. … ►The corresponding eigenvectors

\mathbf{a}_{1},\dots,\mathbf{a}_{n}

can be chosen such that they form a complete orthonormal basis in

\mathbf{E}_{n}

. …

29: 17.10 Transformations of ${{}_{r}\psi_{r}}$ Functions

… ►

17.10.1

{{}_{2}\psi_{2}}\left({a,b\atop c,d};q,z\right)=\frac{\left(az,d/a,c/b,dq/(abz% );q\right)_{\infty}}{\left(z,d,q/b,cd/(abz);q\right)_{\infty}}{{}_{2}\psi_{2}}% \left({a,abz/d\atop az,c};q,\frac{d}{a}\right),

ⓘ

Symbols:: ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left(\NVar{a_{1},\dots,a_{r}};\NVar{b_{1},\dots% ,b_{s}};\NVar{q},\NVar{z}\right)$ 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, $q$ : complex base and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/17.10.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §17.10, §17.10 and Ch.17

►

17.10.2

{{}_{2}\psi_{2}}\left({a,b\atop c,d};q,z\right)=\frac{\left(az,bz,cq/(abz),dq/% (abz);q\right)_{\infty}}{\left(q/a,q/b,c,d;q\right)_{\infty}}{{}_{2}\psi_{2}}% \left({abz/c,abz/d\atop az,bz};q,\frac{cd}{abz}\right).

ⓘ

Symbols:: ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left(\NVar{a_{1},\dots,a_{r}};\NVar{b_{1},\dots% ,b_{s}};\NVar{q},\NVar{z}\right)$ 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, $q$ : complex base and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/17.10.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §17.10, §17.10 and Ch.17

… ►

17.10.3

{{}_{8}\psi_{8}}\left({qa^{\frac{1}{2}},-qa^{\frac{1}{2}},c,d,e,f,aq^{-n},q^{-% n}\atop a^{\frac{1}{2}},-a^{\frac{1}{2}},aq/c,aq/d,aq/e,aq/f,q^{n+1},aq^{n+1}}% ;q,\frac{a^{2}q^{2n+2}}{cdef}\right)=\frac{\left(aq,q/a,aq/(cd),aq/(ef);q% \right)_{n}}{\left(q/c,q/d,aq/e,aq/f;q\right)_{n}}\*{{}_{4}\psi_{4}}\left({e,f% ,aq^{n+1}/(cd),q^{-n}\atop aq/c,aq/d,q^{n+1},ef/(aq^{n})};q,q\right),

ⓘ

Symbols:: ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left(\NVar{a_{1},\dots,a_{r}};\NVar{b_{1},\dots% ,b_{s}};\NVar{q},\NVar{z}\right)$ 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, $q$ : complex base and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/17.10.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §17.10, §17.10 and Ch.17

… ►

17.10.5

\frac{\left(aq/b,aq/c,aq/d,aq/e,q/(ab),q/(ac),q/(ad),q/(ae);q\right)_{\infty}}% {\left(fa,ga,f/a,g/a,qa^{2},q/a^{2};q\right)_{\infty}}\*{{}_{8}\psi_{8}}\left(% {qa,-qa,ba,ca,da,ea,fa,ga\atop a,-a,aq/b,aq/c,aq/d,aq/e,aq/f,aq/g};q,\frac{q^{% 2}}{bcdefg}\right)=\frac{\left(q,q/(bf),q/(cf),q/(df),q/(ef),qf/b,qf/c,qf/d,qf% /e;q\right)_{\infty}}{\left(fa,q/(fa),aq/f,f/a,g/f,fg,qf^{2};q\right)_{\infty}% }\*{{}_{8}\phi_{7}}\left({f^{2},qf,-qf,fb,fc,fd,fe,fg\atop f,-f,fq/b,fq/c,fq/d% ,fq/e,fq/g};q,\frac{q^{2}}{bcdefg}\right)+\operatorname{idem}\left(f;g\right).

ⓘ

Symbols:: $\operatorname{idem}\left(\NVar{\chi_{1}};\NVar{\chi_{2},\dots,\chi_{n}}\right)$ : $\operatorname{idem}$ function, ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ 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, ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left(\NVar{a_{1},\dots,a_{r}};\NVar{b_{1},\dots% ,b_{s}};\NVar{q},\NVar{z}\right)$ 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
Permalink:: http://dlmf.nist.gov/17.10.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §17.10, §17.10 and Ch.17

►

17.10.6

\frac{\left(aq/b,aq/c,aq/d,aq/e,aq/f,q/(ab),q/(ac),q/(ad),q/(ae),q/(af);q% \right)_{\infty}}{\left(ag,ah,ak,g/a,h/a,k/a,qa^{2},q/a^{2};q\right)_{\infty}}% \*{{}_{10}\psi_{10}}\left({qa,-qa,ba,ca,da,ea,fa,ga,ha,ka\atop a,-a,aq/b,aq/c,% aq/d,aq/e,aq/f,aq/g,aq/h,aq/k};q,\frac{q^{2}}{bcdefghk}\right)=\frac{\left(q,q% /(bg),q/(cg),q/(dg),q/(eg),q/(fg),qg/b,qg/c,qg/d,qg/e,qg/f;q\right)_{\infty}}{% \left(gh,gk,h/g,ag,q/(ag),g/a,aq/g,qg^{2};q\right)_{\infty}}\*{{}_{10}\phi_{9}% }\left({g^{2},qg,-qg,gb,gc,gd,ge,gf,gh,gk\atop g,-g,qg/b,qg/c,qg/d,qg/e,qg/f,% qg/h,qg/k};q,\frac{q^{2}}{bcdefghk}\right)+\operatorname{idem}\left(g;h,k% \right).

ⓘ

Symbols:: $\operatorname{idem}\left(\NVar{\chi_{1}};\NVar{\chi_{2},\dots,\chi_{n}}\right)$ : $\operatorname{idem}$ function, ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ 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, ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left(\NVar{a_{1},\dots,a_{r}};\NVar{b_{1},\dots% ,b_{s}};\NVar{q},\NVar{z}\right)$ 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, $k$ : nonnegative integer and $q$ : complex base
Permalink:: http://dlmf.nist.gov/17.10.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §17.10, §17.10 and Ch.17

30: 5.10 Continued Fractions

… ►where ►

a_{0}=\tfrac{1}{12},

►

a_{1}=\tfrac{1}{30},

►

a_{2}=\tfrac{53}{210},

… ►For exact values of

a_{7}

a_{11}

and 40S values of

a_{0}

a_{40}

, see Char (1980). …

a posteriori

21—30 of 888 matching pages

21: 12.8 Recurrence Relations and Derivatives

22: Sidebar 22.SB1: Decay of a Soliton in a Bose–Einstein Condensate

Sidebar 22.SB1: Decay of a Soliton in a Bose–Einstein Condensate

23: 12.6 Continued Fraction

24: 16.6 Transformations of Variable

25: Jim Pitman

26: Barry I. Schneider

27: 16.12 Products

28: 1.3 Determinants, Linear Operators, and Spectral Expansions

29: 17.10 Transformations of ${{}_{r}\psi_{r}}$ Functions

30: 5.10 Continued Fractions

a posteriori

21—30 of 888 matching pages

21: 12.8 Recurrence Relations and Derivatives

22: Sidebar 22.SB1: Decay of a Soliton in a Bose–Einstein Condensate

Sidebar 22.SB1: Decay of a Soliton in a Bose–Einstein Condensate

23: 12.6 Continued Fraction

24: 16.6 Transformations of Variable

25: Jim Pitman

26: Barry I. Schneider

27: 16.12 Products

28: 1.3 Determinants, Linear Operators, and Spectral Expansions

29: 17.10 Transformations of ψ r r Functions

30: 5.10 Continued Fractions

29: 17.10 Transformations of ${{}_{r}\psi_{r}}$ Functions